THE UN IVERSITY OF MICHIGAN DEARBORN CAMPUS Division of Engineering Final Report SIMILARITY ANALYSES OF PARTIAL DIFFERENTIAL EQUATIONS Tsung-Yen Na Douglas E. Abbott Arthur G. Hansen ORA Project 07457 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GEORGE C. MARSHALL SPACE FLIGHT CENTER CONTRACT NO. NAS 8-20065 HUNTSVILLE, ALABAMA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1967

FOREWORD Tsung-Yen Na is Associate Professor of Mechanical Engineering, The University of Michigan, Dearborn Campus, Dearborn, Michigan. Douglas E. Abbott is Associate Professor of Mechanical Engineering, Purdue University, Lafayette, Indiana. Arthur G. Hansen is Dean of Engineering, Georgia Institute of Technology, Atlanta, Georgia. iii

TABLE OF CONTENTS Page ABSTRACT vii CHAPTER 1. GENERAL CONCEPTS AND SCOPE OF THE INVESTIGATION 1 1.0 Introduction 1 1.1 General Concepts 2 2. GENERAL TRANSFORMATION METHODS IN FLOW THEORY 5 2.0 General Concepts 5 2.1 Typical Transformations Used in Fluid Mechanics 6 2.2 The Primitive Transformation 7 2.3 Requirements Imposed by the Physical Description 8 2.4 The Generalized Similarity Analysis 12 2.5 Concluding Remarks 21 References for Chapters 1 and 2 23 3. SIMPLE GROUP THEORY APPLICATIONS 25 3.0 Introduction 25 3.1 One-Parameter Group-Theoretic Analysis 25 3.1.1 Absolute Invariants 26 3.1.2 Morgan's Theorem 27 3.1.3 Application of One-Parameter Group Theory to Boundary-Layer Equations 28 3.2 Multiparameter Groups 33 3.2.1 One-Parameter Group Method 33 3.2.2 Two-Parameter Group Method 34 3.2.3 Example of a Similarity Analysis for a Three-Variable Problem 36 References for Chapter 3 39 4. GENERALIZED GROUP-THEORETIC ANALYSIS 40 4.0 Introduction 40 4.1 Definitions and Theorems 41 4.1.1 Definitions 41 4.1.2 Representations of Infinitesimal Transformations 44 4.1.3 Functions Invariant Under a Given Group 49 4.1.4 Extension of Two-Variable Analysis to n Variables 54 v

TABLE OF CONTENTS (Concluded) Page 4.1. 5 Relationships Satisfied by Differential Equations Admitting a Given Group of Infinitesimal Traons format ions 55 4,1.6 The Extended Group Concept 58 4-1. 7T Contact Transformations 61 4 tl. 8 Cofntact Transformation in Terms of the Chiar acter-istic Functio-n 64 4. 1.:9 Conr-tact Transformatiorl of Higher Orders 70 4,2. Simi.l.ari.ty' Analysis of Diffusion Equation 72 4. 2.oL i Aaiysis 72 4.2.2 2Ab)solute Invariants a nd the Transformed. Eql.tations 80 4. 35 Similarity Ana.lysis of Steadyy, Two-Dimensional, LaminLar9Boundary-Layer Equations 87 43 5.L Infinitesimal Transformation and the Char - acteristic Function 87 4, 532 Limitation on the Mainstream Velocity 98 4. 33 Absolute Invariants and the Transformed Dif-, ferential Equation 99 4,4 Similarity Analysis of the Helmholtz Equation in General C-urvilinear Coordinates 103 4.4,1 Review of Separability Conditions Discussed'by Moon and. Spencer 104 4,4,2 Similarity Analysis of the Equation 108 4,~43 Conditions of Similarity for a Given Coordinate System 114 4o 4,44-.Conditions of Similarity for a Given Groiup of TranLs formation 118 4, 5 C:oncluding Remarks 122 References for lChlapter 4 124. vi

ABSTRACT Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of a Generalized Similarity Analysis is introduced and results applied to the equations of boundary-layer flow. A thorough presentation of the application of continuous transformation groups to the problem of similarity analyses (reduction of the number of independent variables of an equation) is given with new and very general methods evolved for determining the nature of transformations.

CHAPTER 1 GENERAL CONCEPTS AND SCOPE OF THE INVESTIGATION 1.0 INTRODUCTION The investigations to be presented in this report are an outgrowth of a continuing study of ways in which partial differential equations associated with problems of physical interest may be simplified through transformation of independent and dependent variables. The original motivation for this study grew out of an interest on the part of investigators in the National Aeronautics and Space Agency as to how rather general methods might be formulated for reducing the number of independent variables of a partial differential equation (similarity analyses). As the investigation proceeded, certain facts became clear. First it was noted that the usual types of transformation for simplifying partial differential equations of physical problems were usually of a rather special class. This led to the question of possibly generalizing the class of transformations noted. As will be seen significant success was achieved in this inquiry. A second main area of investigation centered on finding methods of analyzing very general types of partial differential equations other than specific types such as viscous flow equations, etc. It was soon discovered that little of general nature could be uncovered in connection with the more commonly used methods found in current literature. Most of these methods treated problems for which not only the differential equations but the boundary values were also specified. Transformations evolved were based on both the equation and the nature of the boundary values. There was one exception to this, however, that gave promise of providing a very generalized method of analysis. This was the theory of continuous transformation groups. The method was by no means new. In fact, the basic ideas date back to the last century and are found in the work of the mathematician Sophus Lie. Moreover, the theory of groups has been employed quite extensively in recent times by investigators in the field of similarity analysis. Nevertheless, there was not in the literature a truly in depth study of the applications of Lie's theory to the similarity analysis of partial differential equations. Previous applications were quite limited in scope and it therefore seemed reasonable to find how far Lie's ideas might be pursued in formulating a very general approach to similarity analyses. The results that have been obtained and that are presented here are most encouraging. Very general methods of analysis have been formulated and the way made clear for further work. 1

The approach to be used in this report is the following. For the remainder of this chapter, a presentation of general concepts will be made, followed in Chapter 2 by a discussion of the form of general transformations for partial differential equations of problems having a physical basis. The use of transformation groups for similarity analyses will be introduced in Chapter 3, and basic ideas and applications presented. In Chapter 4, the methods employed in Chapter 3 will be generalized and a set of rather powerful techniques for analyzing partial differential equations will be developed. 1.1 GENERAL CONCEPTS The concept of similarity solutions is very old and dates back to such well-known workers as Buckinghaml and Bridgemano2 Originally, the term similarity analysis implied a procedure for finding some information about the solution of particular physical problem usually short of a complete mathematical answer. More recently, in the work of such analyists as Birkhoff3 and Sedov,4 similarity solutions have incorporated rigorous mathematical techniques which result in solutions in the engineering sense, that is, numerical answers. Of particular importance, the more recent work has yielded solutions to nonlinear partial differential equations which have been intractable to more standard solution techniques. In fact, exact solutions of the boundary-layer equations of fluid mechanics are almost universally based on similarity methods. The various similarity solution methods can be broadly characterized as techniques which employ transformations of variables or parameters, or both. Typically transformations may linearize a problem (for example the Kirchhoff and hodograph transformations), reduce partial differential equations to ordinary differential equations (for example the Blasius similarity transformations), transform the system to one already solved (such as the Mangler transformation), or perform some other reduction of mathematical complexity. Sedov4 formulates a mathematical theory of similarity on the basis that "two phenomena are similar, if the assigned characteristics of the other by a simple conversion, which is analogous to the transformation from one system of units of measurement to another)" Thus, a certain analogy exists between the theories of dimensional analysis as formulated by Buckingham and Bridgeman and the geometric theory of invariants relative to a transformation of variables, a fundamental theory in modern mathematics and physics. Some authors have even found it convenient to define classes or types of similarity. Kline5 defines "external similitude" as the similarity between problems of given class (a transformation in terms of parameters alone) and "internal similitude" as a similarity characterized by a mathematical relationship between points inside a single system of a given class (for example, a transformation which reduces the number of independent variables)o Inherent in any general theory of similarity, however, should be the recognition of specifying, in some sense or other, the complete physical problem to be analyzed. In fact, the term "similarity" implies a comparison between 2

two or more complete and recognizable systems. Thus, it is of prime importance to formulate a "complete" mathematical description of the physical problem under consideration before proceeding with a similarity solution. Gukhman6 characterizes a complete problem as one which is expressed in terms of the governing equations together with a body of information termed the "conditions for uniqueness of solution" which may contain not only boundary and initial conditions in the usual sense, but also possible auxiliary physical considerations such as conservation requirements. To be sure, it is not always possible to prescribe the uniqueness conditions a priori for all problems that engineers are called on to consider. In fact, it appears that much of the work that has been done in the past to develop similarity analyses has been motivated by an initial statement of completeness, or lack of it. For example, the early work of Buckingham and Bridgeman on dimensional analysis require only a recognition of the pertinent variables which apply, without any statement being made concerning the governing equations. Later workers such as Morgan,7 Hansen,8 Krzywoblocki,9 and Wecker and Hayes10 investigated similarity methods by considering the governing equations first, and only examining the boundary and initial conditions as a later step, if at all. Another group of workers developed similarity methods by starting with a complete mathematical formulation and, thus motivated, to examine less complete (and more general) problems; see for example Coles,11 Abbott and Kline,12 and Gukhman.6 An examination of these earlier works show that the initial problem statement, as far as assumed completeness, determined to a large extent the kind of mathematical approach employed. The more information that was known, the more direct was the method developed for finding a similarity solution and, at the same time, the less general were both the methods and the conclusions (as regards "general solutions"). It is not suggested that this dichotomy is necessarily bad. The more general techniques, such as group theory methods, have produced powerful theorems and yield results with a minimum of mathematical busy-work. On the other hand, the group theory methods are difficult for the average engineer to follow because their motivation is mathematical, not physical, and this has inhibited their wide use. Also, somewhat amazingly, the more powerful mathematical techniques have been to a degree more restrictive in some of their aspects (such as the "class of assumed transformations") than the less elegant methods. By recognizing the differences in past motivation, and the resulting advantages and weaknesses, the suggestion naturally arises that perhaps a marriage of the two approaches might be fruitful; this on one goal of the present investigation. In the following chapters, the two viewpoints will be examined in more detail than has been done in the past. In the following chapter a technique based strongly on physical reasoning will be used to evolve a very broad definition of similarity. It will be shown that by intro3

ducing the concept of generalized similarity,* a unified method can be developed for deriving the majority of the well-known transformations employed in fluid dynamics. *The term "generalized similarity' was found desirable because of the popular use of the term similarity to imply simply a reduction in the number of variables of a given problem.

CHAPTER 2 GENERAL TRANSFORMATION METHODS IN FLOW THEORY 2.0 GENERAL CONCEPTS An appropriate change of variables is probably one of the most useful methods available for solving the partial differential equations of mathematical physics. The most general criterion for a transformation is simply to change a given problem into a simpler problem is some sense or other; that is, either to a form which will yield to more standard solution techniques, or possible to a form which has been previously solved in connection with a related or similar problem. Hodograph transformations and conformal transformations are well-known methods for transforming given problems into forms which yield to classical techniques (such as separation of variables). Ames13 classifies the various transformation techniques into three groups: transformations only of the dependent variables, transformations only of the independent variables, and mixed transformations of both independent and dependent variables. However, all three groups have a common goal: to find a relation, or more specifically, a basis of comparison, between different physical (or mathematical) problems. This broad concept of comparison is the definition of the term "generalized similarity. " Generalized similarity might be applied in fluid mechanics to attempt to answer such questions as "Is there any similarity (i.e., basis of comparison) between compressible and incompressible flow problems, axisymmetric and planar flows, or in general, any more complicated and a less complicated flow?" By contrast, the usual term "similarity" as used by Birkhoff,3 Morgan, 7 Hansen,8 Abbott and Kline,12 and others is defined in terms of independent variables of a problem. Thus it may ultimately refer to a physical similarity within a given problem, such as similarity of velocity or temperature profiles. Much of the previous work on similarity was motivated by the desire to develop simple methods for reducing the number of independent variables. Out of this previous work came the realization that each of the proposed methods was based on an assumed class of transformations and the recognition that more general classes of transformations might lead to the solution of a wider class of problems. The first part of this chapter is concerned with seeking the most general class of transformations for particular types of problems. The two viewpoints ~ discussed in Chapter 1 will be examined. First, the mathematical theory of transformations will be reviewed and it will be shown possible to postulate the so-called primitive transformation as the most general form of a class of transformations (under a certain assumption). Second, a separate approach, based on postulating the "complete physical problem," is examined for the special case of laminar boundary-layer 5

flows and it is shown that this approach also yields the primitive transformation. The second part of the chapter deals with the development of a technique for employing the primitive transformation to find the form of the variables for a wide variety of generalized similarity problems. 2.1 TYPICAL TRANSFORMATIONS USED IN FLUID MECHANICS As a means of developing some insight into the question of "the most general class of similarity transformations" and the concept of "generalized similarity," Table 1 was compiles of examples of as many different types of known transformations as could be found to represent the field of fluid mechanics. TABLE 1 Transformation15 Transformed Independent Variables Similarity 5(x) = x, r(x,y) = y 7(x) Meksyn-GUrtler (x) = I u(x)dx, l(x,y) = ul(x)y/ 4 0o von Mises 5(x) = x, r(x,y) = *(x,y) Crocco o(x) = x, r(x,y) = u(x,y) Mangler (x) = fX ro(x)dx, r(x,y) = ro(x)y 0 Stewartson,(x) = f al(x)pl(x)dx, q(x,y) = fx al(x)p(x,y)dy Dorodnitsyn (x) = f p1(x)dx, (x,y) = fx p(xy)dy O O Similarity Rules of High Speed Flow (see, for example Ref. 16): (i) Prandtl-Glauett (ii) GWthert j(x) = x, r(y) = By (iii) von K~rmgn Transonic 6

By comparing the various transformations in the table, an interesting conclusion can be made: all of the transformed independent variables are of the general form e = i(x) and r = r(x,y) (that is, one of the new variables is a function of only one of the original variables). This realization raises the question "Is the general transformation t(x), q(x,y) the most general class of transformations that need be considered for physical problems?" In an attempt to answer this question, it will first prove useful to review the mathematical theory of transformations. 2.2 THE PRIMITIVE TRANSFORMATION The following theorem can be found in the mathematical literature of general transformation theory (see, for example, Courantl4): THEOREM: An arbitrary one-to-one continuously differentiable transformation = - (x,y), r = r(x,y) of a region R in the x,y-plane onto a region R' in the g,q-plane can be resolved in the neighborhood of any point interior to R into one or more continuously differentiable "primitive" transformations, provided that throughout the region R, the Jacobian x()= ny - ynxx ~(x,y) differs from zero. A "primitive" transformation is of the form = e(x), r = r(x,y) Now one-to-one transformations have an important interpretation and application in the representation of deformation or motions of continuously distributed systems, such as fluids. For example, if a fluid is spread out at a given time over a region R and then is deformed by motion, the motion of the fluid is described by the coordinates in the physical R-plane. If the fluid motion in R is characterized by the coordinates x,y, then the corresponding motion in the transformed region R' is characterized by coordinates ~,. The one-to-one character of the transformation obtained by bringing every point x,y into correspondence with a single point 5,r is simply the mathematical expression of the physically obvious fact that the fluid motion in the physical R-plane must remain recognizable after transformation to the transformed R'plane, i.e., that the corresponding motions remain distinguishable. 7

Physically, since most "practical" transformations of interest for solving physical problems should be one-to-one,* that is, have a unique inverse (except possibly at a finite number of singular points), it appears the primitive transformation or the resolution into primitive transformations, should be considered in the search for "the most general class of transformations." For the sake of completeness, the following two properties of the primitive transformation are noted: (i) If the primitive transformation i= i(X)9 r' = r(Xy) is continuously differentiable, and its Jacobian = ~J'~ - xTy - y x 6a(xy) =xry ynx differs from zero at a point P(xoyo), then in the neighborhood of P the transformation has a unique inverse, and this inverse is also a primitive transformation of the same type. (ii) For primitive transformations, the sense of rotation in the x,yplane is preserved or reversed in the,d-plane according as the sign of the Jacobian is positive or negative, respectively. In summary, the primitive transformation appears to be the most general class of transformations that need be considered, purely from a mathematical viewpoint, as long as it is required that a unique inverse of the transformation must exist. In the next section, the question of the most general transformation will be approached from a fundamentally physical viewpoint and it will be shown that again the primitive transformation appears to be a requirement from conclusions based on uniqueness arguments. 2.3 REQUIREMENTS IMPOSED BY THE PHYSICAL DESCRIPTION The question of the most general class of transformations will now be *An exception to this rule; for example, is the von Mises transformation of Table 1, which is singular along the x-axis. However, this transformation is used for computational (not physical) reasons, and hence its use is motivated by a completely different line of reasoning than that under consideration here. 8

examined from a more physical veiwpoint. Some mathematics will be involved, of course, but the physical problem will be kept near at hand and frequent reference to it will be made throughout the analysis. Because in what follows, by definition, involves a particular physical problem, it will be convenient to introduce such a problem. As an example, consider the following equations describing the two-dimensional flow of a laminar incompressible boundary layer: 6u +v U+8V = o ax by (2.1) au 6au 62u dp u ax v y - 2 - dx or an equivalent single equation in terms of the stream function; at a_. at aV a V =. (2.2) by axay ax ay2 ay3 dx For convience in the development, it is assumed that all of the variables are nondimensional. The objective is now to consider very general transformation of variables and see what conditions must be met by this transformation so as not to violate any known physical properties of the problem. Let us begin by specifying a transformation of variables. For the independent variables, no restrictions at present will be placed on the assumed form; thus the form is specified simply as = (x.y), Q = r(x,y). (2.3) The dependent variable, (the stream function) is also transformed to a new function, say T. Previous work on fluid flow tranformations has often assumed that the two stream functions, 4 and T., should be the same at corresponding points and hence that streamlines in one plane are tranformed into streamlines in the other. However, this restriction is found to be unwarranted in many problems and will be avoided here. Instead, the relationship'between, and E will be specified in some weak sense by the form () = g(x,y)t(x,y). (2.4) 9

It should be recognized that this assumption is not trivial and other forms could be considered. Nevertheless? Eq. (2.4) represents a more general case than usually employed and attention will be restricted to this form in the present report. The next step is to carry out the transformation of Eqs. (2.3) and (2.4) on the left-hand side of Eq. (2.2), Application of the transformations (2.3) and (2.4) to the boundary-layer equation is not new and was considered previously by Coles15 and others. However3 the generalized similarity technique was not under consideration by the previous authors. The results of the transformation of Eq. (2.2) is given in Table 2. For convenience in interpreting the physical terms after transforma~tion, transformed velocities U,V have been defined by U = V = The transformed side of Eq. (2.2) given in Table 2 is essentially unmanageable in its present form. While it is unclear at the present time what conditions are either necessary or sufficient to ensure any particular mathematical behavior, it is interesting to consider an argument proposed by Coles.15 Coles' argument is based on the a priori requirement that the transformed flow outside of the shear layer (i.e., for large values of y) should conform both physically and formally to the original flow. Thus3 he argues3 since the physical flow is bounded for large yr and is, in fact, at most a function of x, then the transformed side of Eq. (2.2) must behave in a similar fashion. His argument is then that the substantial derivative terms and the U2 terms become at most functions of 5 for large y. whereas the remaining terms behave either like y or y2. His conclusion is to require the 2 and q2 terms to vanish identically3 which can be accomplished by requiring that g = g(x)9 5 = (x), and qy = 7(x). At the present time3 about all that can be said is that this assumed form is one possible form of a transformation which will preserve a certain sense of physical correspondence between the given and transformed flow; this form of the transformation will be employed throughout the remainder of the present chapter. In summary, the postulated general transformation3 based on the physical mode., is: (,)= g(x)k(xy) (2.5) = t(X)3 = y(7(x)) 10

TABLE 2 Transformation: Y(5,y) = g(x,y)V(x,y); l = Y(x)y), ~ = r(xty), U = V = -_ a dp, 6 a2P 6 a2~ a3~ 6U +U J(..1 dx ay axay axa) X a6y2 = +) y 9,2 -_ - 3 __ =Y U + X Y2 ~3g2 +,,T / ).2 W -, 7/g,rl)+J(riy/g,, ~) J(+y/g, ) + u2 U w'Y/ +V2 Y g g g __ 2 gy yT'y + 2____ q —.LK-,j -2 f +2Q Y) +(i) 4.] + a _ y ( Y) - g [ Y g2 \+(g y a-2u [2 2 t 62V 3 82U rl + 2 7 + 2 < 2 g y2 g g +U W ng )g(gy/gA )] g3 + TV J(/Yg)+gJ(gY/gA1 11

It is of interest to note that the resulting form of the independent variables is a primitive transformation as discussed in the last section, but it has been shown by considering a particular problem and relying on physical arguments that it may be possible to restrict the form of ~ to be a linear function of y for this particular class of problems (boundary-layer flows). Thus, the two approaches, mathematical and physical, have lead to the same conclusion regarding a postulated "most general class of transformation." Of course it is necessary, as previously pointed out, to emphasize that there may be some cases of practical interest which will lie outside the realm of the mathematical conclusions presented here. However, it has boen found that all of the cases listed in Table 1 are satisfied by the transformiation of Eq. (2.5) (with the exception of the von Mises transformation previously discussed). It is now worthwhile to return to Table 1 for a moment. Recall that from Table 1, it was noted that a wide class of different transformations were all primitive transformations. Further, note that one of these transformations the similarity transformation, has been shown to yield to simple general analysis for its derivation for particular problems. In fact, at the present time there are two types of similarity analyses that are founded primarily on simple transformation theory: the free parameter method of Hansen8 and the separation of variables method of Abbott and Kline.12 It is thus of interest to speculate on the following question: Would it be possible to derive all of these "generalized similarity" transformations by the same technique that is used to derive the similarity transformation? The Generalized Similarity Analysis was developed as an attempt to answer this question. This new method is based on a single assumption concerning the admissible class of transformations of the independent variables, namely that the assumed transformation should be one-to-one (that is, have a unique inverse). On the previous pages it was shown that this requirement would lead to the primitive transformation, and further that a particular primitive transformation may be obtained from physical arguments for a particular class of problems, namely 5 = S(x), ~ = y7(x) for the boundary-layer flows. Of course the generality of the present ideas goes beyond a particular case, such as boundary-layer flow analysis and should be appliable to a wide class of problems. 2.4 THE GENERALIZED SIMILARITY ANALYSIS The development of the generalized similarity analysis was motivated as an extension of the method of finding similarity variables (i.e., the reduction of the number of independent variables) to the problem of finding a transformation of variables which will convert a given physical problem into an alternate problem under certain prescribed conditions~ For example, the prescribed condition for a similarity solution is that the number of independent variables must be reduced. By contrast, the prescribed condition for the Mangler transformation is that the axisymmetric boundary-layer equations are 12

to be transformed into the planar form of the equations, and so forth, with similar statements for the rest of the examples in Table 1. There are three distinct steps to the generalized similarity analysis. These steps are: (a) The general mathemathical theory of transformations states that any continuous one-to-one transformation can be resolved into one or more primitive transformations of the form:= (x), y = r(xy). Thus, a primitive transformation form is assumed a priori, where it is recognized that the general analysis has the possibility of being repeated more thar once, depending on the particular problem at hand. (b) The given equation, transformed under a primitive transformation to the new independent variables (tr) is required to satisfy the state requirement; for example, that for a similarity analysis, the transformed equation should be a function only one of the new variables. (c) Simultaneously the boundary conditions for the given equation are required to be satisfied when expressed in terms of the transformed variables. These three steps will be shown to completely and uniquely determine the explicit form of the new variables ~(x) and r(x,y) if, in fact, the original problem and associated boundary conditions do admit a generalized similarity solution. As an example of the method, two problems will be examined in detail. First the classical Blasius flat plate boundary-layer problem will be solved to give a relatively simple motivation of the basic ideas. Second and more difficult Mangler transformation will be derived as an example of the broad applicability of the method. Example 1 The boundary-layer equations for a steady, two-dimensional laminar flow with a zero pressure gradient (Blasius flat plate flow) can be written in the following form in terms of the stream function: _ v (2.6) yxay xy ax ay2 ay3 13

(all variables are dimensional, where v is the kinematic viscosity) with the boundary conditions y =: O X = yy ax 0(2.7) y + co' a + constant = uo by To make the problem statement complete, it is necessary to state the given requirement for the transformation x,y + ~,: Requirement: under the transformation, the given equation and boundary conditions, (2.6) and (2.7), are to reduce to a function of a single independent variable (i.e, the similarity variable). The three steps of the generalized similarity analysis are carried out in order as follows: (a) Assume a primitive transformation = j(x) and j = ~(x,y). Since a boundary-layer problem is under consideration, it was shown in the Section 2.3 that it is sufficient to choose the more specific transformation for ~ as ~ = yy(x), and further, for the present case of a similarity solution, ~ can be assumed in the simple form 5 = x without loss of generality because the final result, by definition, is to be a function of the single variable q(x,y), and not both 5 and i.* For the dependent variable *(x,y), the transformed variable ~(,') is assumed in the form (O = g(x)4(xy) as discussed in Section 2.3. Thus, the assumed transformation for the dependent and independent variables becomes *The question naturally arises, why not assume the form ~ = y, ] = (.x,y)? This alternative is easily eliminated for the present case by carrying out the corresponding analyses in an analogous manner as that presented here; it is then found that this assumed form will not satisfy the given boundary conditions.

= g(x)4(x, y) (2.8) = x, fl = yW(x) Performing the transformation (2.8) on the given Eq. (2.6) yields the following results: ay 6 g ay2 6a2 g a39 f3 rly ay3 bTl3 g -1 1 ___ ax + + 6x g2' g 6~ g 62 _ =J ( f"xy g, 62~ T'ry 62 f2rlyrlx axay 6rl g2 bE b g ba,2 g where primes are used to indicate total derivatives. Substituting these results into Eq. (2.6) yields: a3T -gY 1 rxy6 2 ag2w 1 82Egt 1 V ( + = ~an3 \ g n gJ Vany ar2 afl gqy af2y g fly with boundary conditions: n = (x,o ) o:o ~n y(X O) =0; - g + X(X) fl~~~fl~(XyO~~)+ ~ T T~ =(xo) ( = (xY) + 0': _ Yy(xny+) + constant = uO. (b) Imposing the stated condition of similarity by requiring that! = 1(n) 15

that is, that the transformation must reduce the number of variables, and noting that y = y(x), the transformed equation becomes: VY",' - - - g T = 0 (2.9) g27 2 2 with boundary conditions = 0:'y = O, -E- +'7 = O -g (2.10) + 0:.' -+ u U g (c) So far the functions g(x) and y(x) are unspecified. However, only particular forms of these functions will satisfy both the boundary conditions and the stated requirement that Eq. (2.9) must be a function only of i. Examining the last boundary condition (2.9) yields y,'(co) 7(x) + constant = uo g(x) The only way that this can be true is if 7(x) = uog(x) Also, the only way that Eq. (2.9) can be a function of the single variable P is if the coefficients of T and its derivatives are constant. Hence, letting g 1 _= g' = constant = cl g2 7 uog3 then integration of this ordinary differential equation yields g = [-2cluo(x+xo)]l/2 16

where xo is an undeterminable constant of integration,* and letting the arbitrary constant c1 take the value cl = -v/2 yields g = [vuO(x + o) ] 1/2 Y = uo[vuo(x + XO)]-/2 Evaluating the coefficient of,2.: = cl and 7 g27 72g oe Thus this coefficient is identically zero** and the final form of similarity solution is: Tt, + 1Be = 0 (2.11) (O) = T' (O ) = 0,? (X) + 1 In summary, it has been shown that a unique choice for the transformation variables can be found for the similarity solution of the Blasius problem by carrying out the three steps of the generalized similarity analysis. The next example will show that the method can be applied to a much more- generalized problem, that of finding the conditions for which an axisymmetric boundary layer is similar to a planar boundary layer. *The meaning of xo is clear from the physics of the problem, its magnitude is determined by conditions at the leading edge of the plate and cannot be derived from boundary-layer theory alone. Thus the appearance of xO is a sufficient proof that the Blasius solution is a correct asymptotic downstream solution for laminar flow over a plate which holds independent of the initial conditions specifying the plate leading edge. **The coefficient of Iy,2 in Eq. (2 9) is not independent of the relation 2 gl/g2y = cl. This can be seen as follows: for simplicity, let y = 1/8 and g = l/h. Then g,/g2y = -h'8 = c1 and letting (g,/g2-y7/y72g) = (-h'8+8'h) = c2, then 1'h is a constant, say C3. The two equations h'8 = -c1 and 5'h = c3 may be cornined to yield 58" = (cl/c3)'12 = 0 which has the solution 8 = lax + b ll/ (1-/c3) for c1 / C3 and 8 = aebX for cl = c3. This provides a general solution, however we cannot evalute cl or c3 without another condition. The conclusion, as found above, remains the same; the required condition must come from the boundary conditions. 17

Example 2 The well-known Mangler transformationl6 serves as a good example of the broad meaning of the idea of generalized similarity because the transformation answers the question "Under what conditions is an axisymmetric boundary-layer flow similar to a two-dimensional planar flow?" The answer lies in finding a transformation between the variables describing the two types of flow. The generalized similarity analysis is formulated in the same way as the simple similarity analysis of the preceding example; that is, by specifying the equations (and boundary conditions) and the stated requirement to be satisfied by the transformation. The governing equations for a thin laminar axisymmetric boundary layer are: (r) (rv) _ (2.12) ax 6y au + v ~u dul 62u 6u + v u = ul + v (2.13) ax by dx ayd where ro = ro(x) is a given quantity (ro specifies the body shape relative to the axis of symmetry) and v, the kinematic viscosity, is a constant. The general boundary conditions are: y =0: u,v =O (2.14) y+ 00: U+ ul() The problem statement is completed by writing down the required mathematical form of the desired equations, namely: Requirement: under the transformation u,v e- U,V and x,y + I, the transformed equations are to be in the form of a planar boundarylayer flow; that is, of the form -U+ = o (2.15) 6U we + v a = U1 -~ + v - -(2.16) 18

The three parts of the analysis are as follows: (a) Assume a primitive transformation. Again, since the problem involves the boundary-layer equations, it is sufficient to assume the same general form of the transformation as given in Section 2.3, that is = g(x)*(x,y) (2.17): =,(x), =- y:(x) where, for the present case, the stream function *(x,y) is defined by the equations u =, v = ro by rO ax so as to satisfy Eq. (2.12). For the transformed flow to be in the form of planar equations, the transformed stream function F(5,q) is defined by u v Using these definitions and the transformation (2.17), the velocities and their derivatives become u(x,y) = Y U((, ) rog v(x,y) = 1 g' r(,) + Sx V((,r) _ nx U(5,q) rog g rog rog 6u = u ay rog an a2u a22 6y2 rog 6a2 a -x (_Z_)U + r rog ro g19

Substituting these expressions into Eqs. (2.12) and (2.13) yields, after rearrangement: J(T) au+ - O (2.18) x ( aU + V aU dU = (yrog) v(rog) v 3u + r0) (U2-U2) -g' -- (2.19) [ro0 It is found that the continuity Eq. (2.18) is invariant under the transformation (2.17) as long as the Jacobian J(, ) = ySx differs from zero. Thus Eq. (2.18) does not force any requirements on the transformation. However, in comparing Eqs. (2.16) and (2.19), it is seen that for the latter equation to fulfill the given requirement of being identical to the planar form given by Eq. (2.16), it is necessary that x = 7rog (2.20) and, og) (U2_u2) _- g = 0. (2.21) (ro) A sufficient condition for Eq. (2.21) to be satisfied is for _2 = constant = cl (2.22) r0g and g = constant = c2. (2.23) Although Eqs. (2.22) and (2.23) may not be necessary conditions for the satisfaction of (2.21), they at least provide one solution for the given require20

ment and this is usually satisfactory from an engineering veiwpoint. Since there are no further requirements to restrict a choice for the constants cl and c2, they are normally chosen for dimensional reasons to assume the values cl = 1 and g = 1/D where D is an arbitrary reference length. Hence, the final form for the transformation becomes: ( = D t(x,y) (2.24) D e= - dx (2.25) o D2 r ( x) (2.26) D (b) It has already been shown that for the present case, the unique form of the transformation, Eqs. (2.24), (2.25), and (2.26), is determined by requirements on the differential equation alone. Thus, the boundary conditions do not provide any additional information and, in fact, they are found to carry over directly as follows: 0 = O: U,V = O (2.27) + oo: U + U( ) The analysis is thus complete, Eqs. (2.24), (2.25), and (2.26) being known as the Mangler transformation. 2.5 CONCLUDING REMARKS It has been shown that the primitive transformation appears ot be the most general class of transformations necessary to provide a transformation with a unique inverse. This result was obtained from the general mathematical theory of transformations,'but was also supported'by a physical argument for such cases as boundary-layer problems which showed that the transformation E(T3) = g(x)4(x,yy) e = 5(x), n = y7(x) 21

is possibly sufficient to ensure proper behavior of the equations. The generalized similarity analysis was then introduced to solve a wide class of problems which could be formulated as a comparison between two given subproblems. Two examples were given, the Blasius similarity problem and a derivation of the Mangler transformation, however all of the cases given in Table 1 can be derived in s aimilar fashion. A few comments can be made concerning the role of the function g(x) appearing in the transformation of the dependent variable for the boundary-layer equations. In certain cases, the value of g(x) will be uniquely determined by the problem. For example, g(x) is proportional to fx and j (x) for the similarity and Meksyn-Gkrtler transformations, respectively, and g is found to be constant for the Mangler transformation. However, in some cases, the choice for g is arbitrary; for example, g may take on any value for the von Mises transformation. Further, for the case of comparing compressible and incompressible forms of the boundary-layer equations, different choices for g lead to different, but nevertheless useful, results: Stewartson chooses g = constant and Dorodnitsyn chooses g = Ty = 7(x). Coles discusses the significance of g for the difficult problem of compressible-incompressible transformations of the boundary-layer equations for turbulent flow in Ref. 11. In summary, the ideas presented in this chapter are based on, or more appropriately, motivated by a physical description of a problem which is in some sense complete. Depending on the particular case under consideration, completeness may imply a knowledge of all necessary boundary conditions, or possibly only a statement of a particular requirement of the transformation. In any case, a fairly specific problem formulation is implied. In the next chapter, a different technique will be examined which focuses attention on a more narrow application; the simple similarity problem (in the sense of reducing the number of independent variables). This technique, the group theory method, being less encumbered by statements of broad generality, will prove to yield very elegant and powerful mathematical results for finding similarity solutions for a wide range of applications. 22

REFERENCES:FOR CHAPTERS 1 AND 2 1. Buckingham, Eo, "On Physically Similar Problems: Illustrations of the Use of Dimensional Equations," Eo Phys. Rev., 4 (1914). 2. Bridgeman, P. W,, Dimensional Analysis, Harvard University Press, 1921. 3. Birkhoff, G., Hydrodynamics, Princeton University Press,.950. 4. Sedov, L. I., Dimensional and Similarity Methods in Mechanics, Academic Press), nnc, 1960o 5. Kline5 S. J., Similitude and Approximation, Theory, McGraw-Hill Book Company, 1965. 6. Gukhman, A. Ao, Introduction to the Theory of Similarity, Academic Press, 1.965. 7. Morgan, A.J.A., "Reduction by One of the Number of Independent Variables in Some Systems of Partial. Di.fferenti.al. Equations," Quarto Applo MathO, 5 (1952). 8. Hansen, A. G0. Similarity Analyses of Boundary Value Problems i.n Engineering, Prentice HalL, 1964. 9. Krzywoblocki, M.oZovov Association of Partial Differenti.al Equations with Ordinary Ones-, and Its Applications, Inmto. SympO on Nonlinear Differential Equations and Nonlinear Mechanics; J. P. LaSa.lle and S, LaLachata, Academic Press, L963o 10. Wecker, M. S., and Hayes, W. D., "Self-Similar Fluids" AFOSR TN 60-894, 1960o J11 Coles, D. E., "The Turbulent Boundary-Layer in a Compressible Fluid," Rand Report R-403-PR, 1962. 12. Abbott, D. E.o and Kline, S. J., "Simple Methods for Construction of Similarity Sol.utions of Partial. Differential Equations, AFOSR TN 60-1163, Report MD-6, Department of Mechanical Engineering, Stanford University, 1960o 135 Ames. W. F,, Nonlinear Partial Differential Equations in Engineering, Academic Press, 1965 23

REFERENCES FOR CHAPTERS 1 AND 2 (Concluded) 14. Courant, R., Differential and Integral Calculus, Vol. II, Interscience, 1956. 15. Sun, E.Y.Ch., "A Compilation of Coordinate Transformations Applied to the Boundary-Layer Equations for Laminar Flows," Deutsche Versuchsanstalt fur Luftfahrt, Report DVL 121, 1960. 16. Liepmann, H. W., and Roshko, A., Elements of Gasdynamics, John Wiley and Sons, 1956. 24

CHAPTER 3 SIMPLE GROUP THEORY APPLICATIONS 3.0 INTRODUCTION In this chapter we will consider one of the more widely used methods of reducing the number of independent variables of a partial differential equation, the group-theoretical method. The foundation of the method is contained in the general theories of continuous transformation group that were introduced and treated extensively by Lie in the latter part of the last century. G. Birkhoff2 was one of the first to apply this concept in searching for similarity solutions of partial differential equations. Subsequently, Morgan5 investigated quite thoroughly the mathematical theories involved and essentially completed the development of the method now generally referred to as the group-theoretic method.3 In the classical group-theoretic similarity analysis, a one-parameter linear or spiral group of transformation is employed.* The variance of a given partial differential equation under the transformation group and the introduction of new invariant variables, enables the number of independent variables to be reduced by one. Once the theories involved are understood, the process of obtaining similarity solutions is considered to be the simplest as compared with other techniques. Extension of this method to multi-parameter linear and spiral groups of transformation has been developed in recent years, and will be discussed in this chapter along with the one-parameter approach. 3.1 ONE-PARAMETER GROUP-THEORETIC ANALYSIS In this section the topic of group-theoretic analysis will be introduced by restricting attention to one-parameter methods. The material will have the twofold purpose of showing how the one-parameter method is used in similarity analyses and will provide a background for material in the remainder of this chapter and Chapter 4. By no means does the one-parameter similarity analysis plumb the depths of Lie's theory of continuous transformation groups. As will be seen, only two simple types of transformation groups will be considered-the linear and spiral groups. In the next chapter, a far more general method is presented that permits the construction of a greater variety of transformation groups. This section primarily reviews the work of Birkhoff2 and Morgan.5 *See Ref. 3 for a general introduction to group theory and this method. 25

3.1.1 Absolute Invariants The first step in introducing the concept of one-parameter transformation groups is by establishing the group property of a transformation of coordinates. In general, if a transformation is defined by xi fi(xl..x; a'.,an) then there exists (m-l) functionally independent absolute invariants SJ(x',...,xm) where j=l,...,(m-l). By definition, an absolute invariant has the property:J0(X' o.. x M) = ~ (X',.,Xm). (3.2) Follwing Morgan,5 we restrict ourselves to the one-parameter group* Ta: Xi = fi(x. xm;a) (3.3) YJ = hJ(yT,...,yn;a) where i=l,...,m (m>2) and j=l,..o,n (n>l). For the subgroup Xi = fi(x',..,xm;a), there exists (m-l) functionally independent absolute invariants ak(X'5-o9.xm). k=l..(m-1). (3.4) For the group Ta as a whole, there are (m+n-l) functionally independent absolute invariants. We therefore add the following to the list: gi(x' gOxm;yt,.. yn) (3-5) =l1 o o en *The variables xi and yi will shortly be identified as independent and dependent variables. For now, they are just general variables. 26

3.1.2 Morgan's Theorem Assuming that yj are the dependent variables and xi the independent variables in the one-parameter group of transformations Ta: Xi = fi( xT,;a) (3.6) yJ = hJ(y',.. yn;a) the system of partial differential equations of order k 7'k k n 6(xl) o (Xm)k is invariant under this group of transformation. Tan if each of the j is "conformally invariant" under the transformation Tk This means that tntiakyl e akyni j(X',.o.,Xm;y',...Yn; k y, a a(xt)k k = )xm;y9.0.y a k;a) x ~j(x'~.o.xm;y'. yn;.; (. (3.8) In particular, if F = F(a) = 1, ij is said to be "absolute invariant" under this group of transformations.* The above result leads to the very important theorem of Morgan:5 Theorem: Suppose that the forms sj are conformally invariant under the group Ta] Then the invariant solutions of ~j = O can be expressed in terms of solutions of a new system of partial differential equations 6 k k The *i are the absolute invariants of the subgroup of transformations on the xi alone and the variables Fj are such that *See Refo 30 27

j ly...~,lm_)l = function of (x'I,,, o om). (3 10) An example is given in the next section to illustrate the method. 3.1.3 Application of One-Parameter Group Theory to Boundary-Layer Equations Consider the steady, two-dimensional laminar boundary-layer equations: u Vu + 311) TX= o u u + v u = U dU + v 2u (3.12) ax by dx ay2 subject to the boundary conditions y = 0: u =v = 0 y = o: u = U(x) Group theoretic methods will now be employed to reduce Eqso (3.11) and (3.12) to ordinary differential equations. To this end, two groups of transformation (namely, the linear and the spiral groups) are considered. The method by Birkhoff and Morgan serves as the basis for the analysis that followso Case 1: The linear group. The linear group of transformation is defined as x = A x, y Ay, u A u v = A v, U = Asu (3.13) where ac,..,oE are constants and A is the parameter of transformation. The function U will be considered as an independent variable with the defining relation U = U(x)o 28

Under this group of transformation, Eqs. (3.11) and (3.12) become A -;__ + A 4-2 (3.14) 6x ay and 2 - Aa+a- +- 2C A2a -aU dU a3-2a2 (315) It is seen that the differential equations will be invariant before and after the transformation if the powers of A in each term are the same. Thus, we get - C = c4 - 2 (3.16) and 2aC3 - 1 = 3 + 4 -2 = 2a5 - 1 = a3 - 2a2 (317) Solution of Eqs. (3516) and (3.17) then gives ~3 = a5 = al - 2Z2, 04 = -a2 ~ (3.18) Knowing the relation among the W's, the absolute invariants can be obtained by eliminating the parameter of transformation A. Thus, noticing that Y = Y a = a2/a, U U y2a -1 -2u u x x V V y v -a --- x x U U 29

these combination of variables are readily shown to be invariant under the linear group of transformations and so are absolute invariants. According to Morgan's theorem, Eqs. (3.11) and (3.12) can now be expressed in terms of these invariants; or in other words, these irivariants are possible similarity variables. Therefore, we put _ y u v 2; f(fl) = 2; gh() = m-1 (m=1-2a) (3.19) and h() = cl = (3.20) where f and g are functions of i and h(r) = cl is a constant since U. the mainstream velocity, is a function of x only and thus cannot be a nonconstant function of ~ which would introduce dependence on y. From Eq. (3.20), we get the form of mainstream velocity for the establishment of similarity solutions, i.e., U(x) = clxm. (3.21) We may now make a check, the transformation of the boundary conditions. It is seen that the boundary conditions are transformed to: q = 0: f = g = 0 fU(x) =00: f =x c~ xm Thus, for constant values of ~ constant values of f and g result. It is now a simple matter to transform the differential equations. Thus, Eqs. (3'.1i) and (3.12) are transformed to 1 -m mf 1m f + g' 0 (3.22) 2 and f2 - ff' + gf' = mc2 + vf" (525) 3o

The boundary condition are those previously stated. Case 2: The spiral group. A spiral group of transformation is defined by ~- _ - 032b - 3b x = x + Blb, y = y e2b u = u e v=~e~4b U=Ue~5b (3.24) v = v e4b U = b (3U 24) Following the same steps as given in Case 1, we find that the parameters are related by 3 = = 2 5 = 2-22 (3.25) and, after elimination of b, the absolute invariants are: y e g = (u=3/ 1) (3~26) e 2 e e2 and h(n) = c U(x) (3.27) eiX Again, c2 must be a constant. Thus, the mainstream velocity should be of the form U(x) = c2eX (3.28) for similarity solutions to exist. Checking the boundary conditions, we find that they are transformed as follows: ~ = 0: f = g = 0 = CO: f = C2 31

The last step is to transform the differential equations. Equations (3511) and (3 12) become f+ - rf' + g' = O (3.29) and 3f2 + P1 Bff' + gf = vf". (3 30) The above example illustrates the one-parameter group-theoretical methodo Some further comments are now in ordero In both examples, the steps are as follows: 1o Define the group of transformation and substitute into the differential equations o 2. Require that the differential equation be invariant. Relations among the constants in the transformation are obtainedo 3. Eliminate the parameter of transformation to give absolute invariants, which will become similarity variables~ 4. Check the boundary conditions to see if they can be transformed into constant valueso 5. Transform the differential equations. It is seen that up to step 49 no differentiation is needed. This makes the group-theoretic method an advantage over other methods. As long as in step 2 zeros for all the constants (eog., looo, a5 in the linear group) are not obtained, the partial differential equations are transformable into ordinary differential equationso Step 4 in this particular problem needs not be put before step 5; however, in general, if the unknown function appears in the boundary conditions alone (eog.o the diffusion equation with a boundary condition u(O,t) = U(t) and the form of U(t) is given), it ultimately may save effort to check the boundary conditions first as above. Equations (3.11) and (35.12) can be reduced to one equation by solving Eq. (3.12) for go However, we are illustrating the method and thus retain f and g in two separate equationso 32

Finally, comparison between Eqso (3.18) and (3~25) shows that Eq. (3~25) can be obtained by putting a, = 0 in Eqo (3.18). Thus, the spiral group is sometimes considered to be a special case of the linear group in which ac=O~ 3.2 MULTIPARAMETER GROUPS The method discussed in the previous article using a one-parameter group of transformation is usually applied to cases in which one independent variable is to be eliminated. If two variables are to be eliminated, the method should be applied twice, first reducing the variables by one and then applying the technique to reduce the variables of the transformed differential equations again. The simplicity of the method characterizing this technique is then losto Thus it would be valuable to have a method in which the number of variables can be reduced by more than one in a single application~ Such a method based on multiparameter groups was presented by Manohar4 and later by Ames 1 We will discuss this method but limit the discussion to the method for reduction of the number of independent variables by twoo Generalization to more variables at a time can be made in a similar manner. Before proceeding with the nultiparameter group analysis, we shall first describe a special case where one-parameter groups may be employed to reduce the number of independent variables by more than oneo 3.o21 One-Parameter Group Method* Let a transformation group?ll'be defined by a,_ a2- A_3:ll' xl = A, x2 A 2xl x3 = A X3 (3o31) yj = yj, (j=4,o,n) where A is the parameter of transformation and the a's are constants to'be determined from the condition that the given partial differential equations be constant conformally invariant under this group of transformation. We consider a partial differential equation in which the independent variables are xl, x2, and X3o In general, the group Fll can only reduce the variables by one. However, in the special case in which al = a2 f 0, reduction of the variables by two is possibleo Absolute invariants can be shown to be *Amesl considers the case in which cl=2-2=O as a separate case0 However, since the form of absolute invariants in this case is the same as in rT'4'below, this case is omitted~ 33

X3 (3.32) (axl+bx2 )C(3/51 and f(ri) = aJ ( 333) (axl+bx2 )(j/sl where a and b are any arbitrary constants. 3.2.2 Two-Parameter Group Method Let T2l be a two-parameter transformation group r21: x- A x1, x2 B = 2, 3= A3B x3 (3534) YO = A OB Jo (j=4,...,n) Again, A and B are parameters of transformation and a's and P's are constants to be determined from the condition that the given partial differential equations be conformally invariant under this group of transformation. Absolute invariants are found to be TI X3 (3.35) i X2I33 and fj W ( ) (3.36) xlaj/Ul X Pj1P Next, we consider the two-parameter group r22 where r22: xl = xl + 1A, x2 = Blx2, X3 = e 3AB x3 (3.37) yj = e j B yj (j=4,...,n) Absolute invariants for this group are 34

-X= x3 (3.38) e X2' and CI = OJ 1 pi (3 39) e X2 A third group r23 may be defined as l- - s3 P3B_ r23: x, = A x, x2 = x + 1B, x3 = A e x3 aj PjB (3~40) Yj = A e yj (j=4,0o,,n) 0 Absolute invariants for this group are: X3 r3 3 X2 (3o41) and fj(n) =.42 ti:Cl Pj i(3~42) x2 Finally, let the group r24 be defined as r24: 1 = X1 + a1A X2 = x2=2 + B, X3 =e e3Ae3B3 (3A43) y jA e yj (j=4, o.,n) Absolute invariants for this group are: =X3 (344) 35

and fj(s) = YJ. (3.45) erl Xe 1 x2 3.2.3 Example of a Similarity Analysis for a Three-Variable Problem Consider the unsteady, two-dimensional boundary-layer equations expressed in terms of a stream function t: ___ + at a-v at a-v = aU + U -IF V (3.46) atay ay axay ax ay2 at ax ay3 The boundary conditions are y = o0 ao,. a = ay ax y=o: = U(x,t) One-Parameter Group: Let?ll be defined as rll: t = AallT, A = y A y (3.47) A = A 47, U = AcsU From the invariance of Eq. (3.46) we get 3 = a, a4 = - 2, %5 = C2 - a1 (3.48) 1 1 G3 = 2 1, a4 = 2 0, O 05 = O. (3.49)

Absolute invariants are: = Y (3.50) 4at+bx and f(~) = -_ — _ (3551) N-(at+bx) Since U is a function of t and x, the only possibility for similarity to exist is to assume U = constant = Uo o (3 52) Using these variables, Eq. (3.46) can be easily transformed into an ordinary differential equationo fTtt + a'b f"'~ 2rjf" + + ff" = o (3 53) subject to the boundary conditions r = 0: f = f' = 0 + 00: fv = 1. Two-Parameter Group: Let F2l be defined as 121: t = A'lt x = B1 y = 3B3 - AC4B%4, U = A BU (3 54) From the invariance of Eq0 (3o46), we get 37

1 1 C5 = -a. a3 a, C=4 = 2 P4 = Bs = 1, 3 o=. (3555) Absolute invariants are:,> _nc =.Y f ( (6) t t X Since U is a function of t and x, c cannot be a function of ro Then the only possibility is c to be a constant. Equation (3.46) is transformed into the following ordinary differential equation: f ft + ff + cff"(1-C ) = O (3 57) subject to the boundary conditions = O: f = f' = 0 = o: f' = 1. It can be shown that for the other three groups mentioned earlier, all c's and P's are zero. This means that similarity solutions do not exist for those groups. 38

REFERENCES FOR CHAPTER 3 1. Ames, W. F., Nonlinear Partial Differential Equations in Engineering, Academic Press, 1965) pp. 141-144o 2. Birkhoff, Go, Hydrodynamics, Princeton University Press, 1950, Chapter 4. 3. Hansen. A. Go, Similarity Analyses of Boundary Value Problems in Engineering, Prentice-Hall, Inc,, 1964, Chapter 4. 4. Manohar, Ro, "Some Similarity Solutions of Partial Differential Equations of Boundary Layer Equations," Math, Res. Center, Univo of Wisc., Techo Rept. 375, 1963. 5. Morgan, AoJ.A., "Reduction by One of the Number of Independent Variables in Some Systems of Partial Differential Equations," Quart. Appl. Math, 3, 250-259 (1952). 39

CHAPTER 4 GENERALIZED GROUP-THEORETIC ANALYSIS 4.0 INTRODUCTION In the last chapter, the application of the simple group-theoretic method developed by Birkoff and Morganl was treated. Extension of this technique to multiparameter transformation by Manohar12 and Amesl was also discussed. As it turns out, these methods are probably the simplest to apply of the principal techniques. Whereas other methods may involve solutions of differential equations or other fairly complex mathematical manipulation, the grouptheoretic method require straightforward algebraic procedures. One drawback of course, is that boundary conditions are not taken into account until the analysis is largely completed. However, checking for satisfaction of boundary conditions is a simple matter. From the mathematical point of view, the method developed by Birkhoff2 and Morganl4 considered two particular groups of transformation, namely, the linear and the spiral groups of transformation. Application of this method of similarity analysis to a given partial differential equation therefore answers only the question as to whether similarity solutions exist for these two groups. Since there is no proof that these two groups are the only two possible for similarity solutions to exist for a given partial differential equation, it is still necessary to raise the question: Given a partial differential equation, what are all possible groups of transformation that make similarity solution possible? Are there groups other than the linear and spiral groups To answer questions of this kind, we shall develop in this chapter a systematic procedure in searching for all possible groups of transformation to a given partial differential equation. The procedure is based on Lie's theory of "infinitesimal continuous group of transformation" and his concept of contact transformation. Although both concepts were introduced by Lie in the latter part of the nineteenth century, there has been little application to the solution of nonlinear partial differential equations. The present technique therefore can be considered as extension and application of Lie's theories of infinitesimal and contact transformations to the similarity solutions of'boundary value problems. The technique itself, instead of the mathematical theories, is emphasized in the development to follow. For a complete treatment of the theory, the reader is referred to the books by Lieg-ll and other references.3515 In the first section, definitions and theorems are presented briefly. This is followed by three examples illustrating techniques. The diffusion equation 40

is treated first0 The goal of the analysis is to show how all the possible groups of transformation for a given partial differential equation (linear or nonlinear) may be foundo By requiring that the given differential equation be invariant under a "infinitesimal' transformation," the functional form of the so-called "characteristic function" can be determined. Once this is done, all possible groups of transformation can be obtained by the method developed in this chapter0 The second example shows the application of this method to problems containing an arbitrary function that is to be determined as part of the analysis0 The equation discussed is the steady, two-dimensional laminar boundary-layer equations o In the third example, the role of coordinate systems on similarity analyses is investigated by analyzing the Helmrholtz equation in general orthogonal curvilinear coordinates~ The problem encountered is the determination of the characteristic function mentioned earlier0 By requiring that the Helmholtz equation be invariant under an infinitesimal transformation3 a differential equation is obtained for the determination of the characteristic function, The equation contains the metric components (scaling factors) of a curvilinear coordinate system0 The equation determines whether or not similarity solutions exist for a given coordinate system; or, given a group, what are the conditions under which the scale factors, h s, should satisfy so that similarity solutions exist0 41l DEFINITIONS AND THEOREMS 4.o1l DEFINITIONS A group is said to be continuous if.'between any two operations of the group, a series of operations within the group can always'be found of which the effect of any operation in the series differs from the effect of its previous operation only infinitesimally. Thus. the transformation x? = alx + a2y (4.1) y' = blx + b2y is continuous if a], a2, bl, and. b2 are any real number0 For example, if alo, a20, blo, and b20 are values of al, a2, bl and b2, respectively, that carry (x1,yl) into (x{,y{), a sequence of values of these parameters can be found to affect the same result0 Each transformation can be made to differ from the previous one infinitesimally0

The concept of infinitesimal transformations comes as a natural consequence of the definition of a continuous transformation group. An infinitesimal transformation is one whose effects differ infinitesimally from the identical transformation. Any transformation of a finite continuous transformation group which contains the identical transformation can be obtained by infinite repetition of an infinitesimal transformation. Let the identical transformation be X1 = ~(x,y,ao) = x (4.2) Y1 = *(x,y,ao) = y where ao is a particular value of the general parameter a. Then the transformation X1 = ~(x,y,ao+65) (4.3) Y1 = *(Xyao+&E), where 6b is an infinitesimal quantity, defines an infinitesimal transformation in a broad sense. A slightly more restricted definition will shortly be given based on the concept of be being an "infinitesimal" i.e., a quantity such that higher orders of b6 than be itself may be neglected in a given operation. Equation (4.3) can be expanded in Taylor series which gives x1 = (xy,ao+s ) = (X,,yao) + + (4.4) ~ 1! a ao 2! (a ao and Y1 = *(x,y,a) = *(x,y,ao+6e) = *(xy,'a) + ( a)a a 2 + (4.5) Equations (4.4) and (4.5) can be written as 42

xi = x + bc + + o(4o6) and Y, = Y + qbe + o (4~7) where I = (xyyao~ ) (a)ao (4.8) and = (xy,ao) = (4.9) The expression ~(x,y,ao) and *(x,y,ao) in Eqs. (4o4) and (4-5) can be replaced, respectively, by x and y according to Eqo (4~2). Now since ao is a fixed value of the parameter, a, both i and 9r will be function of x and y onlyo Also, since be is infinitesimal, we shall neglect higher order of 5c and write: x, = x + ~(x,y)b5 (4010) and. Y -= y + (x,y)bc 0 (4o11) This tPransformation s:hal:l lbe defned as infinitesimal transformation0 It can'be shown that any one-parameter group, G, 9 contains only one unique infinitesimal transformation~ The definition can'be extended to a transformation containi:ng any number of variables ('but only one parameter)o Geometrically, an infinitesimal transformation transforms a point (x,y) to a neighboring point a distance i(x~tx)2+(zyzy)2 = J(12+22) 82 (4.12) in the direction 0 where cOS e = (4013)

sin O = -(414) 4.1.2 Representations of Infinitesimal Transformations Let f(x,y) be a generally analytic function of x and yo The effect of an infinitesimal transformation x, = ~(x,y,ao+b5e) (4.15) and Yi = 4(x,y,ao+5E) (4.16) on f(x,y) will be to produce the quantity f(xl,yl) which, upon expanding in Taylor series,5 becomes f(xl,yl) = f[((x,y,aO+&E), r(x,y,aO+5e)) f(x,y) +y ( y-+ ~ lyj 2T{ + E52 62f + 2g 62f + f2 _2f _ + o o o o 5~n nf nf n-1 z nf + I 1n 6 1xn Yn xn y + Tn af T (4.17) ayn Lie introduced the "symbol"* *It can readily be seen that Uf = (af (xlyi)/aa)ao and in particular, Ux = g and Uy = o 44

6f af Uf = + 1 a (4.18) It should be noted that this symbol itself is not a transformation. It "represents" a transformation, however; and is determined by ito It can be easily shown that the higher order term in the Taylor series expansion for f(xl,yl) can be written in terms of U as: U2f = 2 2f + 2 2f + 2 f (4.19) aX2 axay ay2 Unf = nf ( a)n- nf + +n nf 3xn5lby + oyo + -- n (4 2o) where Unf represents operating on f by the operator (x,y) a + T(xy) (4Q21) ax ay for n timeso Equation (4o17) then'becomes f(xl,yl) = f(x,y) + - Uf + U2f + (422) By using this expansion, it will be possible to deduce the actual equations of the transformation generated by a given infinitesimal transformation0 Some examples will make this point clear. Example 4.lo Consider the infinitesimal transformation represented by = x + Uf 6f + (4o23) X1 = x + ~(x,y)be and Yl = Y + ~(x,y)~ 45

where (x,y) = -y and r-(x,y) = x We now seek expression for xl and yj as one-parameter transformation groupso This can be done quite readily as follows: First take f = x, then using Eqo (4.23), Ux = -y U2x = -x U3x = y U4x = x (4o24) Substituting into Eqo (4o22), we then get be be2 5e3 5C4 x1 = x - 1 y- Y + y + x o 2To 3 4O = 2 bE4 _ y(8~23 + 5~ 0 2~ 4~ - ycose - + (45) 3' 50 x cos be - y sin be o (4~25) Similarly, if f = y, Eq. (4.22) then gives 46

Y1 = Y + x- 21 Y' 3'x+ 4.-T y + + o = x 5E + - ) 53 5 + y~l -E2 +5C 2 45~ =x sin b6 + y cos E. (4,26) The results correspond to a form of the rotation group: = x cos a - y sin a Yl = x sin a + y cos a for which ao = 0 in our analysisa Example 4.~2. Consider the infinitesimal transformation represented by Uff =f Uf clx c + c2yay X1 = x + ~(x,y)bS and Yz = Y+ +(x,y)5~ where (x,y) = clx and r(x,y) = c2y we proceed as before to get expressions for xL and Y1: For f = x, we get 47

Uf = clx 2 2 (4.27) U f = cix (4.27) U3f = c x Therefore, Eq. (4.22) gives C1= c~'2 + _3E3_ = x(l + + Xc-l + 0 2 ~0 = Ce o (4.28) Similarly, for f = y, we get Y1 = ye C2E (4.29) If we let ao = 0 in our analysis and define A = ea (Note, ea~+5 = e ) then Eqs. (4.28) and (4.29) become X1 = Alx, yl = AC2y (430) This is seen to be the linear group. Example 4o5. Consider the infinitesimal transformation represented by Uf = cl a + 2:Y- a. (4,31) 48

We may easily obtain: xl = x + clc (4.32) and Y1 = Y + c2 5 10 2'7 yec285 (4.33) This is seen to be a case of the spiral group: x1 = x + cla Y.L = ye 2a 4o1o3 Functions Invariant Under a Given Group A function f(x,y) is said to be an invariant of a group if it is unaltered by the transformations of the group; therefore, for the function to be invariant, we have f(xLy1) = f(x,y) (4[34) It follows that the function is necessarily invariant under an infinitesimal transformationo From Eqo (4o22), we have 5C 5c2 f(xl,yl) = f(x,y) + Uf + Uf + + (4,35) But if f(x,y) is to be invariant, the following condition must be met: Uf = U2f = U3f =,,o =o 0, (4-36) However, since the condition Uf = O (4,37) 49

implies U2f = U3f = 0.. =, (4.38) Eq. (4K37) is the sufficient condition that Eq. (4.34) is true for all values of x, y and 5c. Hence we may state the following theorem. Theorem 4,l. The necessary and sufficient condition that f(x,y) be invariant under the group represented by Uf is Uf = 0. In other words, the necessary and sufficient condition that f(x,y) be invariant under a one-parameter group is that it be left unaltered by the infinitesimal transformation of the group. To determine the invariant function, f, it is sufficient to solve the equation Uf = + f 0 (4.39) ax y 4 The equation can be solved using the method of Lagrange4in the theory of linear partial differential equations. The procedure in the case of three variables, x, y, and f, can be written as followsi To find the general solution of P1 E + P2 = R (44) where P,, P2 and R being functions of x, y, and f, solve dx dy = f (4.41) P1 P2 R If the general solution of this system is ul = cl and u2 = c2 (4.42) *The system, Eqo (4o41), has only two independent solutions. 5o

then.1(U1,U2) = 0 (.43) or Ul = 2 (U2), (4.44) where g1 and 02 are arbitrary functions of ul, u2, and u2 respectively, will be the general solution of Eqo (4o41)o As an illustration of this method., consider the equation xf f + f = x o (4.45) Based on the theorem, we consider the system dx _ dy _ f (4.46) xf yf xy This system of equations has two independent solutions xy _ f2 - c and x - = C2 o Therefore, the general solution to Eqo (4~45) is of either of the following forms: ~1(xy-f2, Y ) = 0 (4047a) or x = W2(xy-f2) (4.47b) It is seen that application of this theorem to the solution of Eq0 (4K39) for the invariant function, f, as a function of x and y forms a special case 51

as, R = 0. Equation (4.40) becomes P f + P2 6 = 0 (4.48) clearly, one solution of this equation is f = constant and we may write ul(x,y,f) = f = constant (4.49) The general solution of Eq. (4.48) is therefore (cf, Eq. (4.44)) u2 = P3(f) = ~3 (constant) = constant (4*50) where u2 is a solution to dX - dy (4.51) P1 P2 We therefore get the following theorem: Theorem 4.2. To find the general solution of eaf -- 1 a = 0 (4.52) where 5 and ~ being functions of x and y, solve dx = dy (4.53) Let solution of this equation be expressed as 2(x,y) = constant. (4.54) 52

This function is the invariant function for the infinitesimal transformation* represented by af 6af Uf = (+ K 55) Since Eqg (4-53) has only one solution depending upon a simple arbitrary constant, it follows that a one-parameter group in two variables, x, y, has one and only one independent invariant. Example 4~4~ Consider the rotation group represented by y6f 6f Uf = - y + (4o56) ax 6y where (x,y) = -y and. T(x,y) = x.Solution to the equation dx dy -y x gives x + y2 = constant0 (4 57) This is the invariant function of the rotation group, Example 4-5~ Consider the linear group Uf = cx f (4+ c58) ax ay -where c = clx and,g(x,y) = c2yo Solution to *The function f is an invariant of the one-parameter group obtained from the infinitesimal transformation~ 53

dx dy c1X c2y gives = constant (4.59) x02 C1 which is the invariant function of this group. Example 4.6. Consider the spiral group representation Uf = cl aaf *cy 6f (4.60) where = c1 and C = c2y Solution to dx dy Cz c2y gives cY = constant (4.61) eCt which is the invariant function of this group. 4.1.4 Extension of Two-Variable Analysis to n Variables In the case of n variables, all the theories corresponding to two variables can be generalized following the same pattern. For example, if a function of n variables f(xl,...,xn) is invariant under the infinitesimal transformation x' = Xi +.i(Xl, e..Xn) G (i=l,C.,n) (4.62) 54

then a necessary and sufficient condition is again Uf = o (4.63) which, in the case of n variables, takes the form Sm(Xlo,Xn) + xl. + Sn(xi,.o.,xn) = 0 (4.64) Following the same reasoning as in two-dimensional case, the invariant functions can be obtained by integrating dx1 dx2 dxs dxn dx1 dx2 dx3 = o. n (4.65) Since there are (n-l) independent solutions to Eq. (4065) a one-parameter group in n variables has (n-l) independent invariantso The invariant functions are therefore Q2m(xl,.oo,Xn) = cm, (m=lo,n-1) (4.66) and are the solutions to the system of equations given by Eq. (4.65). 4.1.5 Relationships Satisfied by Differential Equations Admitting a Given Group of Infinitesimal Transformations In this article a basic theorem on the determination of relationships satisfied by functions admitting a given group of infinitesimal transformation will be given. This theorem is comparable to the theorems of Morgan discussed in Chapter 3. It is this theorem which lays the groundwork for the reduction of the number of variables in a partial differential equations. Consider now a function F F= ky O Yn k-) (4 67) the arguments of which, assumed p in number, contain derivatives of yj up to order ko Such a function is known as a differential form of the k-th order in 55

m independent variables. Designate the arguments by zl,...,Z p, e.g., Z1 = X1 Z2 = X2 Zp-1 kyn zp a (x)k (4.67a) Thus, Eq. (4.67) can be written in a simpler form as F = F(zl,...,zp) (4.68) The function F(zl,...,zp) is said to admit of a given group represented by Uf = l(zi,...,Zp) afZ + + p(Zl...P) a (4.69) if it is invariant under this group of transformation. Therefore, the function, F, admits of a group if UF = 0 (4.70) or, F1 + + 2- +' + e (4P71) It was shown in the preceding article that there are (p-l) functionally independent solutions, or invariants, to this equation. If they are denoted by rm = m(zl,...,Zp) = constant, m=l,..,p-l. (4.72)

Equation (4.71) must be satisfied, i.e., -1 - +,,, +p = o(473) az, azp Now, if a change of variable is made in the function F, given by Eq. (4.68). from (zl,...,zp ) to (fl,-l.,pl-,zp), we then get F(zl,o0.,zp) = t(1,.o,TBp-i,Zjp). (4.74) The condition given in Eq. (4.71) will still have to be satisfied, Thus, UF = P t -2 + oo + az a+Z2 azp = 1 a + 2 +.0 + 0- O. (4075) UZ = Z2 l zp Since 4 is a function of T1,~..,9p-jzp, the chain rule of differentiation may be used to get UF = 1 + + a6 ap -1 + 6* 6Zp 1 6z, alp-1n azl aZp azl + t2 (a ~+ 0 + t p + a_ t 6'rl, aZ2 a6p.1 az1 azp aZ2 (at E + a rlp-1 +1 a1 )....... 1 zaz azn azp + eo5 + 6* _1 0,0qp-1 + Zp 6Zp + tn 6 8 \P-1 Zp 0=. (4.76) 57

Since 1,. *,pp- 1 are invariant functions, all the brackets equal to zero [cf. Eqs. (4.72) and (473) ], the following important conclusion then results: at ( 1,e., p _-, Zp ) = 0. (4.77) azp This means' is independent of zpo Equation (4.74) can then be written as F(zl,. o,Zp) = (l,. o.,Tpp- 1) = 0. (4.78) Therefore, if a differential equation F(zl, o..,Zp) = 0 is invariant under the infinitesimal transformation, it must be expressible in terms of the (p-l) functionally independent solutions of the partial differential equations Uf = 00 Like Morgan's theorem given in Chapter 3, this result is the foundation upon which the technique given later in this chapter is basedo 4.1.6 The Extended Group Concept Consider the one-parameter group of transformation xl = ~(xy,a); Yi = *(x,y,a), (4.79) Suppose y is regarded as a function of x. Then if the differential coefficent p(= dy/dx) be considered as a third variable, then under this group of transformations it will be transformed to P1 where P. =.. = X(x,y,p,a) o (4.80) dxj ax ay It can be easily shown that the general transformation 58

xi = ~(x,y,a); Y1 = *(x,y,a); P1 = X(x,y,pa) (4.81) form a group. This group is known as the extended group of the group given in Eq. (4.79). Now for a change be in the parameter a, we may express Eqs. (4.79) as X = X + be ~(Xy) a+ + +.. (4.82) 1! 2! ax \y The transformed coefficient Pi will be dy + dx + l ady ) + 00 PiP1: P, = P +' x + + (4,86) Thereafore, the extended infinitesimal transformation given by X, X + Y.+ = Y + oboo T (4.85) p = p + W) UF= - + -- (y +,,,, (4.86) iwheree ax 5 ay 59

Extension of this concept to higher order derivatives can be made by the same reasoning. Example 1. For the rotation group represented by Uf = + xaf (4.88) 6x 6y it can easily be shown that the extended group of infinitesimal transformation is xl = x + S(xy)5c, Y1 = Y+ +(x,y)~C, Pi = P + (x,y,p)5e (4.89) where 5 = -y = X (4.90o) and, from Eq. (4.87), O = + (p p -p2 = +p2. (4.91) ax 6y ax/ ay The symbol of this extended group of transformation is therefore 6f 6f 1 f Uf = xy x + (l+p2) - o (4.92) ax ay ap Example 2. For the linear group, the extended group of transformation can be represented by Uf = cxf + Y + (C2rC,)P (4.93) aUx ay ap 60

Example 3. For the spiral group, the extended group of transformation can be represented'by Uf = c af +c2yaf +c2paf 0 (94) 4.1.7 Contact Transformations The theories developed up to this point are not complete~ If a given partial differential equation F = F(z,...Ozp) = 0 (4~95) is invariant'under the infinitesimal transformation represented by Uf, then = 6F 6F UF - t1 - + +. -P ap aF = 0 (4.96) azi izp and the differential equation can'be expressed in terms of p-l invariant functions. These invariant functions are solved'by using the system of equations., dzl = _ dzp....p (4.97) as discussed in article (4o14). In. case a particular group of transformations is given, as in examples in the previous sections, the function.1 oo9op in Eqo (4.97) are known. The invariant functions can then be found by solving Eq. (4.97). These functions are the similarity variableso However, the results only give similarity solutions for this particular group, What is more important, is to derive the transformation groups and not specifying them0 We s'ha ll now lay the groundwork for this type of analysis. In the present section, Lie's theories of "contact transformations " are introduced0 These theories make it possible to express the transformation functions, 1, ~~~,ip in terms of a single function, known as the "characteristic function." Basically, Lie's theories of contact transformation deals with the transformation of a differential equat:lon in a general, highly abstract, way. The abstractness and the complexity of the theories prevent any extensive discussion. Here, we merely state some of Liens theorems without proofo This is sufficient for purposes of application. 61

Lie9 defined a "lineal element" of the plane to be the ensemble of a point and a straight line passing through that point. In two-dimensional Cartesian coordinates, the coordinates of the lineal element consist of the coordinates, x and y, and the slope p: of the line. Thus any transformation in x, y, and p may be considered as a transformation of the lineal element. Consider now a curve defined by y = y(x) having a slope dy/dx = p at the point (x,y). Let the variables x and y be transformed by x, = X(x,y) (4,98) Y1 = Y(x,y) The curve is transformed into a curve having a slope Pi at (xl,yl) defined by aY aY p -+ -p ax 6y Equation (4.99) can be considered to be a transformation that along with Eq. (4.98) transforms an element (x,y,p) into (xl,y1,p1). Lie called this transformation an "extended transformations " A property of an extended transformation is that it transforms the differential equation dy - pdx = 0 (4.o100o) into the differential equation dyl - pldxl = 0 (4.101) *The differential equation (4.100) is transformed into the differential equation (40101) under the assumption that the Jacobian of the transformation (4.98) is nonvanishing. 62

We may say that every family of lineal elements (x,y,p) satisfying (4100) is transformed by an extended transformation into a lineal element (xl,yl,pl) satisfying (4o101o) For three-dimensional space, x, y, z, an "element" is defined by five quantities, namely, x, y, z, p, and q where p and q, like y' in two-dimensional space, are the slopes of a straight line passing through that point. The family of surfaces containing a given element would be defined by a single relation between the five quantities, that is, by a partial differential equationo Lie therefore defines a "contact transformation" on a transformation of the lineal elements of the plane which leaves the Pfaffiian equation dy - pdx =0 (40102) invariant; that is, a transformation x, = X(x,y,p) Y: = Y(x,y,p) (4103) Pi = P(x,y,p) is a contact transformation if the Pfaffiian equation, Eq. (4o102), is transformed. to dyl - pldxl = 0 (4104) after the transformation~ The above definition of contact transformation i:s extended by Lie8 as follows When ZXL,ooo,Xn,Pi,ooo,Pn are 2n+l independent functions of the 2n+1 independent quantities z,xl, oO,Xn,P1, ~oPn such that the relation dZ - PidXi = p(dz-pidxi) * (4o105) *PidXi =i — PidXi, ioeo, the repetition of indices will indicate summation over all values of the index~ 63

(where p does not vanish) is identically satisfied, then the transformation defined by Z = Z(z,x1, o',xnt,P l,.,Pn) Xn = Xn(z,xl,~ *,xn,pl,*',Pn) Pn = Pn(znx1. e oXnP1,.* *,n) (4.106) is called a contact transformation. The contact transformation, Eq. (4.106) will transform a partial differential equation in z,xl.oo.,Xnpl,..,opn into one in ZXi,...oXnyPi.. Pn and also the solution of the first partial differential equation into the solution of secondo In the next article, the property of a contact transformation, Eq. (4.105) will be used to derive expressions for the transformation functions in terms of characteristic functions. 41,o8 Contact Transformation in Terms of the Characteristic Function First, let us consider the infinitesimal transformation z = Z+bEab Xi = xi+ cEi, Pi = Pi +5 E= i (4.107) where,, Si and j(i are functions of z,x,o...,Xn,pl,P..,pn and E is a quantity so small that its square and higher powers may be neglected. We now wish to find the functions %, i, ci in terms of a single function of z, xi, Pi, known as the characteristic function. From the definition of a contact transformation dZ - PidXi = p(dz-pidxi) (4.108) we get 64

az d + axi p+ +- a dpi - Pi - dz + dr + a-i dpd Oz 6Xj 6pi i z ~X-r r 5Pr = p(dz-pidxi) a (4o109) Equating coefficients of dz, dxi, and dpi on both sides, we get az Pi =a P az az i axi = o 6Pr 6apr az axi P -P~ axr r PPr (r=l, D D 0 9:n) o (4 1lo) For the infinitesimal transformation defined in Eqo (4o107)., Eqso (4-110) become 6: p P. i,r aZ 1. az Pi =Pr aPr - ~ 6 Pi pr = - aPr (4 111) 6xr ixr where the relation p = l+ca is used (c, arbitrary)~ Now, Pi is independent of xr, and zo Therefore, if we define a characteristic function W such that w = Pii o (4.112) Equations (4.111) will yield 65

aw -pr =bir i = r x =-aPr + Jr =P (4.113) xr az We then get 6w api r and a aw (4.115) aw Next, consider the infinitesimal transformation Zi = zi + mi(zv)x p V) Xk = Xk+BE;Gk(Zvx 1,P ~) n where mi, 0k and itk are functions of xloo.,xm;zl,...,zn; and pl,P2,o. Pm (= azn/6xm) and c is a quantity so small that its square and higher powers may be neglected. We now want to express the functions mi, QC, and k (i=lo.,n;k=l,,..,m) in terms of Wi(i=l,,.,n), the characteristic functions. From the definition of contact transformation we have dZi - PkdXk = Piv(dzivP1d) (4.117) *By adding to both sides of the second equation in Eq. (4111) the term api apr S66

we get azi 6Zi Zi V az + dx,, + ) pXk dzv + dx + Xk dpp; Civ (dz-pidxQ) o (4 118) Equating. coefficients of dzv, dxv, and dp, we get azi p aXk aZipi Pk,p pi~ dZ i aXk = v apv k iap, For the infinitesimal transformation given in Eq, (4,116), Eqs. (4.119) become a (mi-Pkk) = Riv azv a (mip k ) = -PvRi~ + n~ ax, a (mi-Pak) = Sliv (4,120) apv, where the relation pi. = -iv+,Ri, is used, Let us now define the characteristic function Wi as Wi = Pkk - mi (4,121) Equations (42120) then become 67

6wi aZV =p R iv + w = c%1iv (4.122) apc, From these equations we then get 6wi 1 a- V ~ u 3 *pV (4.123) Having expressed mi, cx, and it1 in terms of the characteristic functions Wi, we now prove a very important property of mi and ab. For n=l, Eqs. (4.123) reduce to Eqs. (4.114) which are expressed in terms of a single characteristic function. For n > 1, however, there is a very important property involving mi and qc.15 Now, differentiating the third equation in Eq. (4.120) with respect to P, we get ~jj~ ~P (mi-pik - ~p i vO) or 1V (4.124) p v (mi-Pik~k) = -6iv (4.124) The third equation in Eq. (4.120) can be written as p (mi-p czk) = -6ip * (4.125) 68

Differentiating Eqo (4 125) with respect to pV we get v p (mi-Pkk) = i (4126) apapp( a1p Since the left side of Eqs, (4o124) and (4o126) are the same, we can equate their right sides and get iv 6pp ip (4127) If -we set i=v7p, EqO (4,127) then gives 64 = 0 (4.128) apP This means C,; is independent of pPo Next, the third equation in Eq, (4:120.) gives 6mi. ami kPk ap = apk1 =k5k- - = a -, = 0o (40129) This equation means mi is independent of pvo As a result, the infinitesimal transformation defined in Eq. (40116) should be modified to: Zi = zi +cEmi(zv,x4) Xk = Xk + 5Ck(zv,x ) Pk pk + b k(ZinX sP + (4.130) - P p i(z,x pV) where mi, Ck and are defined in Eqso (4o 123)o Equations (4o12) also suggest that the dependence of Wi on p" is linear-a very far-reaching and important conclusion which will be made use of later0 69

4.1.9 Contact Transformation of Higher Orders Let us now consider the extended infinitesimal contact transformation defined by Zi = zi+gbemi(zv,xl,pV) Xk = Xk+'6Ck(ZV,x~,pV) k = Pk k pi i i jk Pjk + ~jk(ZvXI'p'P ps) =~ jk e + jke(Zv,'pPs'P st (4)st where ak and mi are independent of pv for i > 1, and 6zi p =,a etc. (4.132) jk 6xj)xk We now want to express ri and -iA as functions of the characteristic function jk jk2 i Wil The functions mi, Ck, and j k are expressed in terms of Wi and given in Eqs. (4.123). By definition, dP PjkdXj. (4.133) Substituting the infinitesimal contact transformation from Eq. (4.131) into Eq. (4.i33). we get dk = Pjkdua j +,kdxj (4.134) Subtracting both sides of Eq. (4.134) by the quantity cjpXijkdxs, we then get d(k-Pjk j ) = jkdXT - ajsjkdXs. (4.135) 70

The second term on the right side of Eq, (40135) should be dropped since in iTk derivatives of order higher than the second are omitted. We then get Kjik = L ( pk-Pk ) (4136)'k ~ dxj or, in terms of aj and Trk iPj k Kxj +z s Ep_4V j p raj + ps + 2 pV jk Iax; a Ps j apv 1 43 A 6xj ks (4o137) To express -Tk i:n terms of the characteristic functions Wi, we have to put expressions for n and adj from Eqso (4K123) into Eq, (4,137)o Similarly, for the third-order function Ti we get jk w get I d a i i ) (4138) jk dx jkt t or, - jki k - PV e v Pz ~jkL + P Pa P6c P bc P_ +(-a-: a P + - P + bc (4 139) In order to express T'jkR in terms of the characteristic functions, we have to express T(k and ae in terms of Wi, as in Eqso (4.13'7) and (4.123), and substitute into Eqo (4o139)o The same principle can be extended to higher order, e ogo jk m dxr (4,140) 71

or jkm -+ jki pV + jk + Ai~m = a +a Pm Z av 1m +Pac PPcm pd Pbcdm jk f + ~f p f pa + a Pdpa Pac tcm Pcd +Xm + ~p C Pm c (P bcm Pca d Pbcd (4141) 4,2 SIMILARITY ANALYSIS OF DIFFUSION EQUATION We shall now look at the diffusion equation from another point of view, namely, the searching for all possible groups of transformation that will reduce the diffusion equation to an ordinary differential equation. In applying such a technique to a given differential equation, it may turn out that for some or all of the groups other than the linear and spiral groups, the boundary condition cannot be transformed although the partial differential equation can'be transformed into an ordinary differential equation. For such cases, we are at least assured that the groups of transformations that remain are the groups possible for the given boundary value problems. A similarity analysis of the diffusion equation from this point of view is apparently not covered in the literature. The one-dimensional form of the diffusion equation in rectangular coordinate is chosen because of its simplicity. Extension of analyses to equations expressed in other coordinates can readily be made. 42.ol Analysis Consider the diffusion equation au v 62u= O (4.142) at ay2 on which an infinitesimal transformation is to be made on the dependent and independent variables and derivatives of the dependent variable with respect to the independent variable. The infinitesimal transformation is 72

t' = t + 65~(t,y,u,p,q) y' = y + 5be(t,y,u,p,q) u' = u + be5(t,y,u,p,q) p = p + bEcl(t,y,u,p,q) q' = q + 6cT2(tyuypq) P.2 = P22 + 85C22(t,y,upqplP,P 12,P22) (4,143) where, in terms of the characteristic function, W, aw = 6W op 6w -tj = aw aw at + P au - (22 62 W q2 22W +e 2 +2p12a + ay2 ayau au2 ayap w up + 2P22 (-W+ q aW pl2 + 2pl222 * P2 62W P + P22 -- + P22 oThe characteristic function, W, is a function of t, y, u, p, and qo We note that 8u 8u _ ~2u ~u P= t' 4 = P22 - P12 = (4o145) 73

It is shown in article 4.1.3 that the necessary and sufficient condition that a partial differential equation F(t,y,u,p,q,pl2,p22) = 0 invariant under the group of transformation represented by Uf is UF = O which for the diffusion equation, is U(p-vp22) = 0 (4.146) or, expanding the expression by employing the operator U: e( ) 6( ) 6 ( ) + ( ) 6( ) 5;t + +y ~u +I1 ap + 8q2 + Tj( T) + t22 ) + t2 a( ) = 0 (4.147) + i P22 6Pi2 where the parenthesis represents the differential equation P - VP22 = 0 (4.148) Carrying out the operation in Eq. (4.147) yields: rl - vT22 = 0. (4.149) Upon substituting expressions from Eq. (4.144) into Eq. (4.149) yields aw aw a2 — w + vq2w - a- P + v 62W+ 2vq a + v a2 )t 6u 6y2 6yau 6u2 + 2p2W v + 2p2q 2W v + 2p 2 + p ayap P uap ayaq auaq ___2 6aW p2 62W 6W + vpa2 -W + 2pl2p + + p = 0 (4.150) ap2 apaq v aq au Equation (4.150) is seen to be a linear partial differential equation in W(t,y,u,p,q). Since W is not a function of P12, the coefficients of the terms involving P12 and p22 should be zero. We then get 74

2W= o (4o151) ap2 a2w a2W p a2W 62W + 62W + p 62W = 0. (4o152) aya~p 6up v apaq Equation (4.151) indicates that W is a linear function of p. Thus we can write W = W1(t,yut,q) + pW2(t,y,u,q) o (4o153) Substituting this form of W into Eqo (4,1l2), we get aW2, aw2 p aW2 -0 4154) Since W2 is not a function of p, the coefficient of p in Eqo (4o154) must'be zero, We therefore obtain two equations,'namely, W2 = 4155 +W2 q 2 o (4.156) ao Equation (4o155) indicates that W2 is not a function of q, ioe,, W2 = W2(ty,u), and, so Eqo (4.i56) can'be'broken into the two equations, NW2 = 0 (4,157a) ay a nd aW2 = 0 (4.157b) au This means W2 is independent of'both y and u and, as a result, W2 = W2(t) (4.158) and the characteristic func'tion now takes the form 75

W = Wl(t,y,u,q) + pW2(t). (4.159) Putting this form of W into Eq. (4.150), we get aW, p 6W2 2W2W + 2vq 2 W 2W 6t 6t 6Y2 6yau q u2 + 2pa2w+ pq + 2 W2W = p 0 (4.160) ayaq auaq V aq2 Since both W1 and W2 are independent of p, Eq, (4o160) can be separated into three equations, corresponding to the coefficients of pO, pi, and p2o We then ge t p 0 _awl 2 _ q + v2q + 0 (4.161) at ay ayau U 1 a6' + 2 22W1 + q a2W1 = o (4.162) pt ay2 q _u-q p2: aW =0 o (4.163) From Eqo (4.163), W1 is linearly dependent on q, therefore, it becomes Wm = Wll(t,y,u) + W12(ty,u)q o (4.164) Putting this form of W1 into Eqo (4.162), we get dWP + 2 wW. = + q. (4.6 o ) dt ay aU Both W2 and W12 are independent of q, therefore, Eq. (4.165) becomes dW- + 2 = (4.166) dt ay:WT = 0. (4.167) au 76

From Eq. (4.167), W12 is independent of u. Also, since W2 is a function of t only, Eqo (4.166) shows that W12 depends linearly on y, i.e., W12 = W121(t) + W122(t)y o (4o168) Equation (4.162) then becomes dW2 + 2W122 = o. (4.169) dt We will make use of this equation later, The characteristic function, W, now becomes W = Wll(t,y,u) + tW121(t)+W122(t)y}q + W2(t)P o (4.170) Putting Eq. (4.164) into Eq. (4.161), we get -W11 (121 t dW122 3 q + v alWI + 2vq a 2Wl t vq _2 a O W _ d + + 2 v 2q +vq - 0 at dt dt ay2 ayau au (4.171) Since Wj1) W121 and WLa22 are independent of q, terms with different powers of q are grouped and their coefficients are put equal to zero. Three equations are obtained: 0q _ N + v =2W10 o (4,172) - 6t + y2 ql. _W dW322 y + 2v - AW = 0 (4,173) dt dt ay3u Eqq ~4W17 = 0h (4t174) au2 Equation (4,174) shows that W11 is linearly dependent on Uo Therefore, Wmm = W111(t,y) + W112(t,y)u o (4o175) 7'7

Equation (40173) then gives dWl2I dWd22 y + 2v W_12 = 0 (4.176) dt dt ay Therefore, W112 can be written as W112 W1121(t) + W1122(t)y + W1123(t)y2 o (4.177) Equation (4.176), then becomes dW dW + 2W22 + 4vyW1l23 = 0 W (4.178) dt dit Since all the W's in Eqo (4o178) are independent of y, we get dWt + 2vW1ll2 = 0 (4.179) dt dWi22 + 4vWl323 = 0 (4. 180) dt Putting W11 in-to Eqo (4.1172), we get W-J1 m + V 2 r at ay2 fdaW12I + dW-122 y a dWa. 23 y]u L dt dt dt + 2vW1123u = 0 o (4o181) Equation (4.181) can be separated lnto~ uo: _ -w + v a2-l= 0 (4.182) at ay2 78

ul: - dW2 + 2vW1123 = (4o183) dt dt dW1123 = 0 (4.185) dt From Eqs. (4Q184) and (4.185), WIL22 = C 1 (41 86a) and W1123 = C2 o (4.186b) From Eq. (4.183), W1121 = 2VC2t + C3 o (4.187) From Eqs. (4.179) and (4.180), W121 = 2vClt + c4 (4,188) W122 = 4vc2t + c5 o (4,189) From Eq. (4,169), W2 = 4vc2t2+ 2t + 2c 6 (4o190) The final form of the characteristic function is therefore 79

W(ty,u,p,q) = W1l1(t,y) + (2vc2t+c3+c 1y+c2y2 Ju + [2vc t+c4+ (4vc2t+c5 )y q *+ [4vc2t2+2c5t+c6 ) (4.191) where W11l(ty) is any function satisfying Eqo (4o182), ioeo, 68awtll a; I=- 0 (4.192) The characteristic function, W, given in Eq, (4,191), will now be used to determine the absolute invariants, 4~2,2 Absolute Invariants anrid The Transformed Equations With the characteristic function, W, obtained as in Eq. (4o191), we now make use of the general theory to find the absolute invariants, From Eqo (4o65), the following relations are obtained: dt dy du -i~~ 50~~~ C(4193) where the transformation functions ~,., and ~ can be obtained by putting into Eqo (4,1444) the characteristic function, W, given by Eq. (4.191). Equation (4 193) then'becomes dt dy 4vc2t +2cst+ce 2vclt+c4+ (4vc2t+c 5 )y di~~~u; o (4o194) -Wlll (t9 y) (2VC2t+C3+c ly+2C2 y2 )u The number of possible groups are large, due to the fact that all c's are arbitrary and Wlll(ty) is a:n arbitrary function satisfying Eqo (4o182)o Therefore, we investigate a few special cases of the parameters. Other groups can be obtained in a similar manner. 80

Case 1. Wlll(t,y) = cl = c2 = c4 = c6 = 0 Equation (4.194) becomes dt dy du 2C5t C5y -C3U The two independent solutions to Eq. (4.195) are Y- = constant and -u = constant; = C3 According to the theories in the preceding articles, the diffusion equation can be expressed in terms of these two invariants, ioe., T Y (4,196a) and f(,) =. (41 96b) The diffusion equation is then transformed into an ordinary differential equation vf" = af - rlf7 2 T:he transformation are seen to'be the linear group of transformations. Case 2, W111(ty) = C1 = c2 = c4 = c5 = 0 Equation (4.194) then becomes dt = du (4,197) c6 C3U 81

and y = constant. (4.198) Following the same arguments as in Case 1, we get the absolute invariants as = y and f(r) = u_ ( C) and the diffusion equation is transformed to vf" - Pf = 0 The transformations are seen to'be the spiral group. Case 3. Wlll(t,y) = c2 = C3 = c4 = = C6 = 0 Equation (4o 194) becomes dy du (4.199) 2vclt -clyu and t = constant, (4.200) The absolute invariants are = t and f(O) = u and the transformed equation is 2nfY + f = 0 82

Case 4o Wlll(t,y) = c1 = c2 = c5 = 0 Equation (4.194) becomes dt _dy du ( 201) C6 C4 -C3U The absolute invariants are C3. = y C4 t and f =uet6 c6 The diffusion equation becomes vf" + 4 f + c f = 0 C6 C6 The above four cases are examples of cases where the solution of Eq0 (4i194) is straightforward, ioe.o the two independent solutions can be solved by simple pairing of equations, The following cases are those in which the two solutions have to be solved in a sequence of steps. Case 5o Wlll(tY) = c1 = C3 = C4 = C5 = C6 = 0 Equation (4,194) now'becomes dt dy du4 4vt2 - 4vty -(2vt+y2)u The first of the two equations gives the solution Y l= (4o203) t kl. Next, replacing t in the last two terms of Eq. (4.202) by kly [based on Eqo (4,203)], we get (2vk.y+y2) dy = du (4o204 ) 4vkzy2 u 83

The solution to Eq. (4.204) is uyl/2 eY/4Vkl = constant or, using Eq. (4.203) again, uy1/2 ey2/4vt = k2 (4.204a) According to the applicable theorems, the absolute invariants are, from Eqs. (4.203) and (4.204a) = and f(rQ) = uyl/2 ey2/4vt The diffusion equation is then transformed into 2f" _ ~f' + 3 f Case 6o Wl1(t,y) = c2 = C3 = C4 = C5 = 0 Equation (4.194) becomes dt = dy = du c6 2vclt -clyu The first two terms give y- tvc t2 k, (4.205) C6 where k1 is the constant of integration. Combining the first and the third term and making use of Eq. (4.205), we get -1 (klt + e t3) ue6 = k2 (4.206) where k2 is the constant of integration, Using Eq. (4.205), we get

6 (yt - t) ue = k2. (4.207) Equations (4.205) and (4.207) give the invariants, I= y- VCe t2 C6 and cz (yt - vc 1t3 f(P) = ueC ( t3) The diffusion equation is transformed to f" +.c qf = 0 c6V Case 7. Wll(t,y) = c2 = c3 = c4 = c6 = 0 Equation (4.194) becomes dt dyy du 2c5t 2vct+cs5y -clyu (4.208) By following the same steps as in the two previous cases, the invariants are found to be Y 2'vcl iJ 7tJ C5 and c L y Vt f(rT) = ue5 The diffusion equation is then transformed to vf" + 1- f' = 0 o 85

In all the above cases, Wlll(t,y) was taken to be zero, This is not necessary as we shall show in the following two cases, Case 80 Wlll(t,y) = valt + 1 a,2 C = C2 3 = c4 = cC = C6 It can be shown easily that this functional form of Wlll(ty) satisfies Eq. (4.192). For this case, Eqo (4.194) becomes dt dy du 2cst = 1 2c5t cy~-valt - 2 al Using the same method as in previous cases, we find y and f(hr) = u + al(2vt+y2) 4c5 The diffusion equation is transformed to vf" + 1 ~f' = 0. Case _92 Wa1o(tY ) = aO+valt + - aly C1 = C2 = C3 = c4 = C6 = 0 The only change made in this case is the addition of a constant term, ao, to Wlll(tsy)o The invariants in this case are 86

and f( ) = u + aL.(2vt+ya) o nt 4c5 2c5 The diffusion equation becomes vf " + 1 -f = 2 2c5 It is seen that as a result of the additional constant term, ao, one more term is added to f () and the transformed equation, as compared with Case 8. 4.3 SIMILARITY ANALYSIS OF STEADY, TWO -DIMENSIONAL, LAMINAR, BOUNDARY-LAYER EQUATIONS This article is the second application of the theories given in article 41,o The example treated in the previous article concerns the diffusion equationo The present article differs from that example in that an unknown function is involved in the differential equation which has to be determined. This factor introduces complexity in the process of searching for the possible groups of transformation. The method developed in this article can be used for similar problems. We shall see that the boundary-layer equation may be transformed from a nonlinear partial differential equation to nonlinear ordinary differential equation by groups of transformation other than the linear and spiral groups, However, in transforming usual boundary conditions, the linear and the spiral groups are the only two groups possible. 4.3.1 Infinitesimal Transformation and the Characteristic Function The laminar, incompressible boundary-layer equation, in terms of the stream function t, can'be written as at a2 =V a2 ue duee + a (4,209) 6y axay ax ay2 dx ay3 With the boundary conditions y =. x =y y = 0' at = ue(x) 8y 87

where all the quantities are in dimensionless form by the transformation U UoL x - ~ = WL ee L' ~ R Ls - us=e Re = U X7L Ly, Th 4 where x', y', Jr', and u' refer to dimensional quantities and L and Uo are reference length and velocity. Equation (4~209) can be put in a shorthand form as P2P12 - PJP22 = ~ + P222 (4.210) where P.1 = ~0~ ~e6 etc,, Pi x P2 = a P12 = etc, and = uedue o(4211) dx Now, let us denote the following differential expression by F: F = P222 + ~ PaPi2 + PiP22 o (4.212) The equation F = 0 will be invariant under the infinitesimal transformation 88

xi = X + bcl(x,y,tplp2) Y' = Y+ECQa2(x, y,p,p1,P2 ) r' = + tbem(x,y,JfP1Y,p2) P1 = P1 + bec1 (x,Y,fP1,P2) P~ = P2 +b8EC2 (x,y,,P1,jp2) Pl2 P12 + En12 (XY,I,P1,P2,P1,P12,P22) P22 = P22 + 6E:22(x,Y,yf,Pl,P2,PllPl2,P22) P~22 = P222 + ECT222(XYPl,P2P,Pll12,P22,P11, o o,P222) (4213 ) if UF = 0 (4,214) or, in expanded form, aF aF aF aF aF aF aF C,5; +a y+M7- Tp + Ct2 5y + m + + Pj + c imn (p 2o (4o215) Putting F from Eqo (4.212) into Eqo (4o215) we get ~2P12 - P2T12 + T11P22 + PJlt22 +'a.+ t222 = 0 o (4.216) The next step is to express all the transformation functions, c1,, l2, etc,o in terms of the characteristic function, W. The functional form of W is then determined by Eq. (40216). Now, from Eqs. (4,114), (4o137) and (40139), 6p, t1 =aw - — P2 aw aw aw aw K2 = - P2 maw w m=p (4217) Pi api 89

and =2W Y 2W 62W 22W +2pllW 62W ) -12 = P2 ( +w2fYL) pi(w 2w + PP + P2 +P axa6y2xa~ aay a ap1ay apla // + P2j1 62W WY +P2 -2W+ 6axap ava P2P aP2 P1 + (P22 W + Pi+ -P 1+2Wp (4.218) xaP2 ap2 aplaP2 p2'-22 + 2Pi2 + P2 + 2p2 +P2 6y2 6y6 *2 6aYaP 1 6*6pl/ + 2P22 /W + P2 +2W +p12- + 2Pl2P22 ayaP2 6a*p2 ap aplaP2 + 22 aW P (4.219) 9o

- 22 _ 22 4T22 5J22'222 _22 22 P2 - P42 Pbc2 ax2 az ap [A~alp a6Pbc faolt at')t Cat + P22t ( t -P + az PZ + ap P + e + P22t6X Z P L+ P bc,bC2 3W + 2-2P2 +2 + 2pJ 2 +W P2 2a3W + 2P22X~ ~P2 + 2 ~X2~P-a b 2 = 2P22Kx2_p2 + P 2 + 2pW + 2P22 3W + Ps~~~2 a~ + P22~2 r12 + P2{ w 62 w _ P \w lP a6Xp2 2z + p2 3w 2p2 3W w +2123W 6X2 2_ P2 6Z3 P2 + 2P22 + p + P +a3W + 2Pi2P22 Kaa X a2P2 jat2 aP2ap2 + Pe2 ___1+P221 k6X26P2 PP -Z+ P22 +2 2p + 2p + 2pp 3w W( 6 P3W 6 2P2 a + P2 3W +2w _ _ __ 1 apiP2 2 ap2 aJ d2W + Pe2W + 2P22 +P2 a2 +p2 + +22 22P22 22+ +16P3 P22' +P 2pe 6pl 6 xW 6aW + 3W + 2 6SW + 63 w + 2 63 + 62w9 2P~2 aw a —r:awa% ~

+ P222 2 ( 82 + p P2) +2Pi2 8P1 + 2p22 _ +aw + P221 2w + a2w __ 2w_ _2w P22' 6X2 6P L a6P2 zp1 P P2P 2 P22 62W + 62W P2 + Y2WP l2 + 2W P22) 6ax2ap azapi ap2 ap26PJ P222 P2 P2+ 12 + (4o220) Now su:bstituting Eqso (4,217)- (4220) into Eqo (4o216) and eliminating P222'by Eqo (40210), we get foP+ fp2 + f2Pl2 + f3P22 + f4P11P12 + fPe22 + f6P,2.P22 + fPsP P12P22 + f9P232 + floP122 + flruP1 + f12P11P22 + f 3P12 + f14P221 = 0 (4,221) where W _P aw + P2 62W _2W 62w f = P2 6y 26+ P2 PI 6 PIP2 axay axay ay2 ayaVf 63w 653w PW ~3w 3w + 3 3P - 3P2 P2 + W P a + 2p a + p3 3W + ~ W + 6p2 ~W + 6W) y'+ 2%' bybp,2 6$;2p (2 3 3 ) a =. -2P2 - P 3 + P2 (4.223) 92

Q) ~~~~~~~~~~~~~~~~~~~~~~~~CO + + II ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ii r w QI!Q I e~~~~~~~~~~~~~~~~~" QQL QJII/I Id R)~~~~~~~~~~~~~~~~~~~~~~~R ~~~~+ Ql/ QJ/ U1 td Q, + t d QJ / Fd M C. C, Ol Q RD 02102 Q020 H I0 020-' 01 Fb02~ ~~~~~~~~~~~~~ + +0Fd P0 P0 C) R) 020-' ~~~~~~~~~~~~~~~~~~~02F + ~ -+ P0 \ H d QJ QJ W/ Id c02 Pw H H +M 02 P0 + RDt P0 *1 P0 P0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Co P0 P0E P0ID - 9 5 > + 02a1 a2/ H 021 (D (d f\> Fd \~02 H) + 02P0 H~~~~~~~~~~Q/ \0 I 02 P c +' I 021 02 QJ Id-0Q + 9 P0 9 J i t t < Fdi 0 P0 C Y-~~~~~ 021CO P0 P0 CO F~~~~~~~~~~~~~~~~~~~~~~ d 02d Q,/ + ~ ~ ~ ~~~~c, Q, c5~, c Q} QJ QJ aJ @ QJ QJQ,, Q/ Q,/ Ov H co p NI CO R Q.,/-0 P0 P0 - d 0 P0 Fd CO FO~~ 0 d 0202 d P0dP0 0 CA)4 P0 w, J + F, O + I P0 P0t < ay < |ty + P0 + +)IU'< la t I QJ0 02 0_d N, Q,/ Q,,, at P0 + *' I Qf/ N) Q, + Q t~~~~~ 2QJ P P0 tE-:E -E- Q., + 02E -g~ P0tlau > 02 Q 02 w Q+ QP0 0,0 t t v v t t *02 + RD * t Fd ~~~~-d Fd QJ ~~~~Fd02 F FdP0 RI CA)O P0 02 RD + d Fd02 0 — H~~~~~~~~~~~~~~~~~~~~~~~~~~~+F CO Hd Q,022/2 0 N~~~~~~~~~~~~~~~- P RD) RD RD RD R RD 0 ~ o m -~ 0> x-'i Pco N

fro = 2 W + 2p2 + 2Pe2 - + 2P22 (4.232) a~YP a1 a~P azaP f =P2 (<w + P 2W (4.233) aplay apla f12 = p2 w(4234) a3w f3 = (40235) 2W + 2 W 2w 2Ww f +4 - - P2 + P w 2 + P22 (4.236) ayapi Pl 6P P l P1P2 J Since W is a function of x, y, t, pi, and P2, Eqo (4.221) is satisfied identically only if all the coefficients are zero. Therefore fo0,oo,f14 are all equal to zero, which gives fourteen equations fo = f, =.. = f14 = 0 (4O237) from which the functional form of W is determined, From Eqso (40234) W can be separated into two terms as W(x,y,4,pl,p2) = W1(x,y,y,p?) + W2(x,y,y,p2) 0 (4~238) Equation (4.226) implies W1 is linearly dependent on Plo Therefore, Wl(x,y,*,pl) = W ( x, y, )p1 + W12(x,y,) e (4.239) Equations (4.232), (4.233) and (4~236) then give the same relation as follows:;L+ P2 =8~~ 0 (4.240) 94

Since W11 is independent of P2, it means W11 is independent of both y and r and this is a function of x alone, Equation (4.223) is seen to be satisfied identically. W now becomes W = W1 (x)pl + W12(xy,4) + W2(x,Yyr,P2). (Q4241) Substituting this form of W into Eq. (4,227), we get 2p1 3 $ -3w2 + W2 + P2 W2 (4242) 6P2 63Y26P2 66P 2 \ ay2ap2 atap2 2 Since W2 is independent of PL, Eq. (4~242) gives 2W2 - o (4,243) ap2 and thus 62w - a (4,244) 6*6P2 Equation (4,243) implies W2 is linearly dependent on P2, Also, from Eqo (4,244), the coefficient of P2 is independent of T. The characteristic function, W, can be written as W(xy,y,pl,p2) = Wj(x)p + W21(xy)P2 + W3(xy,*) ( (4.245) Substituting this form of W into Eq. (4,244) we get aW3 + P2 (w3 W21 + d = O o (4.246) ay 6afr a~y dx Since W3S Wl and W21 are all independent of Pa2 Eq. (4,246) is separated into two equations, namely, 6W3 - a (4,247) ay 3W3 _ aW21 + dW1 = a, (4,248). aby dx 95

Equation (4.247) shows that W3 is independent of y. Since both W21 and Wll are independent of *, Eq. (4.248) means W3 is linearly dependent on Jr. Therefore, W = Wll(x)p1 + W21(x,y)P2 + W31(x)t + W32(x) o (4249) Employing this result in Eq. (4~248), we get dW11 + W31 21 (4.250) dx ay The left-hand side of Eq. (40250) is a function of x alone, which in turn means W21 is linearly dependent on y, i oe., W21(x,y) W2ll(x)y + W212(x). (4.251) Equation (4.250) then becomes dW11 + W31 - W211 = 0 (4,252) dx which will be used later. The characteristic function now becomes W(x,y,Y,p1,P2) = Wll(x)pl + [W211(X)Y+w7212(X)1P2 + W31(X)> + W32(X) (4.253) This form of W will satisfy Eqs. (4.228)-(4o231) and (4.235) identically. We have two equations left, namely, Eqs. (4.222) and (40225). Substituting W from Eq. (4.253) into Eqo (4.222) gives'W14 + p2 dW211 + P2 dW31+ I(3W2 l+w3) = o (4.254) dx dx which can, in turn, be separated into two equations: 96

dW211 dW31 = o(455 dx + d = o (4255 )'Wii + 0(3W211+W31) = 0. (4'256) Similarly, putting W from Eqo (4.253) into Eq. (4.225) gives j dh Pir Ld I1 Y. + W3 d12 P2 dW32 +P2 Q y L ddxL dx dx dx dx + dW212 + p(W211-W31) 0 (4.257) dx which then is separated:into dW31 - o (4.258) dx dW32 = O (4,259) dx dW W2 + W = 0 o (4.260) dx Equations (4,258) arld (4.259) slhiow that W31 and W32 are constants and so from Eq. (4,255), W211 is also a constant. Equations (4.252) and (4.260) give the same result, namely, Wiz' being linearly dependent on x. To conclude, the characteristic function W can be written as W(xyy,plp2) = (aO+alx)pl + [g(x)+bly]p2 - [co+Cli]| (4.261) where ao, al, bl, co, and cl are constants and g(x) is an arbitrary function of x, One relation relating 0 [defined in Eqo (4.221)] and the constants can'be obtained from Eqo (4.256) as (ao+aax)~' - (c13-bl) = 0 (4.262) 9'7

and also, Eq. (4.260) furnishes another relation among the constants, which is al - bl - cl = 0 ~ (4.263) The absolute invariants and the restriction on 5 will be discussed in the next section. 4 3o2 Limitation on the Mainstream Velocity In the introduction the point was made that the type of problem involves the determination of an arbitrary function, 5. For such cases, not only the characteristic function W must be of the form given by Eq (4Q.261) and Eqo (4~263) satisfied, but the arbitrary function $ has to satisfy Eq, (4.262) if the given differential equation is to be invariant. From Eqo (4o262), (ao+alx)Yo - (cl-3b1)o = 0 Upon integration, for al O, ao = 0 = (constant)x(cl-3bl)/(cl+bl) (4o264) From Eq. (4o211),~ 1 due (constant)x(Cl-bl)/(l+bl) (4.265) 2 dx -we then get ue = (constant)x(cl-bl)/(cl+bl) (4.266) This is seen to be the mainstream velocity associated earlier with the linear group. For al = 0, ao ~ 0, c1-3b1 = (constant)e -' - X (4.267) 98

and thus, using Eq. (4o211), cl-3bl Ue(x) = (constant)e Iao z (4-.268) which is seen to be the mainstream for the spiral group. For the general case in which both ao and al are nonzero, the mainstream velocity is found to be (cj+bj)/j(cj+bj) Ue(X) = (constant)(alx+ao) (4,269) which again belongs to the linear group. We therefore conclude that the mainstream velocity, ue(x), belongs either to the linear or the spiral group, i.e,, powers or exponentials of x. No other forms are possible —a point long assumed but not proved. 4.3.3 Absolute Invariants and the Transformed Differential Equation With the characteristic function, W. obtained as in Eq, (4.261) we now make use of the general theory to find the absolute invariants and the transformed. differential equation. We shall expect the invariants to correspond to linear and spiral groups. From Eq. (4.265), we obtain dx = dy = dt (4.270) axl o2 m where the functions aCl, a2, and m can be obtained by putting Eqo (4.261) into Eq. (2.13) which then gives dx dy d* (4.271) ao+alx g(x)+bly co+cl$ Since this equation has two independent solutions, we have two absolute invariants, Let us consider two special cases. Case 1 ao = O, g(x) = O, co = 0 Equation (4.271) becomes 99

dx dy =d (d4?272) alx bly cl$ The two independent solutions to Eq. (4.272) are Y = constant bl/al and = constant. cl/al x The boundary-layer equation can be expressed in terms of these two invariants, Thus, we get the similarity transformation - Y and f(r) = $ (4,273) bl/a1 cl/al x x The mainstream velocity for this case is given in Eqo (4,266) as ue(x) = k c lb )flc+b ) (4e,274) where kl is a constant. Recalling that Eqo (4,263) has to be satisfied, Eqs. (4.273) and (4o274) can be written as n= A -m and f(I) = (4275) and ue (x) = klxm (4,276) which is seen to be in the same form as the so-called Falkner and Skan similarity transformation (see Refo 6). The transformed differential equation is well-known and can be written as f"tI + 1 (m+l)ff" + m(l-f'2) = 0 (4,277) 2 100

Case 2. a1 = O, g(x) = O, co = 0 Equation (4.271) becomes adx - c_ (4.278) ao bly clj The two absolute invariants can be found to be l= Y and f(h) = ~ (4.279) e 2e2 The mainstream velocity is given by Eq. (4.268) as ue(x) = k2eX. (4.280) The transformed differential equation is f,, = P(f,2 - I ff") -. (4.281) The transformation is seen to be the spiral group. The more general cases in which both ao and co are nonzero in Case 1 and co is not zero in Case 2 poses no problem. These assumptions merely introduce an extra constant in x and * in the transformation [cf. Eqs. (4.266) and (4.269)], However, the case in which g(x) is not zero needs further investigation. For this case, Eqo (4,271) becomes: dx dy d_ _(4282) ao+alx bly+g(x) clt+co Following the same steps as in Case 1, the absolute invariants are found to be i=y g(x)dx -" b _ + 1 (4.283) (ao+alx)al (ao+alx)al 101

and f (r) ci*+cO C (4.284) (ao+alx)c /a, The mainstream velocity is given by Ue (X) = k3 (ao+alxfc1-bY(c+b ) (4+.285) based on Eq. (4o269)0 The boundary-layer equation is transformed to an ordinary differential equation as Clb1 f2 _ CI ff f + k-2 cl-bi (4f286) a1 a1 e +bb By setting ue(x) = k3(ao+alx) (4~287) then m = C1-b (4.288) c l+bl Also, Eqo (4~263) has to be satisfied, ioe,, a, =b + cl o (4.289) Equation (4o283) to (4o286) become Tf g(x)dx (4.290) (aoLaix(a +ax (1my(2) f(q) = c i+co / (4o291) (ao+axlx) 102

and f"' + mk2 -_ mf2 + l+m = 0. (4.292) 2 For g(x) = O0 this is seen to reduce to the Falkner-Skan's flow. In order to transform the boundary-layer equation to an ordinary differential equation, g(x) can be any function of x; however, for g(x) # 0, the boundary condition at y = 0 cannot be transformed. We therefore conclude that for incompressible, two-dimensional, laminar boundary-layer equations, the linear and spiral groups are the only two groups possible. The group represented by Eqo (4.282) will transform the boundary layer equation from partial differential equation to an ordinary differential equation, it fails, however, to satisfy the boundary condition at y = 0. 4,4 SIMILARITY ANALYSIS OF THE HELMHOLTZ EQUATION IN GENERAL CURVILINEAR COORDINATES In the present article the role of a coordinate frame relation on a similarity analysis is considered by examining the two-dimensional Helmholtz equation in general curvilinear coordinates, While the present method of analysis applies to both linear and nonlinear partial differential equations, there are at least two reasons for choosing this particular equation for presentation of the present method of analysis. The first is its simplicity and its wide use in the potential theories of a large number of physical problems. The second, and the most important, reason is that the separability of this equation has been studied thoroughly by Moon and Spencer.13 By separation of variables we mean the original partial differential equation is reduced to two ordinary differential equations each having one of the original independent variables as an independent variable. In the case of similarity, the original partial differential equation is reduced to one ordinary differential equation where the independent variable is a function of the original independent variables, The present analysis is a parallel study to that of Moon and Spencer13 on the conditions of separability. Here we want to derive the conditions under which similarity is possible by requiring that the Helniholtz equation be invariant under the infinitesimal transformation. The basic problem is the determination of the characteristic function, Wo The difference between this example and earlier examples is that the resulting differential equations for the solution of W involve unknown metric components of the curvilinear coordinates. These equations form the conditions for the existence of similarity solutions. They can be used in two ways. In the first case where a coordinate system is given, similarity solution is said to exist if substitution of the known metric components in these conditions result in solutions for the characteristic function, W. Once W is known, the searching of all possible groups of transfornation can be made by following the same 103

steps as in previous examples. In the second case where a group of transformation is given for which the characteristic function, W. is known, these conditions give limitation on the metric components for the general curvilinear coordinates. If substitution of the metric components of a given coordinate system satisfies these conditions, then similarity solutions exist for the coordinate system for that group of transformation, In both cases, of course, the boundary conditions have to be transformable for true similarity to exist. 4o41 Review of Separability Conditions Discussed by Moon and Spencer The following is a short review of the separability conditions discussed by Moon and Spencer.13 This is done for comparasion with results to be obtained from the theory of continuous transformations, We consider two dimensional form of the Helmholtz equation given by g/ g(Xi/2 1 x1)x2 gi/2 aL + 81 ga g-/ a) ax~g2 (gl +a1 = ~ (40293) where gll and g92 are components of the metric tensor and g is the determinant with elements gpqo Now, let the unknown function be expressed as a product of two functions: X = U(xlj)U2(x2) (4.294) Substitution of Eqo (4o294) in Eq. (4a293) gives 1f a (gl/2 da (g/2 dU2 + = 0 (4295) gl/2 U1 xl \gal dxl U2 ax2 g22 dx2lJ The most general condition that will allow separability is that gl/2/gj is a produce of two functions: g1/2 = fl(x1)Fl(x2) (4296) gll g/2 = f2(x2)F2(x1) (4o297) g22 104

Substitution of Eqs. (4.296) and (4.297) into Eq. (4.295) gives ~Ld + -2 d(f2 -)1 + f, = O. (4.298) g1/2 Ul dxl dx Uz dx( d In this equation, g, Fi and fi are determined by the coordinate system and are entirely independent of the boundary condition that characterize a particular problem. But in the solution, ~ = U1U2, the U's are functions of both the coordinate system and the separation constants. Now, suppose we differentiate Eq. (4.298) with respect to Gl and ~2, we get F afU1 df dU 1 f2+ -1/2 o a a av a A dU2) +/ + F —- g dx dx a U2 dx2 d x2J (4.299) F1 U f dU1 + F2 r (f2 dU.2? =0 (4 300) U1 dxjl u dx JU F2 U2 dx2 dx2 Introducing the notation rdj(x ) =1 a 1 d dUi iJ i fj(x ) 6xj lUi dxi dxi (4 301) Equations (4.299) and (4.300) becomes fjFlll(xi) + f2F2021(x2) = gl/2 (4.302) frF,,2 (x1) + f2F2022(x2) = o0 (4.303) Equations (4.302) and (4.303) are solved for flF1 and f2F2 and the results are (if s l0) f1F1 = gl/2 ~ (4e304) f2F2 = - gl/2 (4.705) 105

where s = I (430o6) P21 P22 which is known as the Stackel determinant. Comparison of Eqs. (4,304) and (4o305) with Eqs. (4,296) and (4o297) shows that s = X22 (4.307) g22L2 (4 308) Equations (4- 307) and (4.308) are called the first condition for simply separability. Also, from Eqs. (4.304) and (4.305), g2= fl(Xl) F(X2) 2X2=) f f, (x) (4F309) This is possible only if 2 = fl(Xl) o f (x2) (4- 310) s Equation (4o310) is called the second condition for simple separability. It can be shown that the two conditio:ns are both necessary and sufficient for the simple separability of Helmholz s equation. Detail of the proof is given in the original work of Moon and Spencer.13 We now consider an example in which the cylindrical coordinate is considered, for which x = r cos Q, y = r sin (4 311) 106

From Eqs. (4.307) and (4.308), s (4.312) and s = r2 (4.313) 012 If $22 is taken to be 1, then s = 1 and $12 = - 1 (4.314a,b) 2 However, s is defined in Eq, (4.306) as |11 $12 i11 2 s = $21 $22 $21 1. (4-315) For s = 1, one possibility is $i1 = 1 and $21 = 0. (4.316) The second condition then becomes = r = fl(r)f2(Q). (4.317) s This condition is satisfied if fl(r) = r and f2(Q) = 1. (4.318a,b) We therefore conclude that the Helmholtz equation in cylindrical coordinates is separable. 107

4.4.2 Similarity Analysis of the Equation Let us put Eq. (4~293) in a slightly different form as ax i( ) X (h 2 + t ahjh2 = 0 (4319) where the following relations have been used: g/2 ~ hlh2h9 g2 -S 2 - h We now want to find the conditions on the two unknown functions h1 and h2 which make similarity possible, In other words, the conditions thus obtained rwill enable one to decide if similarity solutions exist for the coordinate system under consideration. Carrying out the differentiation and rearranging the terms, Eq. (4.319) becomes 2h2 3 P~ 6-% -22 p~ ap ax, + 0 ( 20 + h -3 + h h2P22 + Gdh.3h3 O (4 320) w.here Pa P2 a P11 P22 2 (321) We now make the infinitesimal transformation: x = xi + 1~i(x,x2,X,p,pP2) 1 Pi _P + ~~ji(xXiX2,,pl,'p2)nl2 P22 (ij = 1,2) (4~322) 108

where the transformation functions ii, i, i and cij can be expressed in terms of a characteristic function, W. as =i = aW (4.323a) P = p -w W (4.323b) api w + aw (4.323c) axi i a6 62W _2W + 2W W+ 2p + a2W + 2p 2W + Pl + Pl + 2pllP2 aw XP2 0ap2a9 Pp2 apl P2 Pi2y-2 +_ P 62W (4,22Pd6 + P12 a_ + P1L a (4+323d) + 2P22 ( a2w + P2 a _ + P a2w + 2p 2 aw -~22 = 62W 6W + 2a p2 + ape aplaPe + P22 aw + P2 (4.32e) The Helmholtz equation, Eq. (4.320), denoted by F = O0 is invariant under the infinitesimal transformation if UF = 0. (4.324) Expanding the operator in full, we get aF + aF +i FF = 0 (4,325) o109

Putting the Helmholtz equation, Eq. (4.320) into this expression, we then get fL(a h2 h + 2hh2( 2 + hlh2 ah2 3h2 h h2 h hh3, - h P F\6xh h X, 6xl 1xl 6lxl 6 X2h N r. E ahl 2 ah2 2 2h 2 Lh2 ah2 + 2hlh2 hl hlh h P2 axl ax2 aXl ax2 axlax2 axl aX2 axax2 h 3hlhh] h2 h + h9 j Ph2 ~ X 0h23 x+ 3h f l + 3x P22 + 22 3 h2+2h h2 h2h2 + h ha ah2 l a2 l+2 a3hh2 a2xl - h2 2 h 2 h2 h2 + L1 ah2 K )x2 xl 2 _ x h 22 P2 6X2 + h2h + l + 2 PxI + h 2 P22 3 fh2& 3 Pch2>l + O(hlh2 h + 5hhh2 a- + hh2 P2 6 2 _ 2 x 2 I2 JX2 X 2:+:chlh2h + 6C22h2h2 = 3 (4h P)22 I h2fp + + h3PP2 f4: x bha + Th 1 5X2_1 fop h h2 + J3 12f + f2P I f3 2h --- h 12 + flP12 + feP222 + f73PP22 + f8 = 0 (44P27) where 110

fo -= X 2 a P2 X +a a 2 3hlh2 a2 h hh3 K2 2 + p 62w + -w (4k28) \XJaP2 aPi a Pa2x2 d2XOP W P(4o329) f = (4-,29) f2 = - hlh23 6W (4-330) f3 = -2hlh a2w a(4033~) 3 2w hh 62W (4.332) f 5 = 0-h h2p W2 (4o332) 6p1P2 f = h2h3h2 a2W- (4+333) f2h6h (2w + -hp a 2 w ) h2W (4.335) w f7 6i h L+h1 x +62 -.0h X X 3 62W + -, O 6 x~ (4~ k~~~~~x~~~h2.35 2hih2 I. 611

YF2~hh 2]2 h hL2 ah + 2hlh2 2 + h1h(2 h2 ah2 6h1 h3 2 Pi L 5axl ax2 ax 2 ax2ax2 3 x axb ax2 1 bsa P 3xixj] P2 + j3~n~a~h h +, 1 3h 2 3h' 11 + 3' -2 + h2 6+ x2a xh 1 + 2bh2 ah2 ah.2 2 62h2 + b-~2. ~ bh~ h. + 2h, ihhh + h ln2 6'P2 6x2. h~ x 6X2 x x 2x lx ax h2 x ax, 2bx ax2 ax, a JPi + __ hWlh2 i * a+ (Pic- w h. 2 h - hlh2 2pa) 3 b ah 2 3 2P 2 + P2 / -,iT * Pa + P2 -) h2 h ax (4,336) Now, P22 in Eqo (4o327) can be eliminated by using Eqo (4.320) which, in shorthand form, can be written as P22 = -(go+gipl1) ( 4 337 ) 112

where go = hjh {hlh2 - h2 3Xl) Pi + (hlh2 Sh- hi + cah'h>, 4(L58) = h 1 (4. 338) 2 h2 91 0 (4.339) h2 By substituting Eqo (4,337) into EqO (4.327), we get HoPll + H 2P2 + HP +3Pl2 + H4 = 0 (4.340) where Ho = fo + 2f6gogl1 f7gl (4o341) Hl = fl - f5go (4.342) H2 = f2 + f6g. (4.343) H,3 = (4,344) H4 = fr8,-go,- + f6gSo o (4o345) Since the characteristic function, W, is a. function of x, x2,, pi, and P2, Eq. (4.340) is true only if the coefficients of each term are zero respectively; that is, Ho = h= = H2 = H3 4 = O 0 (4.346) The set of equations represented by Eq, (4.346), forms the conditions for the existence of similarity solutions,, These conditions can be used in two ways. In the first case where a coordinate system is given and we want to know if simlilarity solutions exist or not. Since now hl and. th2 are known, they can be suibstituted into these co nditions and if there exists a solution for the characteristic function, W, simi:larity solutions then exist, The searching for all possible groups can be made by fo llowing exactly the same 113

technique as given in the previous examples. In other words, as long as substitution of hi and h2 for a given coordinate system into these conditions results in solutions to the characteristic function, W, the coordinate system under consideration possess similarity solutions. In the second case, we consider a given group of transformation and we want to know the conditions h1 and h2 must satisfy for similarity to exist. For a given group of transformation, the characteristic function, W, is known. Substitution of this form of W into the conditions, Eq. (4.346) will result in five equations connecting h1 and'h2. They are the conditions that a given coordinate system must satisfy if similarity solutions exist for this given group of transformation. Two examples are given below. 4.4.3 Conditions of Similarity for a Given Coordinate System As a simple example, consider the rectangular coordinate system in which both h1 and h2 are unity. Then the conditions for the existence of similarity solutions, Eq. (4.346), give a2w _2w__N 2w _ a b' (P7 X Pi pj 2aPX2 Pax2p 5P2 (4.347) -2 62W p 62W 2 62W_ t P2 a W ) + xlaps ap28 ax28pl a8/8P aplap2 (4.348) 62W _= O (4,349) dp = o 0 14P2 - 2cr + P2 + 2 P) +02 242 6X 2 0 (4.P50)

This means if solution to Eqs. (4.347)-(4.350) for W exists, similarity solutions will then exist for this given coordinate frame. The searching for all possible groups of transformation satisfying these conditions shall not be our concern. The only conclusion being sought is whether similarity solutions exist and if they do what are the conditions the characteristic function, W. must satisfy. Equations (4.347)-(4.350) form those conditions and by trying various functions for solutions, the conclusion is easily drawn that similarity transformation do exist. An example will be given below. Consider now the following form of the characteristic function, W, W(Xix2,y,PlyP2) = Wl(xl)pl + W2(x2)p2 + W3(X1,X2,~) o (4.351) For this form of W, Eqs. (4.348) and (4.349) are satisfied identically and Eqs. (4.347) and (4.350) give _ ~w_ + dW2 = o(4352) =a 9 o (4.352) dxl dx2 /2Ws 62W 262 W__ -aW3 - ( + 2p + Pi P6o - (a2W + 2p2 o2W3 + 2 _2 ) (8X2 P2 bX2 P2 6x2 6W2X Nlaw + 0= 0. (4.353) Equation (4.352) gives W1 = c1x1 + c2 (4.354) W2 = c1X2 + C4. (4.355) These are substituted into Eq. (4.353) and we get JoPl + JlP2 + J2Pl + J3P~ + J4 = 2 (4.356) 115

where Jo = a W (4 357a) axlah J, = 2w — (4.357b) ax2a6 J2 = w (4.357c) a62 J3 = 2 (4.357d) J4 a= -oW3 -2 + 2 1c a+ (4.357e) Since all the J's are independent of the p's, Eq. (4.356) is satisfied if all the J's are zero; that is, Jo = J1 = J2 = J3 = J4 = 0. (4.358) The first four conditions in Eq. (4.358) indicates that the function W3, should'be of the form W3 = C50 + W31(Xl,X2)~ (4.359) Substitution of Eq. (4.359) into the last condition in Eq. (4.358) then gives 2W3s + 2e + aW31 - 24c1 =. (4.360) However, since W31 does not depend on 0, Eq. (4.360) is satisfied if simultaneously acc = O (4.361) ~e~ + ~2~ + oWwl = 0. (4.562) 116

Since a is not zero, cl must be zero. Therefore, the final form of the characteristic function is W = c2p1 + c4p2 + C5s + W31(Xl,x2) (4.363) where W3l is any function which satisfy Eq. (4,,362). To solve for the absolute invariants:it is necessary to solve dx_ = dx2 = d_6 -d~~a.~ d=~X~2 =~(4.364) or, using Eqs. (4.323a,'b) and (4.363), =L _!IX2 d, (4.365) C2 C4 -(C5+w31 ) Let us next consider the case where Ws31 is a function of x, alone, then Eqo (4o,362) is reduced to d2 dxaLt W = O. (4.366) dx? The absolute invariants are the two independent solutions of Eq. (4.365) which are X2 ~ i1 (4.367a) C2 f =1 + T W31e dxj (4.367b) f17j)= ~ec2 drC Putt'ing these transformations into the Helmholtz equation and making use of the condition) Eq. (4 366), we get 1 + f" + 2 c4c5 fv + += (4. 68) 2 ( C2/ C2 C2/) (4.368) 117

which is an ordinary differential equation. The above is only a special case of all the possible groups of transformation which will reduce the original partial differential equation to an ordinary differential equation. Other solutions to W and the similarity solution for those groups can be obtained in a similar manner. We therefore conclude that similarity solutions do exist for Helrmholtz equation in rectangular coordinates, in addition to its separability. One final remark concerning the boundary conditions is necessary. When it is stated that similarity solutions exist, we mean that not only the partial differential equation is reduced to an ordinary differential equation, but also that boundary conditions can be transformed satisfactorily. For separation of variables in the usual sense (see the'brief summary earlier), the boundary conditions can always be transformed. This is not true for similarity transformation. In the example just treated, Eqs. (4.367) and (4o368), the only possible similarity solution is when the boundary conditions are given as =O at x2 -2 xl = k C2 -O at X2 -a x = k2 where k1 and k2 are constants. By putting W31 to zero, Eqo (4.367) gives the following boundary conditions for Eq. (4.368), f(kl) = f(k2) = 0 O (4.369) 4.4o4 Conditions of Similarity for a Given Group of Transformation As an illustration, consider the spiral group of transformation where the spiral group of transformation where the characteristic function is given by W(Xi X2 nP1iP2 ) = clpl + C2X2p2 - 3 o (4.370) The conditions for the existence of similarity solution, Eqo (4.346) then become 118

2 ch2 l l hh c h- 3x2J + 2h1h3e2 =0 2cihlh2 6 h3 I + 2C2X2 hL2 ah 2 h2 X2 2 and Go + Glp, + G2p2 + G3~ = 0 (4.372) where h L h2 + ah2 h2 +h2 h2 2 h2 h + c x2 f 2hl1h2 lh2 Xhl 1 2a 2 L x1 x- + hhe 6 - 3h2 ha2 h X2 1 2- 6hi 6h2 h2 ( h2 y 2 2 h 2 (x) fz1(5h~lh2 6; + ha + C2X2 (3hlh2 X + h2 ahX) 2- e2h3h2 + c3hlh2 ( *37 ) - axl 6xl x, h xl 6xl ('cl 3hlh2 hal + hl1 bh6e~ + C2x2 hih2 X2 +A1 X2 2ah 3h2 + cah2 h (4 374)

G2 = cl 2hh2 h- hl + h a2h2 h + h2h2 1 axl ax, aXl ax2 axax2 3h2~ h_ ah2 _ h3 a2h2 > + C2x2 2h.h2'h_ L -h2 ah2 h + h+1h2 2 ax ax2X2 ax2 aX2 h 2 3 ah h2 h 3 a h2x 6ax2 ax2 a-2h / (x) I1 (h5hh2 lh + h 2+ C 2X2 (;hlh2 < + h- 1li) ax, ax, axs aT. 2 c2h, lh2 + c 3hh2j ( 4375) G3 = aCl(3hit2 h + 3hha + Cc2X2 3h.h2 3 2 3 $ h2'h i~~ + ac3h,(h2) 3 "c2 hl-h-2 + c3h 1n C (4.3I76) It shoud be noted that for W in the form gven in Eq. (4.370) the condi It should be noted that for W in the form given in Eq. (4.370), the conditions H1 = O0 H2 = 0 and H3 = 0 are satisfied identically. From Eqo (4 372) we conclude that the G's should all be zero, Thus, we get Go = = G3 = G0 (4.377) Equations (4-371) and (4.377) are the conditions to be satisfied for the h1 and h2 for similarity solutions to exist for the spiral group. As long as the functions h1 and h2 for a coordinate system satisfy these conditions, then the spiral group of similarity transformation exists for that coordinate system. 120

As an application of these conditions, Eqs. (4.371) and (4.377), let us ask if the spiral group of transformation exists for the rectangular coordinate frame. For such a coordinate system, both h1 and h2 are unity. Equation (4.371) becomes c2 = 0. (4.378) Equation (4.377), with c2 = 0, are satisfied identically. Thus, we conclude that the spiral group of transformation exists for the rectangular coordinate frame, since its h1 and h2 satisfys the conditions given in Eqs. (4.371) and (4o377) if c2 = 0. The characteristic function is therefore W = clpl - c30 (4.379) and the absolute invariants can'be obtained'by solving dxL _- (4.380) C1 C3~ and x2 constant (4.381) which gives = x2 and f(q) = c1 ec1 and the Helmholtz equation is reduced to f" + (a + c )f =. By substituting into Eqs. (4.371) and (4.377) the h1 and h2 functions for a cylindrical coordinate (hl = 1, h2 = xl), it is seen that similarity solutions do not exist since they do not satisfy these conditions. Other coordinate frames can be tested in a similar manner. 121

4.5 CONCLUDING REMARKS The method given in this chapter can be summarized as follows: Consider a partial differential equation F(zl,..., Zp) = 0 (4.382) where Z1 = X1 Z2 = X2 ak Zp_1 = (zp Xm)k This equation is said to -be invariant under the infinitesimal contact transformation = Zi + iE i=l p if the following condition is satisfied OF 6F UF = +... + - (4.383) 1 pP 6Zp Since the functions, Si, in the transformation are expressed in terms of a characteristic function, W, Eq. (4.383) is used to predict the form of W. The invariants can then be obtained by solving the following system of equations: dzp _(4.384) 51 5p Finally, using the theorems given in article 4.1.5, the number of variables can be reduced by one using the invariants as new dependent and independent variables. 122

For simultaneous differential equations, the functions in the infinitesimal contact transformation are expressed in terms of characteristic functions, Wj where i = 1,...,m and m is the number of dependent variables. The present method is seen to'be a systematic way of searching for all possible groups of transformation which will reduce the number of variables by one. For reducing more variables, the same steps have to be repeated. 1l23

REFERENCES FOR CHAPTER 4 1. Ames, W. F., Nonlinear Partial Differential Equations in Engineering, Academic Press, 1965, pp. 141-143O 2. Birkhoff, G., Hydrodynamics, Princeton University Press, 1950, Chapter 4. 3. Cohen, A., An Introduction to the Lie Theory of One-Parameter Groups, Heath, New York. 4. Cohen, A., An Elementary Treatise on Differential Equations, Heath, New York, p. 252. 5. Courant, R., Differential and Integral Calculus, Blackie and Son, Ltd.o Vol. II, p. 80. 6. Falkner, Vo M., and Skan, S. W.,, "Some Approximate Solutions of the Boundary Layer Equations," Phil. Mag., 12, 865 (1931). 7. Ince, E. L., Ordinary Differential Equations, Dover,:1956, p. 103. 8. Lie, S., Math. Annalen, Vol. 8, 1875, po 2200 9. Lie, S., and Engel, F., Theorie der Transformations-gruppen, Vols. 1-39 TeubnerLeipzig. 10. Lie, S., and Schef+fers, G., Vorlesungen uber Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipsiz. 11.o Lie, S., and Scheffers, G.o Vorlesungen uber continuierliche Gruppen mit geometrischen und andernen Anwendungen, Teubner, Leipzig. 12. Manohar, R.,o "Some Similarity Solutions of Partial Differential Equations of Boundary Layer Equations,"' Math. Res. Center, Univo of Wisc., Technical Summary Report, No. 375, 19630 13. Moon, P., and Spencer, D. E., Field Theory for Engineers, D. van Nostrand Co., Inc., 1961. 14. Morgan, A.oJoA., "The Reduction by One of the Number of Independent Variables in Some Systems of Partial Differential Equations," Quart. Applo Math., _5, 250-259 (1952)o 124

REFERENCES FOR CHAPTER 4 (Concluded) Muller, Von Ernst-August, and Matschat, Klaus, Uber das Auffinden von Ahnlichkeitslosungen partieller Differentialgleichungssysteme unter Benutzung von Transformationsgruppen, mit Anwendungen auf Probleme der Stromungsphysik, Akademie-Verlag, Berlin, 1962. 125

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