ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report ON THE BEHAVIOR OF A GAS NEAR A WALL; A PROBLEM OF KRAMERS' C. S. Wang Chang G. E. Uhlenbeck Project 2457 OFFICE OF NAVAL RESEARCH, U. S. NAVY DEPARTMENT CONTRACT NO. Nonr-1224-(15) August 1956

The University of Michigan Engineering Research Institute TABLE OF CONTENTS Page ABSTRACT iii OBJECTIVE iii I. INTRODUCTION1 II. THE COUETTE FLOW NEAR A FIXED PLATE 2 IIIo SUCCESSIVE APPROXIMATIONS AND NUMERICAL RESULTS 13 A. FIRST APPROXIMATION 13 B. SECOND APPROXIMATION 13 C. THIRD APPROXIMATION 15 D. FOURTH APPROXIMATION 16 IV. TEMPERATURE DISTRIBUTION NEAR A FIXED PLATE 17 A. FIRST APPROXIMATION 22 Bo SECOND APPROXIMATION 23 C. THIRD APPROXIMATION 25 V. CONCLUDING REMARKS 26 APPENDIX I 27 APPENDIX II 28 APPENDIX III 30 APPENDIX IV 31 REFERENCES. 32 ii

The University of Michigan T Engineering Research Institute ABSTRACT The deviations from the originally linear velocity or temperature distribution due to the introduction of an infinite wall in the plane x = 0 and with a velocity or temperature equal to that of the gas at x = 0 before its introduction have been studied by the same method employed in our previous reports. Formal solutions involving infinite determinants have been obtained. The general features of the solutions agree with the prediction of Kramers. Some successive approximation calculations were carried out. The first approximation results for both problems are in agreement with the classical results of Maxwell and others. OBJECTIVE A theoretical study of the behavior of a gas near a wall. iii

L The University of Michigan T Engineering Research Institute I. INTRODUCTION The problem of the behavior of a gas near a solid wall is of long standing. The temperature and velocity jumps in the immediate vicinity of a wall were first observed by Knudt and Warburgl in their experimental investigation of the heat conduction and friction in rarefied gases. Approximate expressions for the temperature and velocity jumps have also been derived by several authors.2 In two of our previous reports3 we derived these expressions for the cases where there are two solid parallel plates and where, in the case of heat conduction, the ratio of the temperature difference between the plates to the mean temperature of the plates and, in the case of Couette flow, the ratio of the velocity difference between the plates to the mean molecular velocity are small. Kramers4 in 1949 attempted to derive a more exact expression of the velocity jump by studying the modification of the distribution function due to the presence of a stationary plate in a gas moving in the zdirection with a constant velocity gradient dw/dx. Though he was unable to give a complete solution, he did suggest that by writing for the distribution function an "Ansatz" of the form f.= f o x dw Cx C, + Cz + X 0 3 dx given by x(dw/dx) is a faxwelree path, and a constant to be determinedi and given by x(dw/dx), k is a "free path," and ~ a constant to be determined, and by a superposition of solutions of the type X = %(c)e-yx with constant 7, perhaps both the Boltzmann equation and the boundary conditions could be satisfied. We will confirm this expectation completely. Most recently, Welander5 also derived the temperature distribution, the velocity distribution, and the temperature and velocity jumps in the neighborhood of a plate. Welander made the main assumption that the rate of change of the molecular distribution function due to collision is proportional to the deviation from the Maxwellian state and took for the proportionality factor the expression given by the Stokes-Navier approximation for Maxwell molecules. For perfect acconmodationthe following expressions for the temperature and velocity jumps were found: 1

L The University of Michigan T Engineering Research Institute (AT)w = 15 (1.152) T d 8 dx -dw (AV)w = (1.21) dx where dT/dx and dw/dx are the temperature and velocity gradients, respectively, just beyond the transition region (i.e., a few mean free paths from the wall), and ~ is the mean free path related to the viscosity coefficient, by the relation 2[L = _S kT nm -- a rm The numbers in the parenthesis are the correction factors for the expressions derived from the simple theory.2 The problems considered in Reference 3 are closely connected with Kramers' problem. The treatment presented there, with proper modifications, can be adapted to solve Kramers' problem. In fact, the velocity jump Kramers was seeking can be deduced from the results of those reports. However, the apparent differences between the problems, such as the nonexistence of parameters like AT/T and d/X, where d is the distance between the plates, seem to warrant a separate treatment. For simplicity, we will limit ourselves to Maxwell molecules, i.e., molecules interacting with the force law, Kr-'5 K being the force constant. II, THE COUETTE FLOW NEAR A FIXED PLATE We consider an infinite space where the gas is flowing in the zdirection with a mass velocity cz(x) proportional to x. A plate, with accommodation coefficient a, is introduced in the plane x = 0. The problem is to find in the stationary state the distribution function of the gas in the upper half infinite space (x > 0). The Boltzmann equation is* 6f cx = nJ(ff) (1) ax where f is the distribution function, c is the molecular velocity, and J is the collision operatoro We write again f = f(l + h). (2) *We will always use dimensionless velocity variables by measuring all velocities in units (2KT/m)1/2 2

.The University of Michigan T Engineering Research Institute Differing from Reference 3, we take for the zeroth-order distribution the Maxwellian distribution with a mass flow z =- x(dw/dx) where dw/dx is the velocity gradient in the x-direction, i e., we write o = \ )n n mj/2 eC2, (3) where C2 = Cx2 + cy (CZ - x d)2 or C = - (f). It is assumed that dw/dx is a constant, and that l(dw/dx) is small compared to the mean molecular speed [e(kT/m)l/2]. Substituting Equations 2 and 3 into Equation 1, and dropping the term h(6fo/6x), one finds dw ah 2CxCx + x h = nJ(h) (4) where now C may be considered to be identical with c. Equation 4 is the integro-differential equation to be solved. The boundary conditions can be written as follows: lo x = 0 h = 8 dwCC + EC (5a) _? Cx 3 dx 2. x = 0 and Cx > 0 h(0,Cx, C C) = (l-a)h(O,-CxCyCZ). (5b) The first term on the right-hand side of Equation 5a is the viscous effect as calculated by Chapman, where X is the mean free path defined by = - ( nA377i, 1, K being the force constant and A2 = 2t dG sin39 F() and F(@) = /m/27 gl(gG)o The mean free path x is related to the mean free path f defined before by the relation x = 3-. (6) The second term is the effect at infinity due to the presence of the plate at x = 00 A consideration of the symmetry of the problem indicates that this effect should be proportional to Czo The proportionality factor E is independent of the velocity components and it is expected that it will have the form X(dw/dx), except for a multiplicative constant which is to be determinedo In Equation 5b the quantity a is the accommodation coefficient; it means that at the plate a fraction a of the molecules striking the plate will be reemitted 3

The University of Michigan T Engineering Research Institute with the Maxwellian distribution in equilibrium with the plate, while the fraction (1 - a) will be reflected specularly. Equation 5b is the mathematical statement of this facto The problem is, thus, completely defined. To solve it, we first make a change of the dependent variable from h(x, C) to Y(x, C) where h(x, C) = Y(x, C) - 8 X dW Cz + E Cz (7) 5 dx so as to make the integro-differential equation homogeneous, namely, by= n -J(Y) (8) ax aCx The boundary conditions can now be written in the form x = Y = 0 (9a) x= 0 Y(O ) [(l-a)Y(0,-Cx, Cy, Cz) + (2-a) 8 x ~ CaC 3 dx - C Cz] 1 + sign Cx (I) 1 - sign x (.b) 2 2 It is to be noted that Equation 9b for Cx < 0 is just an identity; only for Cx > 0 does Equation 9b give a condition for Y(O, N)o A natural way of solving Equation 8 with the boundary conditions Equations 9a and 9b is to use the Laplace transform. Introducing the Laplace transform Z (s, C) of the function Y(x, C) defined by 00 so Z(s, C) = %'dxe'sx Y(x, ), (10) we have sZ - Y(o, ) - ). (11) Cx With Y(0, C) assumed to be known, we will solve Equation 10 in the same way as the problems in References 3 were solvedg namely, we will expand Y(x, C), and consequently Z(x, C) in terms of rr, m(C ) the eigenfunctions of the collisior operator Jo As discussed in Reference 3, only terms with m = + 1 will enter, hence we write Y = 7 [ar,,l(X)trrl + ari.,tr,-] (lla) r,i 4

IThe University of Michigan * Engineering Research Institute z = [ r,,(s)rl I, + r,-l(s)tr,-] (llb) r;~ Further, since ir -m is the complex conjugate of r ~,m, the reality condition of h, and Lence that of the functions Y and Z, requires that ar,~. a*r, * Thus we need only to solve for ar~,1 or ar,^ In terms of the a's, Equation 8 becomes dar p21-1.dx.. = n Rr,2,i; r.2'+l,1 ar;2~" +,l(x) (12a) r'Q' rdar dar2,1+l, - n Rr'2f+ii; r;21 ar$2~:L(x) (l2b) dx 1* rV' and Equation 10 becomes Sr 1, ar, (0) = n Rr,li; r;i2l CrI 2. (13) rIP where the ar.,i.(0)'s will be assumed to be known for the time being, and Rr,l,; r;lil = [1. r rl1] Cx Xrtt dCe'1 Xr,'~, de.C2 1 $t ~. $r l'1 The expression Xryt' is the eigenvalue of the collision operator J, and'it has the dimension of an area. As before, we separate Equation 15 into two sets, a set of even I and a set of odd I, src,.il - ar,2,1 (0) = n, Rr,2^2.;r21'+il arr2+.,1 (13a) Z_!_j S'ri,21+1,1-a,+i (0) = n ),2+1,1; r;211 r2i (t5b) rtt Eliminating ar,2+1,1, we find 5

The University of Michigan * Engineering Research Institute S2Or,2, 2 R,,,,, = r2 i - n2 V r,21,1; r22'+1,1 r,2 +1,; r',2L'. r',2' r-v r'^" = sar,21,I (0) + n I Rr,2~,1; r2V'+1,1 ar2T2~'+1, (0) ( r. a set of linear inhomogeneous equations for all aOr2,, Since Ro,2,1; r,21+il, = ~ a consequence of the conservation of linear momentum, and since molecules for Maxwell Rr.21+1,l; rt2";1 = 0, unless r = 0 and ~ = O but Rr,2,;0,ll = 0 because kol = O, the set of Equations 14, can be written as So,^z, 1 = ao^2 (0).032js 1 ~ (14a) and Z (s2rr8 L " -Rr n, 2,1;r2+ Rr,2' +1,1;r21'.L ) ~Cr2 2'l rt"~" r- f' = sar,2Ll(0) + n Rr 2l;r1 21 l 2 +ar,1 ( 0) /, ~~ II (>(rI), ( Or') (0,1) (14b) The solutions of Equation 14b can be easily written down. We write, - it (s -pi)(s +P ) o i The pi'S are the aroots of the equation Ae(s) = they are of the form of a constant divided by the mean free patho.Let Dr,2.1 be the determinant.obtained from A by replacing the (r,2,1l) column by the inhomogeneous part of Equation 14bo By splitting in partial fractions, one then can write, except for r = 0 and I = I, 6

The University of Michigan * Engineering Research Institute Ur 2, =1 \i i Dr-2il(s = Pi) (dA \dsjs = Pi 2-1 + Z Dr,2, l(S =-Pi) s Pi ti d i i \ddsj ='Pi Pi s + Pi, (15a) where the determinants Dr 211 (s = + Pi) still contain the ar, l(O)s, Since the Dr, ldepend linearly on the inhomogeneous part of Equation 14b) only one.of the sets Dr z21 (s + Pi) is independent. Choosing this set to be D12 2'I the others are given by* Dr,2,1 (s = ~ Pi) (i) Al 2' xr', D,2, (s + Pi) Di,2i (s = ~ pi), (i) aX,2,1,12,11 where A 21,1;r,22,1 is the algebraic complement of the (l,2,l;r,21,l) element of the determinant A. Substituting the expression for caral, Equation 15a, into Equation 13b one finds Dr2 1 (s = L TSS By taking the Equations 12, (S) ar,2Q+1.(O) + n Rr,22+1l'l;r2z x rQ' Pi) 1 + Dr 2z}i(s = i) 1 s(s - Pi) (d s(s + pi) )i ni dss c Pi inverse Laplace transform one obtains for the solutions of (l5b) ao,2,i(x) = ao,2,1(0) = const. (16a) ar,21, (x) = DrZ.t.l,(s = Pi) ePix +e2 i (ds)= Pi (ds)=~si s= Pi s, (16b) ar, ++i, (x) = ar,2~+1,(O) + n Rr,2.+1,11r$21, _r Pi rr''I Dr, 2Q, (s=Pi) Pix p dA (ds )= Pi - 1 Dr,2,(s =- (e-Pix) 1( x d = Pi d ds -~ (16c) *See Appendix I. 7

The University of Michigan * Engineering Research Institute Equations 16 are the general solutions of Equation 8. So far, we have not really taken into account the boundary conditions. The conditions, Equations 9a and 9b, in terms of the ats are, for all r and i, ar.ei( o) = ~ (17a) ar l(0) = ( Mr;r ar l(O) r' 4(2 - ) w dw 9iNo21. dx 2iNo:.1 where the N's are the normalization constants for the 4's and /> -m > _Ce 1 + sign Cx'* m'' Mr.Lxm r'.Jf' =,,m -- rd e.... The elements Mr,,m;r:A.m have the following properties:* 1l Mr',,,m;r': m =Mr'lm;r,~,m (18a) 2~ Mr,2.,m;r2' m = 1/2 &rr,'6, (18b) Mr,2a!+l,m;rJ,+lm = 1/2 brr',ee' (18c) 3 7 Mr,2 limr;2+r1,mI'1+2L,m;r2'm,= 1/4 8r"6UQ" (18d) a+ Mrr,22+l,m;r;2;m Mr;fmr2t = 1/4 rr,",,e" (18e) The conditions at infinity require that ao,2,2(x) = 0 (19a) Dr$2,l(S = Pi) = 0, or Dl,2al(S = Pi) = 0 for all i, (19b) ar,2+1,l(0) + n Rr,2+x, i;r;. DrP — )l( - " 0,(19c) _________r'~ i ( d S)s = Pi *See Appendix II. 8

The University of Michigan T Engineering Research Institute where in arriving at Equation 19c, we have put Equation 19b into Equation l6c. Each of the Equations 19 is a linear relationship between the a(O)'s. Howevery these are not all independent because of the differential equation 12bo Leet us consider the case where we take N eigenfunctions, among which there are Ne even ones (I = even) and NO odd ones, such that Ne + No = N. Since as in Reference 3 we always will cut off the series eigenfunctions for various values of M = 2r + I with I < M - 1, one easily sees that No > Nea It can be shown that Equations 19b follow automatically from Equations 19c, l1b,.8a, and 18b. Thus the condition at infinity furnishes us with No + 1 linear homogeneous relations for the a(o)'s. The conditions at x = 0, i.e., Equation 17b, can be separated into two sets, namely, 2-a, ar_2,l(0) + 2Ma r, M2,I r2+l, ar;22+,l() x dW i — ^l2Sa& -M (20a) 9iNo21 dx ro 2-a 2iNol0 Mr2loll (2 ar,2+1,l(O) + 2(2-c) i Mr 2+l 1;r;2~ ar ^21;(0) r;' O 2-a) M p4 10pT^^ — sdw- E r-6o ~ (2Ob) - =2 Mr,2i+ll o2l dw _ 2~ roblo (20b) 9iaNo21''0'2' dx.2iNo1ro For N = finite. Equations (20a) and Equations (20b), Ne and No in number, respectively, are independent. Thus, for an approximate solution where we limit ourselves to a finite number of eigenfunctions, we have 2No + Ne + 1 linear inhomogeneous relations for the N + 1 (N ar,,'s and e) unknowns., If we choose to have the condition at infinity (Equations 19a and 19c) and Equations 20a satisfied, then the Equations 20b can not be satisfied. Presumably by taking N bigger, the Equations 20b will be better satisfied. The last statement is confirmed by observing that for the exact solution, where N - oo, Equations 20a and 20b follow from each other. For instance, if we multiply Equations 20a by Mr,2L,+ll;r2jz, and sum over all r and I, by using Equations 18d, one finds I Mr;,21,+l,3r,2,(O ) + art 2t+1,() rt 4 dw a e 9iNo2 dx Mr 2'+,;3,2, - 2(2-a) 2iNol- r'o R'' giNO21 11#0 2 I 9

I The University of Michigan * Engineering Research Institute which is c/2(2*a) times Equations 20b. Making use of Equations 19,.one finds ar,^2,1(x) Dr,2~,1(s = - Pi) 1 Ts is = - Pi (21a) &r, a.. (xp = - n Rr,22+.,;r;22:1 p rti r'~'I i Dr. 2 2(S = - I Pi) e-pi)r (21b) \dAs = - Pi For the actual solution for the constants e.and the ar,, 1(O)'s it is most convenient to call (i) br,2l,1 Dr,2A,l(s = - Pi).. (22) dA KdSS _= Pi By the relationship between Dr,2f,l and D1,2,l, it follows that b i,21, (1 ) ( i2) r.,2,2 1, ( i I Y t,21 1-11 (23) where bl,2 1 is the set of independent variables to be solved. Eliminating the odd a's between Equations 21b and Equations 20a, using 21a, 22, and 23, one arrives at the following set of equations for the determination of E and bj i 1 for all i, Mr2.'l.1~' l _ + E Q b.Pi ALI. r2 1 1. 2-a 4.dw 1. 2=- -9iNo- - d ro' 20 9iN021 dx (24) r-su)ol i with.i - mi' Ror.,2i^ i n,1,,,,2 I i ".'2 Ir' -.''R' 1A;'2..').. (25a) rit' rV.'. (i) Qr,2, = -n E Mr,2 L,;r2Z'+.i,i Rr$2~,+l,l;r.2,, I ^1.'2 1 r.2R3lp.. Y. -T 2 I... II T'(t, j )lo ~t~ 2-a _Al,2_,_;r,2~, 20c \A-1,2.1. l21 I r. 0 and I ~ 1., (25b) 10

I The University of Michigan T Engineering Research Institute (i) Equation 24 is a set of equations Ne in number for the unknowns C and bt,2,1, where i goes from 1 to Ne - 1h It is seen from Equation 24 that both c and all the b s are proportional to the parameter \(dw/dx)o Formally our problem is solved. The deviation from the Maxwellian distribution with a streaming velocity in the z-direction equal to x(dw/dx) is h.= 8? dw CxC + eCz + (ar,erlr,l + ar,,-,r,,1,-) dx L z (26) where all the ar,. + Is are linear combinations of the form e Pxi. This is in agreement with the contention of Kramers as mentioned in the introduction. For distances much larger than the free path, x >> \, h is of the form h = - ~~ d CxCz + eCz 3 dx The general expression for the velocity in the z-direction is -z S f d c f Ui = x w + - al dx 2 0, 3/2 (27) ao,1l is purely imaginary, as can be easily deduced from the symmetry property. Since ao,1, is a linear combination of negative exponential functions in x, the velocity gradient is therefore no longer constant. However, at points more than a few free paths from the plate (i.e.. outside the transition region), the velocity gradient will still be a constant, and, in fact, the same as if the plate were not present The velocity at a large distance from the wall is (E'Zx > % -: dw 2 dx- + -, dx 2 (28) which can be written in Kramerst form, by putting C = 2k0(dw/dx), so that (FZ ) x >> % (x + k7) dw dx where k is called the slip coefficient, a numerical constant to be determined, and T is the free path defined in the introduction0 Thus the velocity is still a linear function of x with the same gradient, but the plane of zero velocity is at x = -k~ instead of at x = 0. The velocity at the wall, which is equal to the velocity jump defined in Reference 3, is._________..^~ n - J-L

The University of Michigan * Engineering Research Institute w = o)=e _ a0 () (29) The velocity jump, (AW)w) used by Welander is defined as the difference between the velocity at the wall obtained from a linear extrapolation of the velocity distribution curve beyond the transition region and the velocity of the wallj ise. (W)w (30) The figure below gives a sketch of the velocity distribution. cz in th absence of the plate xdw dx | xX~~~~/ / Tangent to the ff~/ y~/ ~velocity distri/ /^ bution curve at large x: dx / / - ax,.kQ()) d/ /'// E ao 1 10) - = kQ dw kYQ / / / / 12

The University of Michigan * Engineering Research Institute III-. SUCCESSIVE APPROXIMATIONS AND NUMERICAL RESULTS In the previous section, the formal exact solution of the problem is given. The actual solution consists first in finding the roots (Pi) of the infinite determinant A,. and the solution of the set of infinite linear inhomogeneous equations, Equation 21, for - and b. We will in this section find some approximate solutions by using the same cut:-off process as employed in Reference 3. A. FIRST APPROXIMATION We take into account O, 1,1,l, l 3, Ao 2 The solutions are clearly aoW i, l = ~0 a0,2,1 = 0 The set of Equations 15 becomes one equation for c, namely, -O 52 1 1-1 e 2a 4 I dw 2iNo11 2ct 9iN021 dx giving c 2- 4_ dw 2-Ca dw = -X- = -. 2T a 3 dx a dx The average velocity in the z-direction remains linear, and is given by z = (x + 2-a ) d a dx Hence the slip coefficient is (2-a)/ain agreement with the *results of the older theory; cz is linear in x throughout the whole space, Aw = (Aw)w = 2-a/a T dw/dx B. SECOND APPROXIMATION For this approximation we use five eigenfunctions::0o,2tltl,2,i; ~Otll, lll$O, 3S L. The only nonzero value of p, as solved in Reference 3 is p = l. k

The University of Michigan T Engineering Research Institute The constants to be determined are E and bl 2, given by the two equations 1 Mo,21:o.ii E Q b 2iN2 *_l,, 13 7Y Oil 2.-a 4 dw 2a 9iN0o2z dx 2MlI N2 O 11' E + Ql,2,1bl,2, 2iNoll = 0 where Qo,2, 1 = 2. n M r t+l Rr +, 2 = P j 7 W 355it Ql2,2, l = -_ n lMil;ra,+l, Rr"'+ll;lal + 2-a = P ZI 2Q! 13 2-a 4W 1JTg~ 2a The solutions are 2ca+ 15 _2 + 13 2-a 44 dw 2Ca 4/10t E:.. a 3 dx 2.. 12 20Jwx where we have written in such a way that the last factor is the correction to the simple theory, and b1, 2-a 4 bl 2l =1 = -.... 2 9 2iNo21 dw 1 dx 2-a- 3 2a-. i Using Equations 27 and 21b, the velocity distribution is given by C~~~~a 3 2-za 13 2a )4N07) dw 2o-a 2-a + 12 J dx 2ag 2a 4-Io;.2 _ dw 1 dx 2a.3 2a T e5 x -IaE e^Q \ The slope of the velocity at the wall is bigger than that at infinity by a factor 1 24 1 1 + 2a 6 2.a 5 2a 1 T v which is independent of the density and is equal to 1.08 for a = 1, The velocity at large x is 14

The University of Michigan (')x >> = CX Engineering Research Institute 2-a 1L 0- + 1 i +, _ + a 2-aC 12 dx 2CX 4,v giving for the "velocity jump" 24-a 1+ 3 2 —a 2-a 4IT dw (AW)w = a I 2-a 12 dx 2aQ i~ For a = 1, this gives the value 1.043 for the slip coefficient. at the wall is smaller than (Aw)w; it is The velocity ("E x=O 2-a + 11 2-a - 2a dw = Aw = -1 - -- a.22-a. 12 dx 2a 4 - dw = 0.957 2 d dx for a = 1. C. THIRD APPROXIMATION In this approximation, the functions *o 41, 2 2,.1,1 and'1 2,1 are added. The determinant A has two roots 0.7151 PI = k 1,.342 P2 - k After solving the set of three equations of Equation 24, one finds 2 — 2 44 a 3 0.945 + 3.259 dw dx 0.756 + 2..go906 2-a 2c- + 2.785 2a (2 2 <.R - + 2.785 2a (2.-a'\2 and cz = (x + k) d dx 2-a 21ir a 2-a 0 o,7151 dw 1 ((3^909 + 7.514 a- -)e dx 8l; 0,756 + 2. 90 xa -1. 342 + (1.447 + 2.5352 2)e; 2-( + 2.785 (-a)2 2CX 2!a 15

The University of Michigan * Engin Thus the slope of the velocity at the wall is leering Research Institute w r 2.g 1 4.737 +- 8771 2dw a.2-0 12 2 0 dx. a 12 0756 + 2.906 2- + 2.785 ( 2.a)2 2a 2a For a = 1, this slope is 1.262 dw/dx. The velocity at the wall and the "velocity jump" are easily deduced. For a = 1, they are (cz)x=O = 0.874 dx (Aw)w = 1,126 Q dw dx D. FOURTH APPROXIMATION In addition to the 4'"s in the third approximation we include also tr2,12,, 1,4,1' To,5,1.e The four values of p's are 0.4604 Pl = 1o. o65 E2 x 1.261'. P3 = - l. 610 9 = 4 = The following results are obtained: 2-.a 44 e _- - 3 a 3 dw 437 + dx 3.48 + 33.24 2- + 93.98 2.-a \2 it ) + 1168 2-a + 80 (16, - + 53,76 (. -) (2a./ 27,73 2, + 2a 82.71 ( )2 \2a V + 109o41 (,2 + 53 f276 = (x + kT) d dx a2 — Is-, x 6.6 + [2 6124 + + [26. 24 + dw 1 dx 1427 1 2 —a 2-a + 2 — 2 2-a 5.48 + 27.73 + 82. 7l' + og09.4l(.- 53. 5 76(. 2a \2a/'2a/ 22a I + 41.94 - + 88.55 2-a)2 2 0 O46o a~a j, V-OL a-a 161.72. + 2Ca 326,90 (2 21889 ( )5 + 218,89 2-a I e] e-1. o65 - + [1oll + 8.17 2,..a 1,8 ( 2 a -1.261 X.1978 (2c)2 + 15.76 a e 2-a + 2-a. 2 )6 + 1 2-a\3 -1,610' 2 — +.151,21 -- + 7 (-) I e + [11.98 + 75.61

The University of Michigan Engineering Research InstituteI The slope of the velocity at the wall is 2-a 51.68 + 323.57 + 657-31 (2 + 44291 ( )3 1 2- 22a 1 2o 22., dx u a 21 34. 27.73 + + 82.71 (l) + 109.41 ( + 5376 2a 2a 2-.2c, For a = 1, this is equal to 1.37 dw/dx, The corresponding velocity jumps are Aw = (z)xo = 0.852 T dx (Aw)w = 1.134 T d. dx It is of interest to compare our result with Welanderts value of (Aw)w = 1,21 1 dw/dx. IV. TEMPERATURE DISTRIBUTION NEAR A FIXED PLATE We consider the infinite space filled with a gas which has no mass flow but the temperature distribution of which is To(x) = To + tx, To being the temperature at x = 0 and t is a constant. It is assumed that the change of temperature over a mean free path is small i.e t/T is small An infinite plate at the temperature To and having the accommodation coefficient a is introduced in the plane x = 0. To find the new stationary state velocity distribution function f, we write again f = fo(l + h), where for the present purpose we choose for fo0 3/2 mv2 f no(x) (22mKJ x) e 2kTo(x), where v is the molecular velocity~ The number density no(x) and the temperature To(x) are related by the equation no(x)kTo(x) = constant. To the first order of approximation in Xt/To0 the Boltzmann integro-differential equation reads (c2 5) t +h n J(h), 2 To 6.x c 0, cx where, now both n and T can be treated as constant and replaced by no and To. The velocity c is the dimensionless velocity defined by 17

The University of Michigan T Engineering Research Institute C = -a V C 2kTo The boundary conditions are X 0 oo 2 (5 2 x = oo h = 2 ( c2)cx +: e + 2 ( - c ) (32a) T 2 2 2 x = 0, cx > 0 h(O;cxcy,cz) = B + (1 - c)h(O, - cxCycz). (32b) Equation 32a states that at distances far from the plate, h is composed of three terms, the ordinary heateflux term of Chapman and two other terms responsible for whatever change in density and temperature is brought about by the presence of the plate.One must expect that both Ce1 and E.2 are proportional to xt/Too At large distances, the constancy of the scalar pressure, i.e.,.nkT = constant, should again hold true, even though the presence of the plate might disturb this relation in the immediate neighborhood of the plate. As a consequence, -e and E2 are related, namely, ~ -= c2 o (33) Equation 32b is the boundary condition at the wall. The difference in form from Equation 5b of Section II is due to the fact that the density is now also a function of Xo The unknown constant B is related to the density of the gas near the plate. As in the previous section, we bring the integro-differential equation, Equation 31, to the homogeneous form aY - J(Y), (34) ax ex by introducing Y(x,c) defined by h = Y(x,c) + 2 ( c2)X + 2 ( c2) (35) T0 2 The corresponding boundary conditions become x = oo Y = 0 (36a),x = 0:

The University of Michigan T Engineering Research Institute Y-(O,cx,,cy~lc,) {=.B + (1 c)Y( - cxcy cz ) -(2 t (5 o2) c sign Cx - cae1 - a2 (3 - c2)2 g c2jl + sign( ) CX - sign cx 2 2 2 (36b) Introducing the Laplace transforms, making expansion in the eigenfunctions Vr L,~ and finally eliminating ar, 2, as in the previous section, one is led to the set of equations ) ('S2brr" SI tt - n2 ) Rr. 2+' r2 Rr 2; r +l) ar' +''" -r'. = sar,22l(O) + n Rr,2.+l*;r'2V ar2' (0), rV'Q (57) corresponding to Equation 14 in Section II. As consequences of the conservation laws,, Ro 1;r2 = R1,13;,2 = Rr,2;o, 1 0O Furthermore, for Maxwell molecules Rrs,21,l = ~ Y except for (r,2.) = (0,0) or (1,0)o One finds then easily the solutions ao 0 = const. (38a) a, I = const. (38b) ar,21.+1(x) _ _ \ Dr.21'+ ( = Pi) ePix.s =Pi i Dr, + (s = - Pi) e~PiX /dA ds - Pi (38c) TIt is clear in this problem only m = 0 is of interest; hence, in this section we will drop the index m altogether. 19

The University of Michigan * Engineering Research Institute ar,2 (x) = ar,2 (0) +t n 1 Rr 2r 2~,v+P —'*-1 Dr:+*(S =Pi) (epix 1) Rr.,2*r'p2 - (e - (e r i' i s = P a,1 =0(aPi -!- Dr'2~+l(s = Pi) iX 1 s = -Pi where the notations have the same meanings as before, iand the basic determinant A is A = / S2hrrt18i"n - n2 f Rr, I2s+Ert21 Rr2 t 21r^l" 21 / r.' The boundary condition at oo requires that ao., = 0 (39a) a, = 0 (39b) Dr,2I+I( = Pi) = 0 (39c) ar,(0) nb Rr12+ 2 r,+ 2 +10 Dr,0 r ++( s = - Pi) 0 9) r;' i (dS)s = Pi In the present case, with the same cut-off procedure, it is easily seen that Ne > No; hence, Equation 39c follows Equation 39d and the differential equations for the a's. The boundary condition at x = 0 gives the following relations: ar,2+.(0) + 2a,Mr,21+1r2, ar;2,r(O) 25 x + Mr,2z1;O,..r2 r*. 2 (40a) N11 To' r6 to 2-a+ Noo 2 - N 0 I i L - 20

The University of Michigan * Engineering Research Institute 1 ar,2~(O) + 2(2-a) Mr,2j;r.21,+1 art2,.+l(O) r.' =. 2(2-a) 1 2Xt -e+ 2 rlo (40b) C1 N ll T Mr.,.21l) +..... N' 00o As remarked in the previous section, Equations 40a and 40b follow from each other when all the eigenfunctions Vri are taken into account. Otherwise we have to make a choice as to which set of equations we want to be satisfied. In the present case, it seems natural to take Equation 40a. Introducing b() DOp3(s = P Pi) 7.3 - fd^ \ds. J s =- Pi making use of the relations between the different Dt1s and substituting the expressions of ar,21+l(O) and ar, 2(0) in terms of the b's into Equations 40a, we arrive at the following set of equations N0 r, 2+1MO,0 N o0 r22!+l; +1,i b +)= 2_ 1 2t ri o (41) i where Qo() _:,lo r y Mo,.,;r~2,[, Rr;,,'.r,'2, +..........) Ql,= n1 M r;r V 7 1 ( n2o1 i2 a"+1 rTV r"! 0 a;J I n M;,3,.0.13 WiZ+1 = nO 0 X M lr~2l' Rr.221'r,'",; ""+p_ ri)., r. Z-j Z-j "i 0 "oA3joAs-y P 2-a (+:noir2+1 _ 2a Ao.,3;r,2~+.r. L The Equations 11 are a set of No linear inhomogeneous equations for the N unknowns b i (No - 2 in number), B and 62 (and hence ~.,because of 35b)' Upon the solution of these equations, the complete distribution function is known: h E= 6+- (- c + — -S c cx + ar,(x)r,, r,~ - - 21

I The University of Michigan * Engineering Research Institute where all the coefficients are of the form epx. The physical quantities of interest are given by n = no(x) -l + I (i. + aoo(x') T = To(x) {l 2 Io + ao(xKT. L 3znl/2 o ( For large x, the a's all vanish, the temperature becomes a linear function of x, and the relation nkT = constant is satisfied. It will be seen that this is not the case for x ^ X. The temperature jump, AT, is given by AT = T(x =0) - To(x = 0) - ( ao(O) - The corresponding "temperature jump," (AT)w as defined by Welander, namely, the difference between the temperature at the wall obtained from a linear extrapolation of the temperature curve beyond the transition region and the temperature of the wall To is (AT)w = To [1 - E2] - To - 2 To To get some idea of the magnitude of these quantities, we give numerical results for some "successive approximation" calculations. A. FIRST APPROXIMATION:'o,, *1,1; oo' o l,0 o0,2 The obvious solutions are ao01 = a,1 = 0 aoo = a0 = aO2 - ~ Equations 37 become a ( - N- Mo1,i;oo - 0a,-,- o = ~ (LB Cil M E2 M 2-a 2Xt a G - _N 0o -af i,1o - N 2 TO giving 2a 5J_ xt Ei E2 = = a 4 To 22

The University of Michigan T Engineering Research Institute 1 The temperature distribution is linear throughout the entire space. The temperature jump is AT = (AT)w = Xt 4 2-a 15 T dTo(x) a 8 dx in agreement with the results of Maxwell and others. B. SECOND APPROXIMATION To the functions used in the first approximation, we now add o0,3y I1i2, and V2,oo The determinant A has one pair of roots, namely, p =~ 4. 4 p Equations 37 now consist of three equations: B Mo;o o. —pMo r r21r b N00 M,1;, Njo Mp,1;r,0 +2 - =, r~ r'.l 0 B.-ci 0 N00 - i2 Mi,1;,o + no N0o P r$i " MI, 1;r2r RrJ L0o3bo 3 2-ca Xt a TO b I BNo Mo 3;0o 0 Nooe - - Mo0 s;,o + Nlo 0,'7 Mojrt2 ~r'21o,3'2 ) (P 7 M~,; I.2I Rr;'to3 r ^'2 = 0 Together with Equation 29b, ~1 = 62 we find the following for the complete solution: 2-a t 29 2-a 59O %t 2a 0Jrt =._. a 4 T 2-a. 15 2a + 2ca 4x a, = a11 = a 1, = a2,0 = 0 a073 -.e_ x 2 —C l fT 3/4 t e K X > ='c 4 16 To 2-a+ 15 20 4+ 15 2Ca 4NJ~ 23

The University of Michigan * Engineering Research Institute -^Jx 2-c 1 3 3/4 At e 4 fox c 4 5 T, 2-a. 15 2c 44 T 2-a 1 3I/ alo = -4 - x xt e 4 A To 2a + 15 2c~ 4/4I15 ao72 ~rJ x 2- 4 3/4 NI e 4 % a 8 To 2*- 15 2Qa 44151 The number density and temperature are given by n = no(x) a 4 2-a 15 - 2 — a -5 + -+_._ L39' 2-a 2 9ax lc/1i ca 4 2-a 15,2Q 1 W 5 it j~Yx 2-qC + 15 To' 2a 4N/15t _ 2 %e-4e T4 x r - 1 To 2-a 15 |T 206 44 j15 iS T = To(x) 1 so that nkT = no(x)kTo(x) 1 - 2-a 1 c Jx1Ix 4 e Xt 2xa 15 T1 2a 4415ir 1 Hence, only when x >> k is the scalar pressure a constant. The temperature distribution is no longer linear in x; however, the deviation from the linearity is again appreciable only in a very small region where x is a few mean free paths. The quantity AT is given by 2 14 2-c 15 _ dTo(x) 2ax 4W AT = a 8 dx 2-a 15 2a 4x 24

The University of Michigan * Engineering Research Institute and (AT) = -a 15 -w a 8 2-a 39 dTo(x) 2a l+ dx 2-a 15 2aa 4cyi For a = 1, we have AT = (0.965) 18 T dT( 8 dx (AT)W = (1.021) 1 do(x) 8 dx giving a value of 1.021 x 15/8 for the slip coefficient defined by w/Y dTo(x) (AT)w - 0 C. THIRD APPROXIMATION Three more 2 ~1 and t1. The eigenfunctions are taken into account. following are the values of the p s: They are o04, o. 4798 P1 = 0.8344, 2 =.o 1.293 Pa- A The temperature distribution is found to be T = T(x) 1 + k... 2 c 4 T-oA2 a 4 To 2-2 1 2,05 + 11.65' -t -2a + 22. 09 + 135.96 2- + 1.96 (22-c + 13,96 Ja 2.63 + 13.92 2-a L 2 + 2429 ( ~ ~)~~C~ 22CX -1 ([11.49 + 44.01 2- + 4 + [0.71 + 2..73 -4 + [o.1 + o.536 2.- + 1.09 2a -2.04 (2.)2] e 2 264 ( ] e f 2.64 (5+)2 I e o,4798x - 0.8344x k.1. 293X (2-a)2] e129 2 j r-L'71f.2aO 25

The University of Michigan * Engineering Research Institute so that the temperature jumps are given by AT = 15 T dTo(x) 28 dx oa 1..61 + 9.96 2-c + 20.45 (2)2 + 13.96 (2)3 2a 2c 2a 2-cea- 2 2c 3-a2 2.05 + 11..65 2 + 22.09 (-) + 13596 ( ) 2 1 2a 29( + 1596 ()2a 2.63 + 13.92 2-c + 24.29 (2-c + (2 2a 2a 2 6 and (T)_. = 15 T dTo(x) 2 —a SW -- 8 dx a 2.05 + 11.6 2.05 + 2 +2209 ( )2 + 13*96 2a 2a.+ 396 (2.)3 22a The corresponding values for a = 1 are AT = (0.888) 5 T dT(x) 8 dx (AT)w = (1,150) 15 T dTo(x) 8 dx Welander obtained the result (AT)w = (1,152) 1.5 - dTo() 8 dx V. CONCLUDING REMARKS The outstanding question, which remains to be answered for this problem just as for the problems treated in References 2 and 3, is the question of the convergence of the successive approximation procedure. Practically speaking, the convergence seems quite rapid; only relatively small changes are produced by going beyond the first approximation which is identical with the classical resultso A second question is whether other ways of breaking off the infinite sets of equations, or other ways of selecting the boundary conditions to be satisfied, would lead to appreciably different results. If so, this might throw some light on the convergence question.* Finally, the effect of the choice of molecular models needs further investigation. *Dr. HI. M. Mott-Smith in the report, P"A New Approach in the Kinetic Theory of Gases," Mass. Inst, of Tech. Lincoln Lab. Group Report V-2, Dec., 1954, has investigated other selections of boundary conditions for the heat flow problem between parallel plates and for the Couette flow. 26

The University of Michigan * Engineering Research Institute APPENDIX I PROOF THAT Dr,2, (s = Pi) Ai2 1r,21 D1 2,1(s 1~ - i Let us consider the system of equations j (s25k bjk) lk k = a j and let s = + pi be the roots of the determinant A = /s2bjk - bjk / Then 0. ak = Dk(s = Pi) 1 dA.....s - Pi i \ds = = Pi Dk(s = - Pi) 1 2a, s /'dA\ s + p-i i \s) =- Pi where Dk is obtained from the determinant A by replacing the kth a s. Hence, Dk = amA m where Amk is the algebraic complement of the (mk) element of A. we have the relation column by the For s = + Pi, Amk: m = An Ane for all m, n, k, and l, Making use of this, Dk (S = + Pi).m am (s = P+ i) 6 (i) Ank ni (i )D ank (s = + i) where = Ank (s = Pi) 27.

The University of Michigan E Engineering Research Institute APPENDIX II PROOF OF THE PROPERTIES OF Mr,l,m;r^;m Mr, l,m;r lm - d- e-C2 1 + sign Cx,*r, m Mrl~-r~lf r*I - *rIIt where 1. Since Mr2,m;r 2~ r rm = NrmSll 2 (c ) c ePi(cos ) the same value of m appears in both ** and jr, it is clear that is symmetrical with respect to the two sets of indices. N27 1 2c r,2,m;r2m = i d dx dcc e 0 0 0 - ~ s* (C 0)a x,)r'm (cx,0) where x = cos G. Since *,2,m ir'2pRm is an even function of x, Mr,2I,m;r,2~m = 2 dec r,2,m *r;2'm 2 1r R 2rr by the orthonormal property of the I's. Similarly, Mr,21+lm;rr2' +lm -= 2 rr'6' 3. To find the sum 7 2+,m;r'2,'m 2t' rl t+ +,m r"' we first expand 1 ~ sign cx 28 +r,2~+,m 22 28

The University of M~ichigan Engineering Research Institute in terms of the complete set of eigenfunctions *r',m 1 ~ sign Cx 2 r bry22+lm;r;'m *r'llm r... where Thus, =C dceI 4 ~ ~ r';e~" _sign Cx br,2l+i,m;r.l'm d= fde C2 tr,2a+i,m'"} 2 m -2g" c + sign Cx 1 +!+1,m....2 2 Mr",2I+1 - M2+1,m;r2 "m.r;.2.Im I - 2 2..~$,1lm For all values of C, except Cx = 0, one can write *r,21+l.,m lSisgncx 1- 1 2r+im ltsigncx + Mr,2+l,m;r:22 nmrr,2.nm 2 2 IrQ 2 r' Ie or *r 2+4l,i^m 1 ~ sign cx = + 2 C Mr,2+l,.tr2 m 2 1 sign cx 2 L ~.., r. -'...22,m. 2 rj" rtl It (A) Similarly, one finds 1 ~ sign ex'r,22,m 2 = ~2 7Mr22.rt+m 4rrt 1 + sign cx I-2 -Mr,2~,m;-r'f2:+ im r.21'+,m.' " I. I'v (B)'By substituting Equation B into Equation A it follows that.r,21+lm = 4 t Mr.l.+l mVr'.21: m Mr? t.2tm;r, aa'tl+m *r 21 +l.m ry Hence, M 1 j Mr,21+,lm;r tm Mrt2Y.m;r'2+l',m -= 4 rr,,l t6"'r^ and similarly r41 r~~~~~~~~~l'~~~~~~ 29

I The University of Michigan * Engineering Research Institute APPENDIX III Rr,2,m;r I.Im -= lt r dee'c2 X 4 1m'* ~r c rx = \r t,Lr;J,m;r;Q.m where Lr,I,m;r;fem is symmetrical with the two sets of indices A. m = 0. Table of Lr;rt~' ~r,!r ~0,0 0 0,2 1,2 2,0 0,4 2 1,1 2 0.0 0 0 ~~~1,1 ~~55 0,33 o* 00 0 2,1 4 2 0 0 0- ~ 1 37 350 B. m = 1. Table of Lr,Al;r;2i1 rt1i 0,1,1 1,1, 053,1 2,1,1 1,3,1 0,5,1 0,2,1 72 0 0 0 0 0 1,2,1, 0 0 0 0 2,2,1 7 57 5 2.$21 o o o 1_4,1 -8.2. 44 1 8 - 0 231' 55'~11 ~~~ 50

L Th~e Univers~ity of Michigan * Engineering Research Institute APPENDIX IV m = -c2d 1 + sign Cx.,^ S = / dce _,m 2 - fJ* 2r,i,m r.!'m A. Mry,2 Im;r 2 rra m L r..M 1~;1T 11'9 P. 2 Mr,2l+i,m;r 2 t'+lm B. Table of Mr,;ri, *, 1 2 brr, 81Q (m = 0) 0,0 1,0 0,,2 1,2 2,0 r 522l2 l-t 4_15ic 71 21 1 11 2 17 1 1 1 - 1 1 3 13 2 1 _22 157 2 2,13 __ 1 112 3_ 5 2 1,5.. 511'_ C. Table of Mr, I,;r.~i1 0,7171 11,1 0,*3tl 21"i1,3, 051 0,2,1 1,2,1 1 ~12x 1 21; 1 125it 1 11 2^5 1 4:7 1/70: 1 44J3 1 3/o, 25 28s10x 1 1 1 6 7431 3 14F/5301 4 - WSl35: 2,2,1 7 167 4- 2.-U11 ~~~2P701L3F~~~~~~~~~~~~~ 1 63701 31 851 111 87 2&1l65 t 11 12aJig 19J1 2,1770 7 1 7/T

The University of Michigan T Engineering Research Institute REFERENCES 1. A. Kundt and E. Warburg, Ann. Phys. und Chemie, 155, 337, 525 (1875). 2. J. C. Maxwell, Scientific Papers, II. Cambridge Univ. Press, 1890, pp. 26-78, 681-712; M. von Smoluchowski, Akad. Wis Wis Wien., 107, 304 (1898) and 108, 5 (1899); H. Grad, Commun. Pure Appl. Math., 2, 331 (1949). 3. C. S. Wang Chang and G. E. Uhlenbeck, "The Heat Transport Between Two Parallel Plates as a.Function of the Knudsen Number," Project M999, Eng. Res. Inst., Univ. of Mich,, Sept., 19535 "The Couette Flow Between Two Parallel Plates as a Function of the Knudsen Number," Project 1999, Eng. Res. Inst., Univ. of Mich., June, 1954. 4. H. A. Kramers, Nuovo Cimento, 6, supplo, 297 (1949). 5- P. Welander, Arkiv. for Fysik, 7, 507 (1954). 6. S. Chapman and T. W. Cowling, The Mathematical Theory of Non-uniform Gases. Cambridge Univ. Press, 1952. IP J - 32