ERRATA SHEET Page 5, line 2 below (1/T)(5q/t): omit the word "not." Page 8, line 2 of second paragraph: replace "bt/bt" by "lbq/6t. " Page 9, line 2 of second paragraph: insert "it" after "finite." Page 11, line 2 above Eq. (2.42): replace "k" by "s." Page 14, lines 4, 5, and 6 from bottom of page: replace "K" by "K." Page 19, line 2 above Eq. (4.23a): replace "(To+e2)3" by "(To+~) 3 it Page 20, Eq. (4.25b): replace "at" by "ate." Page 25, Eq. (4.45a): replace "iwk" by "icuK." Page 26, Eq. (4.46b): replace "k" by "k2.! Page 33, Ref. 10: replace "Landan" by'Landau."

THE U N IV E R S I T Y O F M I C H I G A N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Progress Report NON-EQUILIBRIUM HYDRODYNAMICS OF A CHEMICALLY REACTING FLUID Chi Yuan ORA Project 05424 under contract with: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH CONTRACT NO. AF-AFOSR-20-63 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1963

TABLE OF CONTENTS Page ABSTRACT v 1. INTRODUCTION 1 2. NON-EQUILIBRIUM HYDRODYNAMICS OF A CHEMICALLY REACTING FLUID 2 3. FUNDAMENTAL EQUATIONS 13 4. SMALL PERTURBATION THEORY 15 5. ACKNOWLEDGMENT 32 6. REFERENCES 33 iii

ABSTRACT In this paper, two topics are studied: (1) the thermodynamics (and hence the energy relations) for a chemically reacting fluid with heat of reaction effects; (2) the upper and lower limiting speeds of "linearly perturbed motions" in the fluid. The first of the above leads to the study of two types of relations: (1) the usual extent of reaction equation; (2) linear phenomenological equations for the heat flux and rate of reaction in terms of the affinity and temperature (DeGroot, S. R. and Mazur, P., Non-equilibrium Thermodynamics, North Holland Publishing Co. (1962)). It is shown that if heat of reaction is neglected and proper assumptions are made on the affinity then the non-equilibrium energy relations of L. J. F. Broer (Characteristics of the Equations of Motion of a Chemically Reacting Gas, Journal of Fluid Mechanics, 4, 276-282 (1958)) are obtained. The energy relations of E. V. Stupochenko and I. P. Stakhanov (The Equations of Relaxation Hydrodynamics, Soviet Phys-Doklady, 4, 782-785 (1960)) are obtained by neglecting the heat of reaction, further, their K is shown to be a function of the thermodynamic variables. For the limit speeds, the following results are obtained: (1) in the low frequency case, the limit speed coincides with that of Broer, StupochenkoStakhanov and is independent of heat of reaction; (2) in the high frequency case, the limit speed is independent of the chemical reaction. An explanation of this result has been given by Broer in the above cited paper.

1. INTRODUCTION The purpose of relaxation hydrodynamics is to study the behavior of a fluid while chemical reactions are taking place.- The state of such a fluid will no longer be specified by any two ordinary thermodynamical parameters. In addition, the extent of reaction, i, which characterizes the chemical composition of the fluid (see Section 2) must be known. The introduction of this new independent thermodynamical parameter, ~, into the fluid system leads to a problem which is quite different from the usual one. If fact, a new equation, the equation of reacting rate, is added to the system of relations for the conventional hydrodynamics. The small perturbation theory of this new system for a compressible, inviscid, non-heat-conducting gas was first treated by Broer in 1957.1 Later, Stupochenko and Stakhanov, using a different equation of reaction rate, obtained a similar result for some reacting gas.2 In both of the above papers the authors show that there exists two limiting speeds, corresponding to high and low frequencies. In 1962, Coburn used non-linear theory to extend the above results to magneto-hydrodynamics and another limiting speed was shown to exist.3 In the present work, we shall not introduce the electro-magnetic effects but we shall generalize the work of Broer and Stupochenko and Stakhanov to the case of heat-conducting fluid. First we shall give a general survey of non-equilibrium thermodynamics in order to formulate the fundamental equations for the problem. Then we shall relate this system of fundamental equations to the systems considered by Broer and by Stupochenko and Stakhanov. The small perturbation theory for this system will be carried out in detail. The effect of heat-conduction will be emphasized. We shall show that in the presence of heat-conduction, two limiting speeds exist. The limiting speed of low frequency will remain the same as in the non-heat-conduction case, but the limiting speed of high frequency will be different.

20 NON-EQUILIBRIUM HYDRODYNAMICS OF A CHEMICALLY REACTING FLUID In thermodynamics, the non-equilibrium process does not refer to chemical reactions alone, In a much broader sense, it refers to all irreversible processes. In this work, however, we shall be concerned only with two non-equilibrium processes: first, those which involve simple chemical reaction; and secondly, those which involve heat-conduction. Moreover, we shall assume that the medium of the system is a fluid. This fluid contains m chemical components, denoted by al...am, where m is an integer greater than one. A simple chemical reaction can be expressed by the chemical relation -v1al-..o-vja, + v+;La2+l1+...+vmam where vk(k = 1,2,.. m) are the stoichiometric coefficients in the chemical reactiono As an example, we take the reaction 2H2 + 02 + 2H20 In this case, v = -2, v = =, Vm =V3w2 29 al = H2, a2 = 02, and a3 = H20. Let p be the local density of the fluid, pk be the local density of the ak component, and nk be the number of moles of the ak component per unit mass. Now, we shall follow a system of fluid particles along their paths. As a result of conservation of mass, this system of particles is closed. That is, exchange of energy through the boundary is allowed; matter is conserved. The rate of increase of nk along a fluid particle is denoted by dnk/dt, where d/dt is the substantial or total derivative. Using the law of definite proportion in a chemical reaction4, we find that dnk/dt are proportional to the stoichiometric coefficients, vk. dnl dn2 dnm =: V2 * o. m V1: V ~2 V ~~(2.la) dt dt dt Thus, p dnk/dt, the rate of increase of nk per unit volume, are proportional to vk or

dn1 dn2 dnm P dt P Pt t vl 2 * e - vv (2.lb) Here, we shall introduce two important quantities (Ref. 4, p. 4-5) in nonequilibrium thermodynamics. The rate of reaction, V, is defined by dnl dn2 dnm P dt P dt P dt V (2.1c) V1 V2 Vm The extent of reaction i, is defined by d _ ~V (2.2) dt By substituting (2.2) into (2.1c), we find dnk k d (2 (2.ld) dt. p dt where k = 1,2...m In ordinary thermodynamics, the state of a system is completely specified by two thermodynamical parameters. But in non-equilibrium thermodynamics, the chemical composition will also change with time and position in a system. Thus, in order to determine the state of a fluid in such a system, the chemical composition which is characterized by nk must be known. Let$'be any fluid path in the physical space, initiated from a point Po at time to to a point P at time t. This path,cdf, can be expressed in the following parametric form: xk = ~k(t) k = 1,2,3. (2.3) where ~k are some given functions with continuous derivatives, i.e., of class C! and

a 0#0 k=l in the interval to to t. If we integrate Eq. (2.1d) along this path, we find that t t pdt/ p(k(t),t) Note that we removed vk from the integral because vk are known constants. From (2.4) it is seen that nk depend on the extent of reaction, i, as well as the ordinary thermodynamical parameter, p, the density. Thus, the state of the fluid depends on'any two ordinary thermodynamical parameters and its chemical composition nk, or can be characterized by any two ordinary thermodynamical parameters and its extent of reaction, ~. Again, we follow a system of fluid particles and consider the system to be closed as it moves. The first law in non-equilibrium thermodynamics states that the rate of change of the internal energy per unit mass, e, along a fluid path will be de 5q dT P (2.5) dt =t dt where bq/St represents the rate of heat flowing into the system; p is the pressure; and T is the specific volume (Ref.: 5, Chap. II,III). Note that bq is not an exact differential. It depends on the path chosen in the space spanned by three independent non-equilibrium thermodynamical parameters. The second law of non-equilibrium thermodynamics states that there exists an entropy function s associated with the state of a fluid such that the rate of change of s following a fluid particle satisfies the following relation: T d = + b5q (2.6) dt 6t 6t where T is the absolute temperature and 6q'/bt is the uncompensated rate of heat flow per unit mass due to chemical reaction (Ref.. 5, p. 34). Again, 5q' is not an exact differential. But the quantity 1 bq+ bqt 4t

is an exact derivative which is ds'/dt by (2.6). In a reversible process, bq'/St = O, while in an irreversible process 5q'/bt > O. It is worthwhile to note that in reversible thermodynamics, the quantity T t is an exact derivative.7 However, the second law of non-equilibrium thermodynamics does not imply that the quantity 1 '( ) T \st t must also be an exact derivative. Eliminating 6q/St from (2.5) by using (2.6), we obtain ds 1 de + p dT + q (27a) dt T dt T dt T 6t Since s is a function of e, T and nk, we find by taking the time derivative of s along the fluid path, m ds = s de UsA dT + as dnk dt eTkdt enkdt, e,,nl... nk-lk+l- ~nm nkdt.k=l (2.8) Now, e and T are independent of nk. By equating the right hand sides of Eqs. (2.7a) and (2.8), and letting e and s be constant, we obtain m 1 C V as__dnk T bt nk dt2.9) 1b e~,T;nl)... nk- nk+l.~ ~n (2.9) Then, by (2.9), the relation (2.74 can be written as m ds =1 de + p dT dk (2.7b) dt T dt T dt T dt 1 where ~Lk is called the chemical potential of the ak component,4 6 and is defined by

Ssti e, T; nl... nk- lnk+lt ~.nm (2,1) Substituting the value of dnk/dt given by (2.1d) into (2.7b), we have ds = 1 de + p dT + A d. (2.7c) dt T dt T dt Tp dt where m A - PIkvk (2.11) k=l A is the affinity of De Donder (Ref. 4, p. 3-24). Equation (2.7c) may be written as de = Tds dT- A d5 (27d) dt dt dt p dt Now, we shall consider e to be a function of s, T and i, and form the time derivative of e along a fluid particle path. We obtain de ae ds + e\ dTr + ae d (2.12) t s t 6Sdt Tdt st Js,Tdt where s, T and 5 are taken as independent variables. By eliminating de/dt from (2.7d) by using (2.12) and letting s and T be constant, we obtain A = - p (2.13) The affinity A plays an important role in non-equilibrium thermodynamics involving chemical reactions, since A is a controlling factor of the rate of chemical reaction. It has been shown (Rgf. 4, p. 40) that the necessary and sufficient condition for chemical equilibrium, i.e., no chemical reaction, is A = O (2.14)

As p is always positive, this last condition and (2.13) will imply -) = o (2.15) S,T for chemical equilibrium. Next, we shall derive the equation for entropy production.5 Let V be some region in the fluid at time to. When the time changes from to to t, the shape and the position of this region may be changed, but its mass remains the same, as a consequence of conservation of mass. That is, the region forms a closed system. At any instant, the total entropy, S, in V is determined by S ' psdT (2.16) where p, s and T are the density, the entropy per unit mass, and specific volume as before. The rate of change of S as V is moving along, will be solely due to the entropy flowing in through the boundary and the entropy produced from chemical reaction inside the region. This can be expressed by dS d i dtS = d- psdT - J Uda + odT, i = 1,2,3 (2.17) V z V Here, J is the component of the entropy flux in xi direction, where xi are rectangular Cartesian coordinates; a is the entropy production per unit volume per unit time; Z is the surface of the region; da is the area element on the surface; and Ui is the unit normal vector on the surface. Note that we use superscripts, e.g., J', to denote contravariant components, and subscripts, e.g., Ui, to denote covariant components, even though both quantities coincide in rectangular Cartesian coordinates. The Einstein summation convention of summing on a repeated upper and lower index in the range from one to three is also adopted. Conservation of mass of a given region in time leads to d pdT 0 O (2.18) dt V Therefore, Eq. (2.17) reduces to (Ref. 8, Art. 2)

p ~ dT = -J JiUi da + ydT. (2.19) V z V Using Guass' theorem and the fact that V is arbitrary, we have ds - iJi + a (2.20) dt where axi Equation (2.20) actually states that the rate of change of entropy in the system can be split into two parts: (1) entropy due to interaction with the exterior surroundings; and (2) entropy due to an irreversible process within the system. These two rates of change correspond to -6iJi and a respectively. Now, we shall introduce the heat flux per unit area, denoted by Wi, into the energy Eq. (2.5). We have ST/bt as the rate of flow of heat per unit mass along a particle path. Thus, for the region V, the rate of total heat flowing in through the boundary at a certain time can be expressed in the following equation: -bQ = P 852 dT (2.21) Tt V where 6Q/6t is the total rate of heat flowing into V.through the boundary of region V. By definition of 5Q/6t and Wi, it is clear that eQ = -_A WiUida (2.22) St Then, using Gauss' theorem and the fact that V is arbitrary as before, we find that (2.22) implies P - = Jiw i (2.23) Substituting the above value of bq/6t into (2.5) gives

de 1 dT de 1 aiw P dt (2.24) dt P dt By eliminating de/dt from (2.7d) by use of (2.24), we find Ptds 1 iwi + A d (2.25a) or equivalently, ds _ dP = - - Wi _ + A V (2.25b) dt T\T / T' T Now, let us eliminate ds/dt from (2.20) by use of (2.25b). We obtain - aiJ + a = +- i - T (2.26) A sufficient condition for the validity of the relation (2.26) is i i W Ji = __ (2.27) a w-i 6iT A - (2.28) In many fields of physics, when the universe is considered to be finite, can be shown that Eqs. (2.27) and (2.28) are also necessary conditions for the validity of (2.26) (Ref. 9, p. 265). The usual argument involves two assumptions. First, we note that by integrating (2.26) over the whole universe, (, we have - ~DiJidT + jdT = - i KdT + (Wi T + A dT (2.29) 2 8\T T2 T Then, by use of Guass' theorem, Eq. (2.29) can be written as - 'ia dT ~- Wi A (2.30) 9 ITT. A TT.A- 'Ti 6 i-T i-T A O -zn~

where 'is the surface of. The first assumption states that the fluxes Ji and Wi are zero oni, which is the surface of the finite region R, the universe. Hence, JiUda = 0 (2.31) Wi T Ui da = 0 (2.52) Thus, (2.30) reduces to ocdT = + ( rdT + T dT or9 i mit _ AT dT = O (2.33b) T T Note that (2.33b) is a weak form of (2.28). The second assumption states that this weak form (2.33b) implies (2.28). It is then clear that (2.27) will follow directly from (2.26) and (2.28). Thus, these two assumptions will enable us to obtain (2.27) and (2.28) directly from (2.26). Equation (2.28) has proved to be very useful in non-equilibrium thermodynamics. 5,6 The right side of this equation is a sum of the product of Wi with aiT/T2, and of A/T with V. It is well known in the classical theory of heat conduction that the heat flux, W' is, in general, primarily dependent on the temperature gradient, biT. Fourier expressed this idea in a linear formula for heat conduction, namely Wi = KiT (2.34) where K is the coefficient of heat conductivity. It has been noted in physical chemistry that the rgtg of reaction, V, is dependent on the affinity, A, for many processes.' A linear relation between A and V is often assumed (Ref. 6, p. 291, 4, Chap. IV) namely, VT = WA (2.35) 10

where A is a function of the state of a fluid. More general linear relations of this kind, known as phonomenological relations, are extensively used in non-equilibrium thermodynamics.5 These relations are more general because they consider the possible interaction between heat flux Wi and affinity A, and between rate of reaction V and temperature gradient siT. In our case, these relations will be assumed to be Wk = aki biT + ak4 A (2.36) T2 T V = i iaZ+ a- -. (2.57) T2 T where kie constant ki = 1,2,3 (Ref. 5, Chap. IV). As the fluid is considered to be isotropic, the relations (2.36) and (2.37) can be simplified by Curie's principle, which states that (Ref. 5, Chap. VI) ak a4 k O0 (2.38) Besides, Onsager's symmetric principle gives some more information on the coefficients aki. It states that aki = aik (2.39) Thus, the phonomenological relations for relaxation hydrodynamics with heat conduction will take the following forms: wk = aki bi, aki = aik (2.40) T2 V a44' AT (2.41) By comparing (2.40), (2.41) with (2.34), (2.35) we see that k, A are related to aki and a44 by following relations: k gijaik(2.42) 11j =T2 11

- a (2.43) where Sk, is the Kronecker delta, and gij is the fundamental covariant tensor of the xi system. 12

3. FUNDAMENTAL EQUATIONS In the previous section, we have obtained the energy equation for a fluid in which chemical reaction and heat conduction occur. This equation is (see Eqs. (2.25a) and (2.21)) Pds iwi + (3.1) ~dt T T The heat flux Wi and reaction rate V of (3.1) are governed by two phenomenological relations: e = aik 6kT (3.2) T2 V = 44A (3 3) The equation of continuity and the equation of motion remain essentially the same except that the density, p, and the pressure, p, are functions of 5 and of any two other thermodynamical parameters (RefS'.1,2,5, Chap. III). Thus, the continuity relation is atp + aj(pvj) = 0 (3.4) The equations of motion are at(pvk) + aj(Pvjvk+53P) = 0 (3.5) where t = T Note that vk are the rectangular Cartesian components of velocity, and Sk is the Kronecker delta. 13

Equations (3.1)-(3.5) and (2.2) will be the fundamental equations of our problem. By eliminating V from this system by use of (2.2), the above equations form a system of nine equations in the nine unknowns Ti, v' P s and W. We shall first show that this system will reduce to the systems used by Broer1 and by Stupochenko and Stankhanov2, when the effect of heatconduction is.excluded. Substituting the value of A in (2.13) into (3.3), we obtain a4V = Pe (3.6) Since we are considering the non-heat-conduction case, aik in (3.2) are assumed to be zero. Equation (3.1) will, in turn, reduce to following form: ds A (7a) P dt TV (37a) By replacing A by (2.13) and using the definition of V (2.2), we obtain from (3.7a) ds __1e d_ ~~~ds ltae ~ d5 (3. 7b) dt T\ dt Here, we take, p,s and ~ as independent thermodynamical parameters. Equations (3.4)-(3.6) and (3.7b) form a system which was treated by Stupochenko and Stankhanov2 for 44 K = T where K is the coefficient in their phonomenological relation, V = - The function K was not specified in their paper because in the small perturbation theory it would not influence the results for limiting speeds (see Section 4). In order to get Broer's system of equations, we need to expand A in 14

Taylor series with respect to S,_assuming p and s constant, about the equilibrium value of reaction, ~. The series is A = A(p,s,Y) - p e-( O + P 1 -. (5.8) Note that A(p,s,A) is zero, since 5 is the value of ~ in equilibrium state. Therefore, Eq. (3.8) can be written as A = - p (-~) +... (3-9) In Broer's work,l a quasi-equilibrium state is assumed. Hence, (-F) k = 2,3..., is neglected. Thus, to a first approximation, we have A = - P(- (~ (3-10) Substituting the value of A from (3.10) into (3.3), we have d.... = _ a44p (2e, (_- ) (3.11) dt T K T2/i If we let T =a4\ a2i (3.12) T 2 (3.11) will become Broer's phenomenological relation. d - _ _a(_-) dt The function a was not specified in Broer's paper since the limiting cases of high and low frequencies are independent of a (Ref. 1, p. 278). In the following section we shall develop the small perturbation theory for the system of Eqs. (3.1)-(3.5) and (2.2) in order to see how the effect of heat conduction will enter into our problem. 15

4. SMALL PERTURBATION THEORY By eliminating Wj of (3.1) by use of: (3.2), and noting (3.4) and (3.5), we obtain four equations which will be the basic relations for small perturbation theory. They are: the equation of continuity, the equation of motion, the energy equation involving heat conduction, and the phenomenological relation for a simple chemical reaction: atp + aj(pvJ) = 0 (4.1) at(pvk) + aj(pvJvk+5ip) (4.2) ik a ik t j T3 akaiT + T4 (i T) ((4T) + T (Ot~+vJ6j) + j = aT A (4.44) Applying the standard method of small perturbation theory, we assume that: (1) the velocity components, vJ are of the first order in the perturbation scalar which is denoted by e; (2) the state of the fluid is slightly displaced from the equilibrium state, characterized by po,so and oe. Mathematically, these conditions can be formulated as follows: vJ -=,Ev (4.5) P - o + ep (4.6) s - so + es (47) 0 - + Et (4.8) where p,si are the three independent thermodynamical parameters which we shall use in this section; vJ, p, s, and e are the variable parts of vel16

ocity, density, entropy and extent of reaction, respectively, in the above equations. Since the pressure, p, and temperature, T, and the affinity, A, are functions of p, s, and i, we can expand them in Taylor series, with respect to these three independent variables about the equilibrium state, specified by po, so and ~O' We obtain for the zero and first order terms p = p(p,s,P) = Po + EP + Es + - T (4.9) T T(ps,) = T, + a. T P - ET - (4.10) A = A(p,s,t) = A0+ 7 + a Aa -O (4.11a) where Po _ P(pOsOSO) To - T(PoSoAO) A = A(PoSo, 'o) A -- = ~P (PoSo,9 o), etc. apo ap Since poXo,so are assumed to be the values of p,4,s, respectively, at equilibrium, it follows from (2.9) that Ao must.vanish. By differentiation of (2.8), we find -Ape. p, 2e (4.12) oA h2e -. p a (4.13) 6s b6bs -A _ 2e -A - p e (4.14) As p is always positive, the vanishing Ao (see (2.13)) implies that 17

(e) O (4.15) in the equilibrium state. Using relations (4.12)-(4.15) and neglecting higher order terms, we can rewrite (4.11a) as follows: _Ne N - a22ee -- e~ -- A = _E p E s p (4.11b) Differentiating (4.9) and (4.10) with respect to rectangular Cartesian coordinates, xj, we obtain the following relations akp E oakP + E aP aks + e )ak (4.16) Co a)o aTo (417) 6ikT 6aikP + E as iks + e ia (4.1)k Now, we shall use (4.5)-(4.18) to linearize the fundamental equations (4.1)-(4.4). Substituting (4.5) and (4.6) into the continuity equation (4.1), we find that the first order terms multiplying E will lead to the equation _-j atp + P0jv = 0. (4o19a) Similarly, by using (4.5), (4.6) and (4.16), we can write (4.2) as at(~PoVk+2 PVk) + aj(e2PovJvk+e3pvJvk) af - aks f E,,$-ap i amp) -(4.~19b) C)o p 6p6o When the terms involving e2 and E3 are neglected, (4.19b) reduces to PnOatvk + 7 kp - + 7 +a = ~ k (4.20a) In the same manner, Eq. (4.3) can be linearized by substituting (4.7), (4.8), 18

(4.17) and (4.18) into it. First, we note that by substituting (4.6)-(4.8), (4.17) and (4.18) into (4.3), we obtain E(po+EP)3ts + E2(PO+EP)vJajs aik c4_i.~kp + - 1'T i +T ~i } _(T a )3 aiakP + a aiaks + akA aiak+o 1a + a2aa oak poL + + a a + a is (T +E:P) 0 0 6 0 P0 6 0 '0 T0+E1 al 2e 2e)- e - - _ (4.21) To+EPby o W) P - 0 o aaS o a } (4,21 where = aT + a s + - (4.22) If we multiple (4.21) by (To+C2)3 and retain only the terms of first order in E, we obtain the following linear equation + As + T.PoatS + a = 0 (4.23a) where = aikaiak. (4.24) Replacing i, vj, T, and A in (4.4) by means of (4.5), (4.8), (4.10) and (4.11b), respectively, we obtain + = a J 2e - 2e - 2e Etk + V vj TE+~~ v Po P - Po a S/o - Po /o ti >E TO+E iP 0 ) s Po If we multiply this last equation by To+cE (where D is defined by (4.22)) and retain only terms of first order in c, we find 19

a44po a2e P a44po a2e s + a> + a44P e = o (4.25a) To CP To 0 o To To 6 The system of Eqs. (4.19a), (4.20a), (4.23a) and (4.25a) is the linearized form of the basic Eqs. (4.1)-(4.4). Now, noting that all the coefficients in each of the equations are values taken at the equilibrium state, specified by po, so, and toj we will drop the parenthesis followed by the subscript zero and the subscript zero from the coefficients in the rest of this work. Thus, when we write ap P, we shall mean apn to. In this notation, the linearized system becomes atP + p6jvj = 0 (4.19b) P)6tvk + p akP + ks +W aYE = k (4.20b) A Tp + T- As + T3pats + -T A = 0 (4.23b) K p2e p + K + + K = 0 (4.25b) where 44 K -a" T Now, we shall eliminate vJ and vj from (4.19b) and (4.20b). Taking the partial derivative of (4.19b) with respect to time, we find since p is constant 20

atatP + atajvj = 0 (4.19c) Then, we form the divergence of (4.20b) and obtain pgik ika + s gika-ki S + iki0 (420c) (4.2oc) where gik are the contravariant components of the metric tensor. Since gik is independent of time and ajgik vanishes, it is clear from (4.19c) that pgjkatajvk = -_tatP (4.19d) Thus, by eliminating patOjvj from (4.20c) by use of (4.19d), we find -,ttP + 62P Ap + PE As + _ p A - = 0 (4.26) where ott a2 A - gJ aj _ gij (4.27) We shall consider (4.23b), (4.25b) and (4.26) in our future work. It is convenient to write these equations in the matrix form -6tt + A a2e 2e t + K =2 t s =o (4.28) K ~ ~ t+K~y s K 0(4.28) 6TT- TT - 3 6 - -A\5 a s A + Pt A 6a~P as a~t In the following work, we shall be only concerned with monochromatic waves. We assume that each of the quantities, p,s,S is a constant multiplied by e-i('Dt-bkXk), where cc is the frequency, bk is a component of wave 21

vector, and X and bk may be complex.10 Thus, we write p = 'p exp i(cut-bkxk) s = 's exp i(cwt-bkxk) (4.29) = '~ exp i(wt-bkxk) where 'p, 's, 's are constant. Therefore, we find that derivatives of p,s,) are (see (4.24) and (4.27)) atP = - ip, = - i= s, atS = - iwo; (4.30) attP = _- 2p, ats = 2 att = _ 2; (4.31) Ap = - Jp, As = - Js, (4.32) aT =- JT, where J - bkbk; (4.33) and Ap = - Ip As = - Is (4.34) a = - I where I - aikbibk. (4.535) 22

By use of (4.30)-(4.32) and (4.34), the matrix equation (4.28) can be written in terms of c, I, and J as follows: /2 ap J P aP J -P J P 62e 62e 62e _ aK2e - ic + XK s = 0 (4.36) \ T I - T3pic -a I In order that a non-trivial solution of Eq. (4.32) exist, the following condition must be satisfied. 2 - pJ _ apJ _ apJ de K ae K e - i + e = 0 (4.37) _ T I aT I- Tapi aIT Expanding this determinant and arranging the terms in ascending powers of w, we obtain ap a2e aT ap a2e 6T 62e ap 6T aE aT a2e \ap a6as a6 a+ T,6p as a+5 as ap a6 ap a as -p a2e aT T a. ~.2e IJ:K + ( i aT + ~p siJoi ap a62 as a6 as a6ap as ap ap a6 p 62 ap aewcKJT pi (4.38) ( 2e K T 62e T + KI aa a aT + I - T3pJ (o\ 6- KPT3 a e)u 3i + T3po4 We note that the portion in the parenthesis of the first term is just the following Jacobian 23

p a62e aT ap a2e aT a2e p aT _ aT a2e_ ap a6as a6 ap a6ap as a62 as ap a6 ap a6as -a p a2e aT _a T ap 62e 6(= )P, ap a-2 as a a aapj (p,)s,) and the portion in the parenthesis of the second term is another Jacobian ap TapT = a(p,T) (4.40) as ap ap as a(p,s) Thus, Eq. (4.38) can be written in the simpler form p,T - K PT IJ + i (P IJW 6(psA) 6(p,s) + i P a2e ap 2e pCUKpT3J 60~p 6P 6/1 + (KI e T e 2 (4.41) + i.(~2e Ip _ 3 + T3 p4 = 0 In order to study the properties of small perturbed wave motions governed by (4.41), we shall consider the case of plane waves. In such a case, T.,T,. and Vk depend on only xl = x and t. Thus, (4.33) and (4.35) become J = b'b k2 (4.42) I = allbblb - allk2 (4.43) and k is the wave number.10 Hence, Eq. (4.41) becomes 24

- K (i,) llk4 + iall (P,T) k4 a(p,s,C,) 6(p,s) + i Sapa2e a.-p a2ek2WKpT3 +2e 3 L + r ~ 2e~T~1Tea ~ T 2p k + iKp _ a3iT lk2w3 + T3pS4 = 0 Equation (4~44) is a quadratic relation in k2 with complex coefficients. For any given o, we can find two roots of k2. It is important to note in our future work that the following two limiting values of wu,k will be studied: (1) ow approaches zero and k approaches zero; (2) w approaches infinity as k approaches infinity. We assume al1 is finite and shall discuss the solutions of (4.44) in the following three cases. Case 1: all = 0 In view of (3.2), we note that all = 0 means no heat conduction. Equation (4.44) will reduce to ap Wo2k2 + 4 + iok p ae - a e + 2 = (4.45a) 6P 60P 6p 62/ or in an alternative form, ~2 i p2e 6p 62e 6 2ew20 P- atko + iK + a2e 0e = o (4.45b) The two limiting speeds obtained by Broer,' and Stupochenko, Stakhanov,2 can be found from (4.45b). First, when o approaches zero, but w2/k2 remains finite, (4.45b) becomes 6p a2e ap 62e + 2e 2 (4.46a) t ea6p 6p 6a2 6a2 k2

Or, ap a2e ku 6P 2 eP (4.46b) k 2e where X2/k2 is the square of the speed of propagation.10 A second limiting speed can be obtained by allowing w to approach infinity but w2/k2 to remain finite. In this case, Eq. (4.45b) reduces to (4.47) which is the square of the ordinary sonic speed. Case 2: K = 0 In view of (3.3) and (4.25c), it is clear that K = 0 means that no chemical reaction is taking place. In this case (4.44) becomes icuall (pT) k4 - Ca2k2 aT + VP (2-k2 aP 2 = (4.48) L(ps) Dividing this equation by U/T2 and multiplying it by -i, we obtain ai (P,T) k4 - all o2k2 T Tpis3 + k2ic - Tp 0 (4.49a) T2 6(p,s) T2 as ap Now, we shall show that Eq. (4.49a) is a general eigenvalue equation of monochromatic plane waves for an inviscid, heat-conducting compressible fluid. When Fourier's heat conduction formula W1 = salT K= s H, (4.50) is used to replace (3.2) and when the fluid is considered to be an ideal gas, (4.49a) will reduce to the form found in texts on compressible fluid flow (Ref. 11, p. 120).

Some results from thermodynamics for an ideal gas are listed below.7 ap = yRT (4.51) a s = T (4.52) as)p Cv 6aT\ = T (7-l) (4.53) be s P p = RpT (45 4) as p CV where R is the idea gas constant, cv is the specific heat capacity at constant volume, cp is the specific heat capacity at constant pressure, and c c - R = cp - v (455) Therefore, by use of (4.45)-(4.48), we have a(pT) - ap aT a- p aT - T2(y-l) (4.56) a(p,s) a s a aJp Ps Also, a comparison of (4.50) with (3.2) for the plane wave case gives K-= (4.57) Equations (4.47), (4.48), (4.52) and (4.53) enable us to rewrite (4.45a) in the following form KC 4 2 i22 U2 - k - iwc k - 0 (458) p p p T(7-l) =' which has been given by Pai (Ref. 11, p. 120). Again, we go back to Eq. (4.49a), and divide it by Ck4. We obtain ia1+ { C13, 2 aTC + T3p(( t) O 0(4.49b) 27

Limiting speeds corresponding to high and low frequencies can be found by use of (4.49b). When all is finite and X approaches zero, but C2/k2 remains finite, then w/k2 will approach infinity. Hence (4.49b) becomes 2U =p (4 59) k2 which is the-square of the sonic speed. When c approaches infinity, but C2/k2 remains finite, then w/k2 will approach zero. Then, (4.49b) becomes 6p aT c~2 - _ s 6p (4.60) as Substituting relations (4..51)-(4.54) into (4.60) we have = kT (4.61) 'kT for ideal gas. The speed propagation is about two thirds of ordinary sonic speed. Case 3: all O, K ~ 0 and both are finite This is the general case. Equation (4.44) is, of course, the governing equation for such a wave motion. For any given complex w, there corresponds two values of k which represent two modes of wave motion. We let Co = cl + io2 (4.62) k = kl + ik2 where Cl, C2, kl, and k2 are all real. If k is one of the values corresponding to C then - i(ot-kx) = (Co2t-k2x) - i(wCot-klx) Thus, exp (C2t-k2x) is the amplitude and Clt-k~x is the angle. If C2 is positive and x is fixed, the amplitude of the wave will increase as time in28

creases; when W2 is negative, the amplitude will decay as time increases. In the same manner, when k2 is positive and t is fixed, the wave will decay as x increases and when k2 is negative, the amplitude of the wave will increase as x increases. Two limiting speeds corresponding to high and low frequencies can be obtained from (4.44). If we divide (4.44) by k4 and rearrange terms, we obtain T3p - + i ".2 KT3p( 2 C + e a -as- 2) ~ T ' (p1 + ( K 2e T all. + K 62e 6T all as a(p,s). V agas a5T as p K (- Ke T (4.63) P k2 a(p,s,) = 0 2 When c approaches zero, but d52/k2 remains finite, then each term of Eq. (4.63) is finite except the second term, which goes to infinity unless the factor in the parenthesis approaches to zero. Thus, we find for the limit speed a2e -2 + p2e i 2e = O (4.64) 6a2 k2 a6 6~ep, 6p a2 Or, the limiting speed corresponding to low frequency is ap a2e J' 6p 6S 6~b~ - - ap _ 62 (4.65) =2e This is the limiting speed due to Broer, and Stupochenko and Stakhanov. When (c approaches infinity, then each term in (4.65) is finite, except the third term which goes to infinity unless the factor in the parenthesis is zero. Thus, we must have a(p,T) _p aT (4.66) k2 T - p aT.as as This is the limiting speed of high frequency which we just obtained for the case without chemical reaction (see (4.60)). Now, we should point out that 29

the values of C2/k2 in (4.65) and (4.66) are consistent with (4.63) for X approaching zero and infinity, respectively. In the first case, we multiply (4.63) by c and set a2/k2 equal to the value given by (4.65). Then, the second term on the left hand side will be identically zero. The rest of the left hand side will be some finite value times w. If we allow W to approach zero, we will have zero on both sides. In the second case, we need only divide (4.63) by o and replace XU2/k2 by the limit value given by (4.66). A similar argument will lead to the vanishing of both sides of (4.63). In Broer's paper,l he has pointed out that at extremely high frequency, the chemical reaction is frozen, i.e., no chemical reaction is taking place and A must vanish. Physically, this means that the molecules under consideration move with a speed so fast that there is not enough time for different kinds of molecules to rearrange their atoms to form a new molecule. Thus, in the neighborhood of this high frequency wave front, the fluid ceases to react. The disturbance will then act as in an ordinary fluid. Therefore, it is not surprising that we obtain the sonic speed for the high frequency limit in the non-heat-conduction case. Also, it is not surprising that in the heat-conduction case, the limiting speed-of the high frequency in Case 3 (chemical reaction, see (4.66)) is equal to that.in Case 2 (non-chemical reaction, (see (4.60)). If in the high frequency limit, the chemical reaction is frozen, we may consider ~ as a constant. It is interesting to point out that the thermodynamical expression for Lu2/k2 in (4.60) or (4.66) can be reduced to a very simple form. Let us note that p is a function of p and T when t is constant. In terms of our previous notation, we have p d=p + p dT dt T dt J T dt (4.67) Or, considering p to be a function of p and s, we have dp p = ap dp +s (4.68) dt s dt as/p dt Eliminating dp/dt between (4.67) and (4.68), we find 0)P dp dT p ds ) T 1E)dt a dT apd ds (4.69) La T a S dt dT dsp dt Now, we express dT/dt in terms of dp/dt and ds/dt as follows 30

dT _ aT\ dp + aTa ds (4 70) dt a pC dt dt Substituting the value of dT/dt given at (4.70) into (4.69), we obtain )0 = 4 -)+ T >dt - Sd (4.71) LdP/T ap'5 T~/ ~j-]dt LTI 7} -'P Since p and s are independent parameters, Eq. (4.71) implies p =aP ap- a)T (4.72) s p aTJp asp Eliminating ap/6T)p between (4.72) and (4.73) we have ap)T - a~aT (4.74) Therefore, Eqs. (4.60) or (4.66) can be written as 2 6p\ (4.75) 31

5. ACKNOWLEDGMENT The author wishes to express his sincere thanks to Professor N. Coburn for his suggestions and criticisms. 32

6. REFERENCES 1. Broer, L. J. F., "Characteristics of the Equations of Motion of a Reacting Gas." Jour. of Fluid, Mech., 4, 276-283 (1958). 2. Stupochenko, E. V., I. P. Stakhanov., "The Equations of Relaxation Hydrodynamics" Soviet phys. Doklady, 4, 782-785 (1960). 3. Coburn, N., "The Limiting Speeds of Relaxation Hydrodynamics." Jour. of Math. Anals. and Appl., 5, 2 (1962). 4. Prigogine, I., "Introduction to Irreversible Thermodynamics." Interscience Press, N. Y. (1961). 5. DeGroot, S. R., and P. Mazur., "Non-equilibrium Thermodynamics." North Holland Co., Holland (1962). 6. Prigogine, I., S. R. Defay., "Chemical Thermodynamics" Longmans, London (1956). 7. Sears, F. W., "An Introduction to Thermodynamics, the Kinetic Theory of Gases and Statistic Mechanics." Addison-Wesley, Mass., (1953). 8. Von Mises, R., "Mathematical Theory of Compressible of Fluid Flow." Academic Press, New York (1958). 9. Panofsky, W. K. H. and Philips, M., "Classical Electricity and Magnetism," Addison-Wesley, Mass., (1956). 10. Landan, L. D. and Lifshitz, E. M., "Fluid Mechanics," AddisonWesley, Mass., (1959). 11. Pai, S. I., "Magneto-gasdynamics and Plasma Dynamics," PrenticeHall (1962). 33

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