ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR THE HEAT TRANSPORT BETWEEN TWO PARALLEL PLATES AS FUNCTIONS OF THE KNUDSEN NUMBER By C. S.. WANG CHANG G. E. UHLENBECK Project M999 OFFICE OF NAVAL RESEARCH, U. S. NAVY DEPARTMENT CONTRACT N6onr -23222 September, 1953

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN THE HEAT TRANSPORT BETWEEN TWO PARALLEL.PLATES AS FUNCTIONS OF THE KNUDSEN NUMBER I, INTRODUCTION The method of the previous report' for treating transport phenomena by starting immediately-from the Boltzmann equation (instead.of using the equations of motions of the fluid) has been applied to the problem of heat conduction between two parallel plates. We assume, as in the previous report, that the "Mach number", which in this case is measured by the ratio of the temperature difference, 2AT, between the plates and the average temperature T is small, so that only terms of the first order in AT/T are kept. The Knudsen number, d/X, where d is the distance between the plates and X is some sort of mean free path is, however, arbitrary. In this way the transition from the Clausius to the Knudsen regime can be described more completely than in an earlier report.2 Formal expressions for the heat flux and the temperature distribution between the plates can be derived for arbitrary values of d/X. The limiting cases for the Knudsen gas (d/k << 1) and for the Clausius gas (d/X >> 1) can be deduced from the general expressions. All observable quantities like the heat flux and the temperature distribution are functions of the Knudsen number K-= d/X. It is to be noted at the outset that only for small values of K can a power-series development in K be obtained. It is not possible to find a series expansion in inverse powers of K, since K-= co is an essential singularity. The approach to the Clausius regime is therefore more complicated than previously assumed, due to the development of successive types of boundary -layers, as will be explained in Sec. -VI. This insight is the main result of this report. 1C. S. Wang Chang and G. E. Uhlenbeck, "On the Propagation of Sound in Monatomic Gases", Univ. of Mich., Eng. Res. Inst., Proj. M999, Oct. 1952. 'C. SO Wang Chang and G. E. Uhlenbeck, Transport Phenomena in Very Dilute Gases", CM 579, UMH-3-F, Univ. of Mich., Eng. Res. Inste, Proj. M604-6, Nov0 15, 1949.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN IIo FOPMIELATION -OF TIHE PROBLTEM As in Ref. 2, we take the (y-z) plane halfway between the plates..The upper plate, x -= d/2, has a temperature T - AT, while the lower plate at-x = -d/2 is kept at T + ATO We make the assumptions that 1) AT << T,-and 2) the accommodation coefficient is as; i.e., of is the fraction of the molecules that is re-emitted by the wall with the temperature of the wall. We will use the notation of Ref. 1; the distribution function is f =:fol ( tU + )) where c is the dimensionless velocity (unit (m/2kT)l/2)and f is the complete equilibrium distribution. Because of the first assumption the Boltzmann equation becomes a linear integral differential equation for the disturbance h, of the form Cx A x(1) where n is the number density and J is the collision operator *J-c) = AA C, e-'~ ] 1 HI (cX)( ~'*,~- _ - R.,I (2) which has the dimension of an area and has the order of magnitude of a collision cross section. The boundary conditions are for-m;Lated as follows: In the com-plete distribution function f, we distinguish between the molecules going up and those going down.. Calling these distribution functions f and f respectively, where the plus and minus signs are the signs of the x-component of the velocity, we write: The boundary conditions are then:

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Ite )T Ct )] t(1->)f ( x, a) CeXO (>) i- (+ 1 ifo[ - S (c, 9 t (I ~x ) tCX, '~(3b where ~3/2 _. Eq (3a) expresses the fact that the distribution function of the molecules going-up at the lower plate consists of two parts: a fraction (1 - a) of the total molecules specularly reflected. by the plate so that this portion has the distribution function of the molecules going down at d/2 with the velocity component ex reversed, and a fraction a of the total molecules reemitted by the plate with the temperature T + AT. Eq (3b) is a similar statement for the molecules leaving the upper plate. B+ and B. are two constants which take care of the different densities at the two plates, They are to be determined by the following two conditions: a) the total number of molecules per unit area between the plates is nd, and b) 'there is no streaming velocity in the x-direction. In terms of the disturbance h, Eqs (3a) and (3b) are: ~t+(- d )- bS C + ) 1>t(1-o) Bt A (X ) > C (4a) oT (C- + -0 = _ c) Bt (-, ac (4b) and the conditions for the determination of B+ and B- are: a n ddxjd; - C = (5a) ancd (=b)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN There is, furthermore, one symmetry property of the functions h~, namely: It (CX) = - F(-C,,- ) (6) This symmetry property is a consequence of.'the first assumption and dan easily be seen by reversing.the x-waxis while keeping-the temperatures unchanged. From Eq (6), it follows that S(H)) -= - Cxx) (7) Thus Eqs (4a) and (4b) are equivalent to each other provided and Eq (5a) is then automatically satisfied. To summarize, our problem is to solve the linear integral equation, Eq (1) with the boundary condition Eq (4a), namely: herae tht r (C a > where h~ have the symmetry property Eq (6) and where B is to be determined by Eq (5b). The heat flux -and the temperature distribution are then readily calculated..,_ (8) T(x)-T [,-3x)& e (). It follows from Eq (1) and the conservation of energy that the heat flux q is a constant, III. DERIVATION OF THE:LIMITING RESULTS: TEE KNUDSEN LaIMI Before turning to the formal solution of-'the problem stated above, it is instrictive first to derive the limiting results, valid for.small and large values of the Knudsen.number K. When the Knudsen number is very small, in the zeroth approximation the collision term on the righthand side of Eq (1) can be neglected. r;;ti ing:

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 1= = ti. (9a), 0 and 13 - LIB (9b) ) we have: 1) Zeroth approximation: a - o to Ko Ccac):The boundary condition now becomes.:.,t cC ~x)= ~ Ct Bo) t I+ + (lX) Ro (< tx) T " ~O '2 by use of the symmetry property, Thus ( A ~ i) Ro CC} r x ) = ( -C O + 6 1+ C and or together: OCc<; <.~= o (~ rx,)= A ' (lOa) Bo as determined from -Eq (5b) has the value -2, The complete distribution function to the Knudsen approximation-n is therefore: -(3=71~e m >3K._ () { A+ I —CXT (10b) from which it follows that the temperature between the plates is conStant and is equal to T and the heat flux q(O) is

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN T - T* (11) independernt of the molecular model as is to be expected. For perfect accommodation a = 1, Eq (11) is the well'-known Knudsen expression. For a = O, q(0) = 0; in fact it will be found that for c= O, q = O to all approximations as is to be expected. '2) First approximation: In this approximation Yx - CXTO) the solutions of which are: Cso) 1K~ + K (eC) (12) + K where Ki and K are integration constants like Ko. Because of the symmetry property, Eq (6), K1 and K1 are related by the equation K, (c) = - K, C- x). Substituting Eq (12) into the boundary condition Eq.(4a), one finds. K<,(C)(, + ]' (g)) ~4.. and therefore:,(,=(CX T ar I, BI t14 X it, A (g0)(Xt CX) B1 as determined by Eq (5b) is found to be: BI = zr AL using the bracket symbol: LA, )] = [), A] = | dC (). Hence the first-order disturbance function h1 is:

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN R, ~xX):= _-(x_~~_za_ cc, L) c + (13) + xc (X + C) The first-Order correction to the heat flux, q(l), is -' -)', LI) Ar CK ] Since the square bracket [A, A] is..always negative, the ratio q(l)/q(O) is always negative, For a 5 1 this redue's. to the expression given in Ref. 2.* For elastic spheres and Maxwell molecules, the square bracket has been eval' uated in Ref. 2. The results are: For elastic spheres (diameter a); (0) = ---'a z -3 (15) For.Maxwell molecules (force law K/r5):** ' t-7- Lc~~-3)"~B1 - i'~l(9r3m -t (16) F3or CMe eus n)3ael eue oncaaR 1e. + For Maxwell lmoeeules, one-ca als e'"e(aS in Ref. 1) of the eigenvalues and 'eigenfunctions of the collision operator J The h1s are expanded in terms of the eigenfunctions with coefficients art. In this way one find.s: *The results given 'on p, 35 of Ref. 2 for a # 1 are. not correct. **The function F(., e, 5) used in Ref..2 is (2K/m)1/2 F(G), where F(G) is the dimensionleSs function used in Ref,. 1,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN -..+ Tr~ t-t A! A! (17) where = WTrsJ I&r go Cid c tFs 1 O) ((;h)j are the eigenvalues of J. The sums in Eq (17) can be carried out and this leads again to Eq (16) for q(l)/q(O). This serves as a check of the result given in Ref. 2. 3) Second approximation. The calculation goes as before. The function hp is found to be: M = L ( I 5e7- 7 h,) 3- Ls L (RA.) +.L-0( 7Ax V-_TJt ) T 0T( J 1(18) -)% At ix t;(C06)) - J stat jKAt!a t.31 The second-order correction to the heat flux is again found to be independent of x, as it should be. It is given by the following expression: Q?) _\' ()io<>8> (c; z)$h, Cj sp( EX (C,C( Zr ~~~~(19) The ratio q(2)/q(O) is positive, which can be seen as follows: Replacing A in the curly brackets by the integral wr-=A c,\ %;,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN then { } =( J< e c MT'M )JA -TAX LJ( ' ) - [ate Ac c) which is seen to be positive by use of the Schwarz inequality. IV. DERIVATION OF THE LIMITING:RESULTS:.TIE CLAUSIUS GAS LIMIT Even though the approach to the Clausius gas regime is more complicated, so that a development in inverse powers of the Knudsen number K I(or of the density n) is not possible, we will, in this section, attempt to obtain a solution for the Clausius gas limit directly by making such a series development; i.e., we will write oo _ e The procedure adopted will be slightly different from that used in the previous section. We first solve the integral equations for his making use of the symmetry condition Eqs (6) and (7) and of the fact that Cx must be zero, but without taking into account the boundary conditions Eq (4a). The solution will contain a number of arbitrary constants, and at the end of the section we will try to determine these constants from the boundary conditions. It will be seen that the integral equations for the suceessive approximations can be solved without any difficulty. The fact that these solutions cannot be the true solutions of the problem -will be reflected by the fact that the boundary conditions cannot be satisfied exactly. Up to which approximation the calculation can be trusted will then also become clear. Substituting the series expansion for h.into the Boltzmann equation and equating terms of equal powers of n, we have: 1) Zeroth approximation: the general solution of which is: IM C(o) ) (20 ~R, (XI ~~~x) - ~t 9 Cx + d3 C C *2,~~

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where the a's are independent of c2 and cx but are in general still functions of x. The symmetry property, Eq (7), requires that a O) be even in x and that a(0) and a A) be odd in x. The fact that C = 0 requires that a(2) be zero, while al ) and a(0) may be arbitrary. 2) First approximation: CX,a-,- = Ct,. (21) This inhomogeneous integral equation will have a solution only if the lefthand side of the equation is orthogonal to the solutions of the homogeneous equation. Multiplying Eq (21) by e-, cxec, and (c2 -3/2)eC respectively, one sees that Eq (21) will have a solution if dx. (22) Equation (21) becomes therefore: (0) da, ~9(C- S ) Cx = 2,) (25) the complete solution of which is: Ly I t )C t CL( % (c. (24) The form of the particular solution, i.e., the fourth term, follows from the isotropy property of the linear operator J. By, symmetry we have again that a(l) and a("l must be odd in x, and a(1) and a(1) must be even in x. In or1 3 2 4 der to make the particular solution completely definite, we may and will require that this term be orthogonal to the solution of the homogeneous equation. This imposes one condition on g(c2), namely: |~~~~ 50 -CT )=.(25) Then 7x = 0 again requires that al) =O. Putting n) _=:o) 4 ox (26) the equation for g(c2) becomes: ( ct) c)= t a). (27) 10...

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Together with the condition Eq (25), this equation is identical (except for notation; g(c2) has the dimension of a reciprocal area) with the equation solved in the book of Chapman and Cowling. The method is to develop g(c2) in Sonine polynomials of degree r and order 3/2: Omitting the term with r = 0 takes care of'the condition Eq (25). For the development coefficients.Cr, one gets from Eq (27) an infinite set of linear equations, which can be solved by convergent infinite determinants. For further-details we refer to Chap. VII of the book of Chapman and Cowling, and we will note only that in our notation the heat conductivity coefficient v of the gas is given by: L( )xIcq= 'z j.> (29) and that in terms of v: oI= - d (3o) From Eq (8) for the heat flux and Eq (24) it follows that for the heat flux in the first approximation: >(?I) =- T) o -orI (31) The constancy of da(s)/dx follows from the second approximation. 3) Second approximation: Xe}- [ a(l)+ A W A ) t <i x] - { y)) Y( ~vbX I~:"~ co The solubility conditions lead to: and 11

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN from which it follows that a(1) must be a constant so that a(0) and a (0) are proportional to x. The complete second-order solution is: AL3 0 id a ( )3)x By the same argument, a(2 must be zero, The function g(c2) is the same function as in the first approximation. It satisfies the same integral equation and the same auxiliary condition if one puts again: L~)_ dc')3 The coefficient a(2) will again have to be a constant if one goes to the next approximation. Thus we see that the complete solution of the disturbance h will be: where i=o; "3 >,il and JX Ck X =-1 ~ I Therefore a3 = -a, _ na4x:so that: A=1 ~t-)+ k CN (Ct) (32) The a are the constants to be determined from the boundary conditions. Bounday Condition: The complete solution of the problem depends on the boundary condition Eq (4a), namely: + (Cx,- A)- < (tE 6)Xt -+ (1-i) (-9x- )} (5) where the h+ (cx, x) must fulfill the symmetry conditions Eq (6), and the Boltzmann equation, which we write in the form 12

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN C3( _ (34) Since we know that J(h) = a4 (c2_- 5/2) cx, we can conclude from Eq (34): t (z. X) = -r a X~ (C~_2/) I 4i + ~C: C From the symmetry condition it follows that g- (c2, cx) = -g+ (c2, -Cx), while from Eq '(32) it follows that: Developing - (c-, Cx) in Legendre polynomials in cos @ = -X/c, one can conclude that g- (c2, cx) must be of the form: tjCc c4) boC&)t~cx Cc)2 bi. T (tsO) (37) where bo (c2) '.and b2n (C2) are still undetermined functions. Turning -to the boundary conditions Eq (33), it follows from the fact that in the development of(l/2)(1 +-sign cx) in Legendre polynomials the polynomials Pz (cos.9), n O do not appear, that in Eq (37) allthe coefficients b2n (c2) must be zero. Eq (33) can therefore be written as: dAT eA+ B) \+ " Cx _ _ l+ _t. _ (38) It is clear that this equation cannot be fulfilled identically in cx and c2, so that some compromise must be made. -Multiplying* Eq (38) by cxe C2 and '(c - 5/2)cxe'c2 and integrating, one obtains: (39) IB ( - 1; ) = d o + 4 aT - ~: Solved for Band a4, these equations give: XIt is to be noted that 1 and c2..5/2 are the zero- and first-order Sonine polynomial c (2) used in the development of g(c2). 13 polynomial ~9s/ ~a

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN I s l _ 1o Fi dr / ( ^)T h (40) With -these values of a4 andB, one then obtains from Eq (38) by integrating over all angles of C Using Eq (30) for cal, one obtains for the heat flux: and for the temperature distribution TCX)=T(1$43)=ri --.... - I ( + It shod be emphasized that these results cannot be the exact solution of the problem, since only two of the moments of the boundary condition Eq (38) have been fulfilled. It is stil of interest to remark that Eq (40) can be developed in inverse powers of: n, according 'to: and one gets: 3Maxell collected paperst 14

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Ba c) __ _ c, and for i > 0: e,$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 F( — nw-Th4 Since bo(C2) developed in powers of 1/n begins with a term - l/n, one sees that with the Eqs (40) values for.:a4 and B, the boundary condition Eq (38) is in the zeroth approximation identically fulfilled in cx and c2, so that.up to this order at least the development in powers of X/d is consistent,.V... 'GENEA SOLUTIoN In this section the general solution of the Boltz'mann equation, Eq (1), subjectedto the boundary condition Eq (4a) and the condition = 0, will be obtained. For this purpose we expand the function h in the set of normalized eigenfunctions r belonging to the collision operator J for r, e~~ the' Maxwell molecules: OX')ce where yt t + t (JiNri is the normalization constant, P1 is the Legendre polynomial -of order 1, and Sj)/2 is the Sonine polynomial of degree r and or. der I + 1/2. The Boltzmann equation becomes then a doubly infinite set of linear differential equations for the expansion coefficients art: As a consequence of the symmpetry property of h and the evenness and oddness property of \1rt for even and odd t respectively, and art's have the symmetry property: cyOfLrL) = T 'jC-x) according to I (43) (41) 1~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The requirement that cx = O means that and the boundary condition Eq (4a) becomes the following equations; YT + _C T,)t. g-.. (46) The choice of the functions *r, has the additional advantage that the physi-cal quantities we -are interested in are then' all expressible in terms of the first few development coefficients ar. For instance: TCx) = T - -1 —: <, (47) o =, o48) pXX = Ou I-T- KL (a 10 K OO) and The conservation theorems lead to simple results for.some of the first arQtS One obtains: 1) From the conservation of number so ao01 = constant, which can be taken to be zero: RL D =O (48) 2) From the conservation of linear momentum: f10 3 WYoo )=aX 2 so p = nkT (1+ const), and the constant must be zero since the ar, with even -2 are odd in x. 16

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3) From the conservation of energy: so =,, = * = b~,. (49) The set of 'equations, Eqs (44), has constant coefficients and a formal solution can therefore be easily obtained. However, for the actual solution, we will simplify the problem slightly by separating the equations for even and odd 1's. Because of the property that: L[ )" ~"] 9j 0 only when - i" = odd Eqs (44) can be written as two sets:, for odd and even I respectively. With the. following matrix notations: \e,A= (eur&) a (l tCX these are: (50) On eliminating Ae, one obtains d%;tla J1,. =.(51) Eq (51) gives an infinite set of linear homogeneous equations for all the ar, 2+l's. The even ar2I's are then obtained from ar2r+1 by Eq (50a). Making the Ansatz:. 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN o 0"X. JL, got where fo = (br2!+1) is a column matrix with elements which are constants, and substituting into Eq (51), one finds: This infinite set of linear homogeneous equations will have a solution if the determinant A = | p'I-ari I | = ) (52) where I is the unit matrix. We see that the roots Pi always appear in pairs with opposite signs. -It can also be shown that the Pi's are all real.* From the conservative theorems and the property of the square bracket, it follows that so that among the roots of Eq (52) there are four having the value zero so that among the roots of Eq (52) there are four having the valgue zero. Hence the general solution of Eq (51) is A= E + 8, X lX + 6,~ X6 _ x (53) where the sum goes over all the nonzero roots Pi# The symmetry condition Eq (45) requires that: )"t = 0 and e(+6)- 6(-1' where the +i refer to the two roots Pfi. Of the elements b(i1+l1 of the matrix B(li), one set for fixed r and I can be taken as arbitrary. We will choose for this set the b(i) Then: *For proof, see Appendix I. 18

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (~) 6t - AC-29 03 where Arl) designates the first minor of the (rA, rA) element of the determinant A in which p is replaced by the ith root Pi. It is easily seen that: l0o= b ) = O for all i The constant matrices 11 and L' satisfy the following equations: 0 = ~~Y6 8, (54) (55) Since Rol rl = Rll, r = O, the rank of the determinant!RRI is less than the number of variables by two. Therefore, so far as the set of linear homogeneous equations Eq (54) is concerned, two of the elements of 31 are arbitrary. To satisfy Eqs (48) and (49) we must have: (6v')JO =C )11 =0 ~ hence '1 must be identically zero. Eqs (55) now become linear homogeneous equations like Eqs (54). By the same reasoning, two of the elements of E1 are arbitrary. These must be assigned the values ($,)o, = ~ ~,),, - 1,, =0) in order that Eqs (54) and (55) be satisfied. The rest of the B,'s must then all be proportional to b1l. The solution of the set of equations, Eq (50) can thus be written as: DoL = i,, +& XM b'' a p)nx (56) The matrices O and 0(i) are known constants independent of bll and b)3, the determination of which depends on the boundary conditions. The coefficients ar2i are given by: A~;s = 1,, x~stBA M X Ap~nX. (57) Before making.use of-the boundary conditions Eq (46), we first have to find expqressions for a+r2+l. The differential equation for these quantities is 19

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN hx;ve wt 5 4 c) =o ~ t~ a- ~ + 2n ~ c,,,p: where Tr +I, 2'+ = L L",P 3x't I In mhatrix notation we have: AX = a + +, oJ with Substituting Eq (56) into the above equation and integrating, one obtains: A =2 ({4 +- <n~ TixA4 )2Y1\t I7 (,,nx + ) ( 8) There is no integration constant The symmetry property requires that any integration constant Cr2~+1 be equal to Cr2i+l. On the other hand, since + + - c++2t c must be zero. Hence both r2+1 nd must be zero. r21+l r2 r2stbl zr2o+l Putting Eq (58) into Eq (46) and calling 2+ 0"C

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN one is led to the following equation: r P T i6+ [ 2[-( p59g )P Eq (59) is an infinite set of inhomogeneous equations for B, bll, and b(3 It can easily be.: seen that' this set of equations determines the unknown constants in the following sense. Suppose the basic infinite determinant A is broken off at some value of r and I; then with this finite set of values for 'r and I, the number of equations, Eq (59) will be just equal to the number of unknowns. We have not been-able to discuss the convergence of this procedure. It is possible to deduce the limiting results for the Knudsen and Clausius gas from the general expression. The zeroth order for the Knudsen gas is especially simple, since in this case all the hyperbolic sine and cosine functions can be replaced by zero and one respectively and since n -can be neglected compared to unity, Eq (59) gives two equations for B"and bll, namely: T I -rl By solving for bld one obtains the Knudsen limit, Eq (11), for the heat flux. We will discuss:the Clausius gas limit in detail in the next section for the case of Maxwell' molecules. -The complex nature of this limit is clearly due to the fact that the density (or the Knudsen number) occurs in the argument of the hyperbolic functions. VI. MAXWELL MOLECULES For~ Maxwell molecules: where according tothe definition of J used here Xr. has the dimension of an area; it is */s/k times the He in Ref. 1, where s is the force constant. Thus 21

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where we define: which is symmetric in-the two pairs of indices. Furthermore, for Maxwell molecules Rr,21;ll are zero except Rlo,11 and Ro0,11. From the conservation theorems, Rr,21+1,00 and Rr2+1,ll are zero, so that all the coefficients of bll in the system of equations, Eq (55), are zero. As a result, all the elements of the matrix B except bll must be zero. The solutions of Eq (44) are slightly simplified, and can be written in the expanded form: cy =o " - b,, o LL ) L03 h 43.3~0 (10= ItX LlollX+t k ltfl IL lo L3 Pt03) sx The p's are the positive nonzero roots of the basic infinite determinant A = I P S St',- Xylzp,', s21X / L,.,, +, L., I,lQ., I =. The set of infinite inhomogeneous equations for the determination of B, b11, and b(') becomes 03 Tr 2-. T (r-61) =a + ( II+

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN,where We will use the same successive approximation method as in Ref. 1. In the zeroth -approximation ("ideal fluid"), only the eigenfunctions 0oo*,Lo, and * 1.r are used.- In the first:approximation ("'Stokes Navier"),,11 and ro02 are added; in the second approximaton (Burnett") three more functions (o03, 2ao, 12) are used; and so on. It is-easily seen that in the zeroth approximation one does not get a heat flux, and so we pass to.the higher approximations. 1) First approxi.mation: -For this approximation we take.oo,q rlo, ol1, 'rlr and o02'- The solutions for the development coefficients are: i,, = - box = 0 t0oo = m Xkt Loo,.l t ao = n b X*, Lo,Al x where bl1 and the constant B are to be determined by Eqs (61) which reduce to the two linear equations, _ 9.BAT, + _ i l:L b I TLr,, = (t "r —0,, All the coefficients can easily be evaluated-.* Solving for bl,, we find: bA L_ +4I I *Tables for some of-thhe elements of the matrices P, Q, L, and L+ will be found in Appendix II, 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where X is a measure of the free path defined by: XE- (>nA J7h). I Substituting into Eqs (47d) and (47b), the heat flux is found to be q*=~z,nk T-\ 1 ~ ro~"~~r~~ (62) and the temperature distribution is given by x)_Tt- $ _ ( +-0 — ~ct' (63) Thus there is a temperature slip at the wall equal to: ES N4 K^-T-) =(64) These results are the same as those obtained in Sect. IV if these are specialized to Maxwell molecules. 2) Second approximation: For this approximation, one stops with eight *'s: 0ot0 tol0, '1.o' ~ll 02, 'OVs '12, and 420. It is in this approximation that the hyperbolic sine and cosine functions begin to enter. The determinant A is simply: A = — o, ZL Xvi' (Lv., o3) = e. ~ Making use of the values of XrA, and Lrl,r,, one finds where The linear equations for the determination of the constants are: — pII olB 0,t6 Xd Lj 1 i+Lb,, +, T 2. t 2 --- ) — 24

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN so that b,, _~ _ 4 l K i~ \,;s + __ c_S 2. where we have written $ and c for sinh (ndp/2) and cosh (ndp/2) respectively. The heat flux is therefore given by l I0~f iS+The coefficient alo is in this approximation: = X * b,, vb4 dur x, which gives for the temperature distribution Is ft 2-0( + + _ Eq (66) leads to the following expression for the temperature slip: -F=T(z7)-7r-4r)= 6T a __ 2-0 c( 4sf 2-O(.o( 4kKjd -0 o(} 3) Third-approxination: In this approximation three additional functions are taken. They are 0o42 21, and '*13- The basic determinant A becomes: A2_ Ax f\ f rz - 4 IxH176r A' Table I gives the roots p and the coefficients br2+1 in terms of b(3)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Table I i 1 2 3 i 0.4798 Al 0.8344 Al 1.293 Al 1 1 1 b0( i )/b0(i) 03/03 21)/b ( 5.700 -0.4787 0.4036 b(3)/b(i) | -1-519 -0.5244 1.593 15 03 The set of linear inhomogeneous equations, 7q)(6l), now consists of five members, so that the coefficients bl and b are ratios of two 5 x 5 determinants. By straightforward calculation, using Table I and the tables in Appendix II, one finds;that the heat flux can be written as: where -0.5000 - 1.0937 tl -0.5000 - 0.5722 t2 -0.5000 - 0.5749 t3 D = 2.850 - 3.828 tl 0.2394 + 0.2018 t2 -0.2018 - 0.2430 t3 0.7595 + 0.8938 t1 0.2622 + 0.2540 t2 -0.7965 - 0.7953 t3 -0.5000 - 0.6898 tl -0.5000 - 0.5520 t2 - 0.5000 - 0.5696 t3 D, = |2.850 - 3.076 tj 0.2394 + 0.2393 t2 -0.2018 -(0.2335'ts3 0.7595 + 0.6083 ti 0.2622 + 0.2397 t2 -0.7965 - 0.7991 ts and we have written. ti for tanh (npi d/2). The temperature distribution turns out to be: tD,7 l,,,} 26

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN with c = cosh (npi d/2) and E1 = -(.1.274 + 1.342 t2 + 1.325 t3 + 1.392 t2 t3s) E2 = -(1.917 + 1.986 t1 + 1.976 ts + 2.068 t1 t3) E3 = -(0.2973 - 0.0245 t1 + 0.2225 t2 - 0.1196 t1 t2). The corresponding temperature slip is E 5 i, T 3)L ( t xAD In this way one can go on to higher approximations. However In this way one can go on to higher approximations. However, since, contrary to the case of the sound propagation, the successive-approximation method changes, at each stage, the results of the previous stage, and the calculations grow more and more involved, we have not gone any further. For the discussion of the results obtained so far, we have made some numerical computations and plotted three sets of curves. To simplify matters we have assumed that a = 1. Figure I is a family of curves for q/qK against the Knudsen number K = d/x. The straight line gives the initial slope for such a curve expected from an exact theory. The value is taken from Ref. 2, with adjustment for the change of the definition of X. The initial slopes for the successive approximation results can also be evaluated easily. For comparison they are listed below: Initial Slope Exact theory: -0.907 First approximation: -0.226 Second approximation: -0.236 Third approximation: -0.3o6 The convergence is therefore quite slow, as is to be expected for small K. The differences between the successive approximations are quite small, so that Fig. I may already give a good idea of the dependence of the heat flr.n on the Khudsen number. 27

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Figure II is a family of curves for the temperature slip measured in AT as a function of d/). -They have the same general feature as the curves in Fig. I, except that the difference between successive approximations is larger. The initial slopes of these curves have also been computed. They are: Initial Slope First approximation: -0.226 Second approximation: -0.282 Third approximation: -0.390 We have not computed the initial slope for the exact result. Figure III presents plots of the temperature distribution. On account of the small differences between successive approximations, we have plotted instead the difference Tj(x) - Ti(x) measured in AT against 2x/d, where Ti(x) is the temperature distribution for the ith approximation and Tl(x) is the linear temperature distribution, as follows from the first approximation. The rapid rise of the curves near 2x/d = 1 can be taken as an indication of the so-called boundary-layer phenomena. This rise is a consequence of the appearance of the hyperbolic sine and cosine functions, which is also responsible for making development in X/d impossible.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPENDIX I. PROOF THAT ALL THE ROOTS p OF THE BASIC DETERMINANT ARE: REAL For the proof it is easier to start from Eq (44), namely Ax Y&! IIX. CXT, t) k'J Expanding the function l/ex ~rl in terms of the eigenfunctions, N.-tl t i tCY so that In matrix notations, letting \A, (a0f)Rt,+'Q']) where both Po and ~ are symmetric, the above equation becomes The Ansatz A = Be-npx leads to ~ = ~ot 6 Since I is not singular, one can define 0 = St-l t so that we have 29

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Both t-1 and ORo being symmetric, the determinant for p is symmetric and thus all the roots of the secular determinant must be real.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPENDIX II. COILECTION OF TABLES USED FCR NDMERICAL CALCULATION 1) Table of eigenvalues Xra (A' = A2 4/7kT, At = 0.636 At) 2 4 2 r 1 2 3 4 0 0 0 - 3A2/4 - 9A/8 - + 4.1 0 A/2 - 7A'/8 _ i _ ' + 5A- = -1.176A' 2 12 8 16 2 2 Al/2 - 3A/ 2) Tables: for PrL and Qr 01.11 03 21 13 P.~j~i~/4 -3 I 21/4 -. Pr.Q 1 -/4 1 - 1/ dl 4.14 l-L/4 IT4 2J4 ~ J540 4435 2J15 3) Table for Lra r'V \ r,l 00 10 02 12 20 04 r I 11 2 0 0 0 0.5 5 03 -2 2 16 0 0 0 15 J 3 E 35 21 4 2 0 7 0 2 i 13 8 2 2 2> l W$4f32 0 31

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 4) Table for tLrt r r, O1 11 03 21 13 r' 2' 01 1 1 1 3 _ 1167 2 476 11 1 9 1.19 5 3 ~ o q 10 5 -f 20 10 i 03 - 1 14 1 14 4it 5 77 15 21 3 19 1 233 9. 2 4 7 20 47Y 10 42 2804 201 13 -I _ 3 14 9 119 5log 10 43:1 45 V 20 %2135 32

1.0 0.9 0.8 0.7 EXACT 0.6 INITIAL SLOPE 0.5 04 0.3 0.2 r STOKES NAVIER ]: FIRST APPR. (STOKES NAVIER WITH TEMPERATURE SLIP) m SECOND APPR. ("BURNETT") 0 Z1 THIRD APPR. 0.1 FIG. I. HEAT FLUX AS FUNCTION OF THE KNUDSEN NUMBER 0 2 4 6 8 10 12

I.0 0.9 0.8 0.7 0.6 - 0.5 0.4 0.3 0.2 _ I FIRST APPROXIMATION HI SECOND APPROXIMATION lr THIRD APPROXIMATION 0.1 _ _ FIG.2 TEMPERATURE SLIP AS FUNCTION OF KNUDSEN NUMBER I I I I. I 0 2 4 6.8 10 12 d x

0.06 ' FIG. 3 TEMPERATURE DISTRIBUTION IN THE SECOND AND THIRD APPROXIMATION AS COMPARED WITH THE LINEAR TEMPERATURE DISTRIBUTION T (X) AS FOLLOWS FROM THE STOKES-NAVIER APPROXIMATION WITH SLIP. FOR- =10 0.05 0.04 - TT 0.03 0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2x d