DEPARTMENT OF ENGINEERING RESEARCH Page, UNIVERSITY OF MICHIGAN APL/JHU. CIM-503 UMh-3-F Date: 19 August 198 ON THE THrRY. OF THE THE THICKNESS OF WEAK SHOFCK %TAVE,4' C. S. Wang Chang I. Introduction The two oroblems often suggested in connection with the Burnett equttions Pre: 1) the velocity of provagat %ofhith frequency sound waves, and 2) the shock wave nroblem, In the Burnett equations higher order differential quotients annear; thus the annlication of Burnett equation to-most problems calls for a knowled~e of additional boundary conditions. The above mentioned two -oroblems are the simnlest because they are the two rnroblemrs that do not involve boundary conditions.' The first problem has been treated by Primakoff and Tsieen and Sheiamberg, and recently we h.ve considered esneceilly the calculation of the ve.loet.v of sound in helium g.s (1). The nroblem of the thickness of one dimension'a. shock,aves habs been studied bby both Becker(2) and Thomas (3) using Stokes-Navier equations as exact, though Thomas included in his noarer a discussion as to the changes to be exoected when the Burnett terms are taken into account. Becker started with the Stokes-Navier equations and treated the viscosity coefficient,, and the heat conduction coefficient,, as constants indenendent of the temperature., For P narticular value of the ratio /f'A/c,) namely f -r/'J where r is the ratio of the specific heats ~p and cv of the gas, he rwas able-to solve the hydrodynamical equations exactly and obtained an expression for the velocity distribution. From this velocity | distribution the shock thickness was calculated. His results show that for the Mach number, M, slightly larger than unity the shock wave thickness, t, is about a eew mean free oaths, whi~e for large M{ach numbers, say.f.35, t is of the order of 10-7 cm, or of the order of the mean distance between the molecules 1) C. S. kWang Chang On the Dispersion of Sound ineli, FALJUC2 ).. Temker, Jce r.s, f Phs. 12 4F35321-362, 1922,. ( L. Thomas J. Chem. Whys, 12. 41X 9-1$53, 19L;l

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN 2 There are three objections to this theory: 1) The Stokes-Navier' equations were treated as exact, which is against the soirit of the devel.onent theory of Enskog. 2)' The viscositv and the heat conduction coefficients are certainly not constant especially for very strong shocks where the temperature change 'is tremendous, 3) His solution is for the one value of f- =f/3. For diatomic eases with J ' /Sk, this value of f (1,87) is near-enough to both the theoretical and the exprerimental va.rue, so that the calculation seems to be aDDlicable for;air. But the Stokes-Navier equations are strictly true only for mono.tomic gases where there are no internal degrees of freedom. For Dol-yatomic gases the.relaxation time is usually so large that its effect masks the effect of the other two gas coefficients. Thus vwe do not exoect' results deriverd from the Stokes-Navier equations to hold good for air, or any polyyatomic gases unless we are sure the relaxation time is vetry short. Thomast calculation removes the second objecticn. Hte included in his calcul.ation the teemperature dependence of A/ and ). An elastic sphere model. was taken for the molecules, so that both/, and were. taken to be proportional to the square root of the absolute temperature. The equations are still soluble for the same particular value of f. It was found that for M-2 the thickness of the shock wave is about four times the mean free nath,Xm, (the mean free path at the nlace where -the -velocity gradient is a maximum),. For 'the infinitely strong shock, t has the value of 1.74 X m We will discuss his results and calculation further in II. Recently Mott-tSmith* made an essentially different aronroach to calculate the thickness of shock waves, Using neither the Stokes-Nas.vier *Wae are indebted to Commander H. M. Mott-Smnith for showing us his manuscript on "Kinetic Theory Treatmrent of a Shock '.Wave."

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN 3 equations. nor. the idea of Enskog's development, he started directly from the Boltzmann equation and the Boltzmann's equation of transoort. For both monoatomic and diatomic gases, he found that the thickness of a shock wave is invers ly Drooortional to (i- 1) for M - 1 small, and that for infinitely strong shock waves, t is of the order of a few mean free paths. As a summnry' of the existing results on the calculation of the shock thickness, we have plotted in Figure-1, the values of Xb/t against the Mach number M, where A. is the mean free path in the medium before the shock. Curve I i' comnuted from Becker's theory, Curve II follows from the calculation of Thomas excet that t is now measured in terms of X instead of *A. Curve III is calculated by ulsing the formulas of Mott-Smith*. Curves I and II have the same initial slope. For high M according to Becker L/t increases almost linearly with }at, whereas according to Thomas X/t increases much slower with M and annroaches a value of X/t 1.02 for M =. Curve III has' a smaller initial slope and the asymptotic value of X-/t is 0.55. One sees that there is quite a difference between the theories esoecially for strong shock waves. In our opinion, for strong shock waves the calculation of Mott-Smith is probably the most' dependable, but his theory has to be modified since an exact theory should give the initial slope of Becker and Thomas. We propoe to develop a consistent theory for the shock thickness for weak shock waves. We shall follow the idea of the Enskog. development and develop the shock thickness in a power series of M - 1. To the first approximation we shall use the Stokes-Navier equatiofs and stoo at the first power of Xf- 1. We shall see that we get the same results as Thomas if we develop Thomas' final |We differ here from Mott-S&ith in one respect. We found that in comformity with Becker's definition of the shock thickness t = AX/@ instead of Nott-Smiths ' t X /8. For B we have used lott-Smith's value determined from the consideration of the transport of u2, where u is the streaming velocity, and J/ 7/5.

H*3atWZ HOVW 3HJ Hit S'AVM OHS AO SS DNIHI 3H 10 NO IVA W' J38vanN HOtWJ t~ Z. 1: 9*0 I.~~~~~~~~ ~ ~~~~ I..11 I ".~~~~ ~~~~__. _____ v1

result also in power series in!. 1. To the second order we include the Burnett terms and carry the e!4culation to (M - 13, etc. In II we shall restrict ourselves to the first order calculation. We will give Thomas' result and comnare it with ours. The calculation will be, extended to the second and third order terms in III. The last section will be devoted to the application of the result to a monoatomic gas. II. 'The Thickness of Shock Waves from. the Stokes-Navier Equations. We start by writing down the general one dir.ensional hydrodynamical equations for the steady- state,-,,,"X Jr wherep is the density, u is the streaming velocity, p is the pressure tenseor (P- pxx in the present case) and q is the hest current density, E is the total.energy - nkT/(t-l1). These equations can be integrated once immediately yielding, I At= v(1) whereA, adCr nt - te l where A, B, and C are constants. Given the velocity and the state of the gas before the shock, equations (1) determine the conditions after the shock uniquey In the zeroth order (ideal hydrodynandcal equations) one must Dut p - nkT, q:O,

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN | 6 and equations (1) give then the well known shock conditions. The shock wave anpears as a discontinuity in the velocity, the temperature, the density, and the pressure. However, when the first order Dressure tensor and. heat flux vector are taken into account one finds that the shock waves are really not discontinuities butthere are regions in' which the velocity, the density, the temperature, and the nressure undergo large changes. In this order of androximation (Stokes-Navier equations) equations (1) become now: (2a) 7'/ " 3 W / (2b) (2c) The constants A, B, and C can be expressed in terms of the gas quantities /3. * j. CZ./ I

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN 7 Eouations (2) will be exporessed in dimensionless forms. For this nurpose we introduce' the following dimensionless variables, 2- - 7-; We will also need the M.ach number, M.,o where ao is the velocity of sound in the medium before the shock. For the temperature dependence of the viscosity coefficient we will. first use the more gendral expression where s is equal to one or two for the Maxwell molecules (points repelling with a force r. l/r5), and the elastic spheres respeectively. The specification of the value of s wrill be made.later when we apply our results. to definite molecular models. Further we write 7 is the dimensionless constant 5 i. /j (16y ), and A. is a general Maxwell mean free hath in the medium before the shock, defined by

DEPARTMENT OF ENGINEERING RESEARCH Page I. UNIVERSITY OF MICHIGAN 8 nobeing the number density and Q (g) is the so-called transport cross-section. For elastic snheres where a-is the molecular diameter; A is simply the ordinary Maxwell. mean free path. Dividing the eouations (2b) and (2c) by (2a) and expressing the resulting eouations in 'dimensionless variables we obtain: r - 75 Z"" r ~"'~~-' —' - ~ (3a) and ~~' - (' i, rs -- " '_ -:/ ' ' ~?- '.Z,-/ (3b) For fhrther calculations it is simnler to use equation (3a) and the eouation obtained by subtracting (3b) from (3a). These are h/.f 't v-? (4a) 'Making now series exnansions of all the ouantities in power series in y where y stands for M - '1, zr-=/+27 + / -f-'(5) 7?, F-~~~~~~~~~~~) 2'",~~ +'"

The exnansions.for Z't and v start from the constant term 1 while that for X/t starts with the linear term g91 because for y equals to zero there is no shock wave. Substituting (5) into (4a) and equating terms of eoual powers of y one finds that the terms independent of y are identically zero while the first order terms yield: t2= z(f-/) From ('4b) and the above ecuation one finds the differential equation for' l the solution of which is: T'b'7,f e --- - 4(6) where For b =~-o' V1 0. as it should since one is then in the region before the shook. 'The value of 'L at b - oo is given by the term -y of the expansion of the velocity after the shock, v, determined by the shock conditions, boundary conditions. A is the integration constant which remains undetermined because the shock iS not localized. -If desired one can fix A be requiringb that the maxium velocity gradi.en t occurs at b O. This ies, howe vr, arbitrary and

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN. 10 "Wd it is not necess.ry at nresent. Following the earlier workers we define the shock wave thickrnss by (8) In dimensionless notation this becomes: where the right hand side is known and the left hand side involves the parameters gl, g2' etc. which measures the thickness. To the present order of apDroxiination we have therefore, From the velocity distribution (6) it follows that the maximum slooe occurs at the-value of b satisfying the following equation Using this relation and equations (6) and (8), one emn solve for o( and gl 8/ _ / (9) Y/ &'7 shDi We observed that gl is independent of s, the temperature dependence of/ and i. This is not surorising since it is clear that the temperature dependence of,/A and )> will contribute only to the second order terms. Up to this order

|UNIVERSITY OF MICHIGAN 11: the velocity distribution is symetric and the velocity at the maxinmum velocity gradient. is - 2y/(7,1).) By elitminating 'c from (2a) and (2c) Becker arrived at a differential' equation connecting T and u Which can be solved for f = 4t/3 1.87. This 'relation of T'as a function of u was put back into (2b) and an expression for t was obtained which denends only on the unknown ut Making use of this last expression the calculation of the shock thickness is straight forward. Becker did not take into account of the. temperature variation of the viscosity and the heat conduction coefficients, but the essential features of the calculation are not changed by the inclusion of these temnereture denendences. Letting C| and? be,rooortion&l to,T1/, the generalized Becker's relation between -- | and u is, in our notation, and with W=the mean free path before the shock: ~ '. s:A./ '____ This reduces to Becker's result if.s O, while for s = 2 it is the Tho.mas formula. RestrActtng ourselves to s 2, it can be deduced that dwil be a maximum when v is' a solution of the cubic eountion: "'"3'/ Cf/+,j~ /j~ "4z: Thomas comrnuted the value of AL /t for several values of the Mach number using (8) as the defining equation for t'. His results are exact when the StokesNavier equations are considered to be exact, To comnare aith our results we make again series expansionsAin nowers of y, We find R -,~._.... z i O-4L __ _.~. f..'_f.~..Y.....

One sees that, as to be expected, the coefficient of y is the same as gl if in eqauation (9) we nut f 4 (/3. III. Higher Approximations, To extend this calculation so that it is applicable to stronger shock waves, it is necessary to make the develonment t6 higher order terms. However, it is not correct merely to solve for v2, v3, etc. from eouations (4). A theory consistent with Enskog's expnansion idea must then include in the pressure tensor and the heat conduction vector the higher order terms resulting from the Enskog develonment. In this section the calculation will be extended two steps further. A. The second approximation (the Burnett terms). It is clear from the method of development that when one goes to the next annroximation one needs only to take the linear second order Burnett termsinto account. The non-linear terms contribute only to the third or higher order of approximation just as the temwerature dependence of/, and ) does not (2) (2)..contribute to the first order effect. The linear terms of p and q are: lX 4 n - X where the a is and the 6 's are slowly varying functions of T(4). They are dimensionless quantities, and for the Maxwellian model they are cure numbers independent.of T. The two eouations taking the nlace of (4) are: ( - Ol t'h-eT'anb or. -:Ph'en-omea"-" inRarfied- "ases"t by C S. W'ag Chn.and '0'. E., Uhlenbeck. A.'PL/J'tUt OM443 Feb..O '94,

,UNIVERSITY OF MICHIGAN 13 and From (a) it follows that f7 ____ / (10b) Z 28'Z~Z(dv.MlN/ From (10a) it fotenllows that - = 4 -xy (; -)Q'j) -/- 6 'i This enuation together with (solb) ives the differential euation for(11). Sin is gin terms of knoby insection it is seen that the interatin fator of (11) ~ o es hk h s,

DEPARTMENT OF ENGINEERING RESEARCH Page I UNIVERSITY OF MICHIGAN 14 _ j46 YA - Pd s Thus (/ etL~ea)' e, _ -e'~C) ZF l with F standing for the right hand members of (11). But g~~ ~ ~~~ O /~at the maximum slope, hence we'have the subscripnt max. means that all 'quantities are to be evaluated at bm where v~ is zero. g2 is then solved by setting.Y~6 as follows from equations (7) and (8). The newly introduced Burnett terms enter in F only as coefficients of VI, but (vl")max. vanishes; hence the Burnett terms do not contribute to g2, or the shock wave thickness to this order of aporoximation. The ratio g2/gl is found to be: (12)

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN 15 For s=2, (12) agrees with the coefficient of the,y term of the result of Thomas. However, this fact should not be. taken too seriously, since it is due to the definition (8) of the shock wave thickness t. If another definition for the thickness was chosen, the Burnett terms would have influenced the coefficient of y2 since they have an influence on the velocity distribution inside the shock. The velocity distribution is obtained by integrating equation (11) and one finds: _~. 4 ). - 7 2 C'~- - 472.., C( where the integration constant is to be determined from the fact that at bm vr2"'must vanish, In 'the last section we will plot the velocity distributions.' vty~ t jv2y2 both' from the consistent theory and from the Stokes-Navier equations when treated as -exact. B. The third approximation. In this ap nroximation not only should all the Burnett terms be taken into account but also the linear terms arising from f (3)are of importance. Drooning all the termns that will not contribute, we write

DEPARTMENT OF ENGINEERING RESEARCH Page UNIV'ERSITY OF MICHIGAN I16 J 1. z, where %,' &, =. a, l th er a) 8 s a gen eral t Z'"" ~ f forJ m;7 IOceI _ _ _.1. _j_._ slowly varying temperature dependent,dimensionless ouantities. For Mlaxwellian molecules thev are simply numbers. The calculation is,tedious and lengthy but the method is the same. We shall only 'r'esent the final result in the most general form;' - t 4 V+ --- —"..? _ —5 - + ":~,:~

DEPARTMENT OF ENGINEERING RESEARCH. Page UNIVERSITY OF MICHIGAN. 17._ d'(ff,/d,5' '2 ~^ t in which we have put B for IV. Application to Monoatomic Gases. The calculation will now be anDlied to monoatomic gases. We shall limit ourselves to the treatment of the Maxwellian model, the molecules interacting with the r 5 force law. This choice was made not because this model is.any better than other models from the physical point of view but because using this model the task of nlm'rical computation will be much sinmlified. For }M{axwell molecules the cdotostants have the following- values: rf = 5/2 5/3 B - 7/3 O o10/3, 75/8 -43 c~2-2 B,= 45/8 d- 5/3 =,3 -3 -3 ' -5/8 9OM - -; 3 &7 21/16.~ =.3 85~ 117/4 The relt for the shock thikness is e in Table I.

DEPARTMENT OF ENGINEERING RESEARCH Page UNIVERSITY OF MICHIGAN P18 Table I Shock Thickness in Monoatoride Gases (Mqaxwell molecules) 4'the mean free path in the medium before the shock. 2/g1 g3/pl Stokes-Navier -1_ 0.3 49 Burnett -1/4 -1.176 Third order ecuations -1/4 1.271 From. thi.s table one sees that the develoument of the thickness of shock waves in cowers of (M. - 1) converges very slotl;y and is therefore only apolicable foi, Mach numbers which are only slightly bigger than one*'-. It should also be remembered that this calculation is valid only for monontomic gases. The extension to diatomic gases is possible. If the effect of the relaxation time is of t`he same order of magnitude as the effects of /A, and ), one can shock, vf, that one probably would not get fast convergent series for /t. vf is given by 29 -dl/- 2 gl?.. which is convergent only for y 4< 1 ar4, even then the convergence is verry slow. Sinee the consistent theory is certainly to yield the successive terms in the above formula for vf in the successive a.oroximntions, it is seen that so far. as finding the velocity distribution is concerned we can not expect to find better conerhence. The shock thickness is derived from the velocity distribution, and hence the slow convergence of our result could have been antlcipated

DEPARTMENT OF ENGINEERING RESEARCH Pa-e UNIVERSITY OF MICHIGAN 19 take this fact into account to the first order of aporoximation by replacing (1) -1/ in the expression for:p4 by 3/+ 4A where 2c is the "second viscosity coefficient"#. To higher orders not onlyy the expressions for 'c 's and 6 Is will. be changed but also new constants will enter. Because of the slow convergence of the development it seems at nresent hardly worthwhile to go into such calculations. In III it was remarked that the fsct that g2/g1 is not changed by taking into account of the Burnett terms is only accidental. There are certainly second order effects due to the Burnett terms, only these effects are not re.. flected in our calculation of the shock wave thickness because of the particular definition we adopted. To see one of the second order effects of the Burnett' terms we ha.ve nlotted in Fig. II (on following page) the curves with vlY v2y2 against b. Curve I is calculated from the Stokes-Navier. ecuations while curve II is obtained from the Burnett equations. The horizontal dotted lines are the asymototes of the value of vf=uf/u uo to the different orders of aoproximatkms as indicated. The plot is made for y=M - 1 - 0,2. One sees that the difference of the velocity distribution s is apnrec.aable even for such weak shock waves. The author wishes to express her gratitude to Professor GCorge E. Uhlenbeck for his interest in this work and for his helpful suggestions and discussions.

-ID.o - - -.4 -2 a O...4.6.8 O10 A.6... 0.8 ~_____ _ ____ 0. __. _'_ Vf-.ECOND APPROXIMATION' -~ -l~ -l~ l ~ — l l l Vf -THIRD APPROXIMATION _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 10.7 _ _ Vj- FIRST APPROXIMATION:~ ~FI,..F SE " ORDER VELO Y DISTRIBUTI R MACH NUM RS = 1.2, y 2 I I'M STOKES NA EQUATIONS_ II FROM RNETT EQUATO

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