Derivation of relaxional transport equations for a gas of pseudo-maxwellian molecules

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1 Deriation of relaxional transport equations for a gas of pseudo-maxllian molecules Alexander Orlo To cite this ersion: Alexander Orlo. Deriation of relaxional transport equations for a gas of pseudo- Maxllian molecules. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp < jp1: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1992 HAL is a multi-disciplinary open access archie for the deposit and dissemination of scientific research documents, whether they published or not. The documents may come from teaching and research institutions in France or abroad, or from public or priate research centers. L archie ouerte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de nieau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou priés.

2 of relaxional transport equations Deriation a gas of pseudo-maxllian molecules for 9, Institute for High Temperatures (USSR Acad. Sci.), 1319 Izhorskaya St., Moscow Dept. Russia , deried by the proposed iteration scheme of the solution of the Boltzmann equation for type molecules. pseudo-maxllian fiffit + Vf (go4~) f'f( sin x dxed~i gun where go4~ ga(g, x) const, n (*) Address for correspondence: Tashkentskaya St , Moscow , Russia. J. Phys. I France 2 (1992) MARCH 1992, PAGE 229 Classification Physics Abstracts 05.20D Short Communication Alexander V. Orlo(*) (Receied 16 October 1991, reised 9 December 1991, accepted 20 December 1991) Abstract. In this brief note the moment and energy transport equations of relaxational The Boltzmann equation + fiffit V f f'f( f fi) ga(g, x) sin x dxded~i II for a gas of pseudc-maxllian molecules [ii is f, (I) II1 d~ f. We shall sole equation (I) iteratiely with the use of the following scheme: fk+ifit + gun fk+i (go4~) f( fk)( sin x dxded~i V fk II1 fi Similar method was applied in reference [2] to the linearized Boltzmann equation. The first iteration is fifit + gun ii (go4~) ii fo)( sin fi dxded~i V fo gun fo V fo, (2) x

3 fi~afit + gon~a -fi fofit V fo ~ i~ ~ ~ ~'~ ~ i~ ~~ ~ ~~~ where ddt alai + u V, c u. Vu + c (fl ~ polynonfial of molecular elocity and integrate oer elocities. The right-hand corresponding of the resulting equation is expressed in terms of temporal and spatial deriaties of the side parameters of fo, I-e- n, u, T. These parameters do not contain any ~a (4) by I, c, and c~ and integration oer c. equation the tensor of iscous stresses is Since tipfit + T7 (pu) 0, pdudf + VP 0, dpdf + (53)pT7 u 0 left-hand side describes flows for which the iscous stresses change quickly. Equation (6) The a formal solution has 230 JOURNAL DE PHYSIQUE I N 3 where fo n(2~ktm)~~~ exp (-m( u)~2kt) (3) u T and afterwards real parameters of the gas. After introduction of ii and fo here n, ~a equation (2) follows: rewrite as ~~it~ ~ ~~~ ~ ~~ ~ ~ Vln +d (cc ~I) j, (4) kt 3 2kT 2 In order to obtain the relaxation equations for the fluxes multiply equation (4) by the contributions and hence obey Euler equations as follows from equation (4). This follows from multiplication of a~~a ~a~ m cac~~a d~c m d~ obtain in this way fi~a~ fit + gon~a~ -(film) (p6a~ + pttatt~) V~ (pttatt~tt~ + ptt~6a~)?fl (Pal?a (PUfl) (5) Taking into account the Euler equations derie from equation (5) that fi~a~ fit -pea~ gon~a~, ea~ Vatt~ + V~tta (23)6a~V~tt~. (6) If the left-hand side is neglected receie the ll-known linear Naier-Stokes law ~«~ -~oe«~ngo -»e«~. (6'j t t exp -go n(r)dr (pea~),, dt' (7) ~w ~' ~a~ Here the subscript t' shows that the hydrodynamical alues in pntheses should be calculated at the moment t', I.e. earlier than at t. Equation (7) is the sc-called transport equation with

4 (or with memory) [3], but contrary to [2, 3] in the present note the memory kernel is delay explicitly. expressed (m2) a~~a d~ qa + u~~a~ (~ + go) (q«+ u~~a~) ()pua + )pu~ua) + ~~ + )u~) 6a~ + (jp + )pu~) au~j +$7~ ((~ l'~ -~pv«$ +?VfIP gonq«. (8) equation (7). If neglect the left-hand side obtain in the case u similar to our equations (6), (8) re deried by another approximate method Expressions reference [5], but the equation for qa obtained in that paper does not contain the term with in equations be obtained from the Boltzmann equation with the same accuracy as the classical discussions with Dr. A-G. Bashkiro and Dr. A-D. Khonkin gratefully appreciated. Valuable author thanks ery much the referee for his penetrating criticism and ery useful notes The N 3 EQUATION FOR A GAS OF PSEUDO-MAXWELLIAN MOLECULES 231 We do similar calculations to derie the relaxational equation for the heat flux (m2) cc~ f d~. q In order to do this multiply equation (4) by (m2)a~ and integrate oer elocity. Taking into account that hae the equation similar to the preceding one: We simplify the right-hand side of this equation using the Euler equations as earlier: () The solution of this equation be expressed as the transport law with memory similar to 0 the classical Fourier law go (5p2gon) Va(kTm) AVaT. (9) Notice that Apc 53, I-e- the Eucken relation [4] Apc 52 is hence not alid for pseudc-maxllian molecules. ~a~. Conclusion. Relaxational equations for momentum and energy fluxes generalizing ll-known Naier-Stokes and Fourier laws obtained using the first iteration of the proposed scheme of solution of the Boltzmann equation. This short note appears to be another proof of the fact that these Ones. Acknowledgements. that helped the author to understand the problem more deeply.

5 I~OGAN M.N., Rfied Gas Dynalrdcs (Plenum, N.Y., 1969). [ii ZUBAREV D-N- and KHONKIN A-D-, Theor. Math. Phys. II (1972) 601. [2j [3j ZUBAREV D-1i, and TISHCHENKO S-V-, Physica 59 (1972) 285. FERzIGER J.H. and I~APER H-G-, Mathematical Theory of Transport Processes in Gases (North- [4] Amsterdarr L., 1972). Holland, [5j KHONKIN A.D-, Fluid Mech. SOC. Res. 9 (1980) JOURNAL DE PHYSIQUE I N 3 References