Steady gas flows at small Knudsen numbers around arbitrary bodies (asymptotic behavior for small Knudsen numbers of the solution of time-independent boundary value problems of the Boltzmann equation over a general domain) are considered when the Reynolds number of the system is of the order of unity. The generalized slip flow theory developed for the Boltzmann-Krook-Welander equation is extended for the standard Boltzmann equation. From the result, the effect of gas rare faction on the flow (the relation between Boltzmann and hydrodynamic systems) is clarified, and several features of the force on a closed body in the gas are derived

AOKI, Kazuo

SONE, Yoshio

English

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Asymptotic Theory of SlightlyRarefied Gas Flow and Force on aClosed BodySONE, Yoshio; AOKI, KazuoSONE, Yoshio ...[et al]. Asymptotic Theory of Slightly Rarefied Gas Flow and Force on aClosed Body. Memoirs of the Faculty of Engineering, Kyoto University 1987, 49(3): 237-2481987-08-28http://hdl.handle.net/2433/281355Mem. Fae. Eng., Kyoto Univ. Vol. 49, No. 3 (1987) 237 Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body By Y oshio SoNE* and Kazuo AoK1* (Received March 31. 1987) Abstract Steady gas flows at small Knudsen numbers around arbitrary bodies (asymptotic behavior for small Knudsen numbers of the solution of time-independent boundary value problems of the Boltzmann equation over a general domain) are considered when the Reynolds number of the system is of the order of unity. The generalized slip flow theory developed for the Boltzmann-Krook-Welander equation is extended for the standard Boltzmann equation. From the result. the effect of gas rarefaction on the flow (the relation between Boltzmann and hydrodynamic systems) is clarified, and several features of the force on a closed body in the gas are derived. I . Introduction The relation between the hydrodynamic equation and the Boltzmann equa-tion has been discussed by various authors_D-9J In this connection the Hilbert and the Chapman-Enskog expansions are often mentioned. The expansions, however, are not derived in the framework of the boundary-value problem, and the hydrodynamic equations derived have some awkward properties in considering the boundary-value problem.2i, 3> In this paper, taking the time-independent boundary value problem of the Boltzmann equation over a general domain, we investigate the asymptotic behav-ior of the solution for small Knudsen numbers to derive a set of hydrodynamic equations and their boundary conditions that covers some effects of the Knudsen number (gas rarefaction). From the result, the effect of gas rarefaction on velocity and temperature fields is discussed, and several features of the force acting on a closed body in a slightly rarefied gas are derived. * Department of Aeronautical Engineering 238 Y oshio SoNE and Kazuo AoKI II. Asymptotic Solution for Small Knudsen Numbers II - 1 . Analysis and Hydrodynamic Systems Mach number Ma, Reynolds number Re, and Knudsen number Kn, important parameters in characterizing slightly rarefied gas flows, are related as3> Ma~Re Kn. This relation is important in considering the asymptotic analysis for small Knudsen numbers (Kn« 1 ). The linear theory 4>· 5l, where the quantities of O (Ma2) are neglected, is applicable only for very small Re (Re<<Kn). The standard Hilbert expansionD corresponds to the case with Re-H>0. When Re is of the order of unity (the case of our interest), we must take into account that Ma is of the same order of smallness as Kn. In the present paper, noting that Ma is a measure of deviation from an equilibrium state at rest, we investigate the asymptotic behavior for Kn« l of the system where the deviation from a uniform equilibrium state at rest is of the order of the Knudsen number of the system. Owing to limited space, we give only the outline of the analysis. We introduce the notations: T0, Po, / 0, and lo are the temperature, the pressure, the velocity distribution of gas molecules, and the mean free path of our reference equilibrium state at rest; R is the (specific) gas constant; L is the characteristic length of our system ; Lx; is the rectangular space coordinates; ( 2 RT0) ½ (; is the molecular velocity; / 0 ( 1 +¢) is the velocity distribution of gas molecules. The behavior of the gas ¢ is described by the Boltzmann equation: ( 2) where the standard collision integral]( 1 +¢, 1 +¢) is split into two parts: the linearized operator L(¢) and the remainder](¢, ¢). The complete definition of the collision operators is not given here, but no confusion will take place. On the boundary a condition for the reflected molecules is imposedD, 2>: ( 3) Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body 239 where n; is the unit normal to the boundary, pointed to the gas, and </Jw is a given function or is related with </J (?;;n;< 0 ). The asymptotic solution of the boundary-value problem is obtained in the form: <p=<pH+<pK, </JH=</JH1k+</JH2k2+ "·, </JK=</JK1k +</JK2k2+ "·, ( 4) ( 5) ( 6) where </JH, called hydrodynamic part, represents the overall solution, and </JK, Knudsen-layer part, the correction near the boundary. Since we are considering the case where the perturbed distribution </J is of the order of k, </JH .. and </JK .. are of the order of unity. First we determine </JH as a solution of the Boltzmann equation whose length scale of variation is of the order of the characteristic length L of the system [i}</J I ax;=O(</J)). Substituting Eq. ( 5) in the Boltzmann equation Cl) and arr~nging the same order terms of k, we obtain a sequence of integral equations for </J-: L(</JH .. ) =Inhomogeneous term (</JHm-1,"·, </JHI), ( 7) which can in principle be solved from the lowest order. From the solvability condition* of Eq. ( 7) with m ~ 2, we get a sequence of partial differential equations, called hydrodynamic equations, that govern the component functions of the expansions corresponding to Eq. ( 5) of hydrodynamic quantities (veloc-ity, temperature, etc.). Since the hydrodynamic part </JH, obtained without paying attention to the boundary condition, cannot in general be made to satisfy the boundary condition ( 3) [the differential operator is multiplied by the small parameter k in Eq. ( 1 )), we introduce the Knudsen-layer correction </JK, which is assumed to have the length scale of variation normal to the boundary of the order of lo [kn,a</J I ax; =O(</J)) and to be appreciable only near the boundary. Substituting Eq. ( 6) with </JH previously obtained in Eq. ( 1 ) and arranging the terms with the properties of </JK and </JH in mind, we obtain a sequence of (inhomogeneous) one-dimensional linearized Boltzmann equations. • Homogeneous integral equation L(¢) = 0 has the five independent solutions 1, (,. and l:l. 240 Y oshio So NE and Kazuo AoK1 y arpKl ( ) ~,n, ~ =L <PK1 , ( 8) r,n, a:;2 =LC</JK2) + 2]((¢,Hl)o' <PK1) +](<PKI, <PK1) -r, c( as1 ) arpKl +( asz ) arpKl ), ax, O as1 ax, O as2 ( 9) (10) where x,.. is the boundary surface, T/ is a stretched coordinate normal to the boundary, s 1 and s 2 are (unstretched) coordinates within a parallel surface TJ= const., and ( ) 0 denotes that the quantity in ( ) is evaluated at rJ= 0. The boundary condition for <PK"' at TJ= 0 is (t,n,> 0 ), (11) where ¢,..., is defined by (12) The boundary value of "'""'' which is undetermined, is involved in the boundary condition (11). The analysis of the equations under the condition that <PK vanishes rapidly away from the boundary gives conditions among the boundary values of hydrodynamic parts of hydrodynamic quantities and their derivatives6),ioHz) as well as the Knudsen-layer correction </JK, These conditions serve as boundary conditions for the hydrodynamic equations and are hereafter called slip boundary condition for convenience. Here we list the hydrodynamic equations. The non-dimensional hydrody-namic quantities u,, p, r, and w are introduced: ( 2 RT0) ½ u, is the gas velocity, Po( 1 +p) the pressure, To( 1 +r) the temperature, p0(RT0)- 1 ( 1 +w) the density. The hydrodynamic quantities are split into H and K parts and expanded in power series of k as in Eqs. ( 4 ), ( 5 ), and ( 6 ). (um, PH, <H, and w" are defined in <PH by the same formulae as u, etc. in </J (Appendix 2 ), and u,K etc. are defined as the remainders.) (13) au,Hl = 0 ax, ' 04 a) Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body 241 au;n1 __ _l apll2 + _l a2u;Jfl U;nt ax; - 2 ax; 2 ri axJ ' ar:m 1 a2r:m U;n1 OX; = 2 r2 oxJ ' au;112 OWm OX; = -U;nJ OX; ' 1 o c ( au;J/1 au;n1 )J + zr4 OX; <HI OX; + OX; ' or:112 ( ) 01:m 2 ap112 U;n1---::;-- + WmU;n1 +u;112 -0 - -5 U;n1-,,-uX; X; r.JX; _ 1 ( au.HI au;n1 ) 2 1 a2 c 1 2 ) - 571 OX; + iii: + 2 oxJ r2r:112 + zrsr:HI ' (14 b) (14 c) (15 a) (15 b) 05 c) (16) where r, are numerical constants related with the collision operators L and J (App. 1 ). Equations for (u;J/1, r:n1, P112 ) (Eqs. 04 a~c)J are the Navier-Stokes equations for an incompressible fluid, and the successive equations for (u,nm, !nm, Pnm+1, m ~ 2) are the same order differential equations as Eqs. (14 a~c). These sets of equations are derived by a systematic small parameter (k) expansion, where no assumption is made on the form of the velocity distribution function but special attention is paid to the estimate of physical variables so that the analysis may cover physically interesting cases with finite Reynolds numbers. Incidentally, in the standard Hilbert expansion, sets of the first-order differential equations, starting with the Euler equations for an ideal gas, are derived; in the Chapman-Enskog expansion, the order of the differe:1tial equations, starting also with the Euler equations, increases with the progress of approximation. The slip boundary conditions for the hydrodynamic equations (13 ~ 15 c) take the same form as those for the Boltzmann-Krook-Welander equation except for numerical constants. The latter results are given in Ref. 5 for solid bound-ary where neither evaporation nor condensation takes place and in Ref. 13 for interface between gas and its condensed phase where evaporation or condensa-tion is taking place. (For brevity, the former boundary is hereafter called solid 242 Y oshio So NE and Kazuo AoKI boundary and the latter interface.) The slip boundary conditions are as follows : ( i ) On the solid boundary uiH1 -uwi1 = 0, LHl -rw1 = 0, d arm L/12 -rw2 = I -an;, :X; (ii) On the interface Cl 7 a) (17 b) (18 a) (18 b) (18 c) (19 a) (19 b) (19 c) (20 a) (20 b) (20 c) where l; is the direction cosine of a tangential vector to the boundary ; i IL is the mean curvature of the boundary where the sign of each principal curvature is taken negative when the corresponding center of curvat!-lre is on the gas side; Uu,;m, Lwm, and Pwm are the terms of the expansions of the velocity ( 2 RTo) ½Uw; (with Uw;n;= O ), the temperature T 0 ( 1 +rw) of the boundary, and the saturation gas pressure Po ( 1 + Pw) at temperature To ( 1 + rw) : Uu,;=Uu,;1k +uw;2k2+ ···, Tw=Twik +tw2k2+ •••, Pw=Pw1k+Pw2k2+···, (21 a) (21 b) (21 c) Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body 243 (uwi, Tw, and Pw correspond to the deviation from our reference equilibrium state and thus are of the order of k. The higher order terms of k are retained for the convenience of treating the problems where the boundary values are not known beforehand.] ; ko, K1, d1, K2, C1, C4*, C6, C1, Cs, C9, C10, d4*, ds, d1, ds, d9, d10 are numerical constants. For B-K-W equation, C1 = 0.558437, C1= - 0.380569, d5= 0.330345, d9= - 0.223375, K1 = - 0.383161. c; = - 2.132039. Cs= 2.32007 4, d1 = 1.302716, d1= - 0.131574, d10= 0, K2= - 0.795186. c6 = 0.820853, C9= 1.066019, dl = - 0.4467 49, ds= - 0.0028315, ko= - 1.016191. Finally, we list the hydrodynamic parts of the stress tensor1>- 2> p0(6;;+ P;;) and heat flow vector1>- 2> Po ( 2 RT0) ½ Q; (App. 2 ). The component functions of their expansions corresponding to Eq. ( 5 ) are : (22) 5 arHI QiH2= - 4 72 ax;' 5 arH2 5 arHI 1 a2um1 QiH3= - 4 f2 ax; - 4 75<Hl iJx, + 2 73 ax; . (23) The last term of P;;Ha (Q,83) is non Navier-Stokes stress (heat flow) and is called thermal stress. The term before the last in P,;83 (Q,83) shows the temperature dependence of viscosity (thermal conductivity). The results of this subsection (Sec. II - 1 ) are the generalization of the senior author's work (Ref. 5) developed for B-K-W equation. II - 2 . Velocity and Temperature Fields The first order hydrodynamic equations (Eqs. (14 a~c)J are the Navier-Stokes equations for an incompressible fluid. The second order equations (Eqs. 05 a~c)J combined with Eqs. (14 a~c) differ a little from the Navier-Stokes equations of a slightly compressible gas. If ra in the numerical coefficient of 244 Y oshio SoNE and Kazuo AoK1 {)2,HI/ oxJ in the square brackets of the first term on the right hand side of Eq. (15 b) is zero, Eqs. (15 a~c) coincide with the second order equations of the Mach number expansion of the Navier-Stokes equations for a compressible gas. (Noting that the case Ma=ak with a=O( 1) is under consideration, transform the k-expansion to Ma-exp.J The difference is due to the thermal stress in PiiHa• This difference, however, can be eliminated by the replacement : (24) Further, the slip boundary condition (up to the second order of k) does not contain p113 (cf. Eqs. (17a) ~ (18c), (19a) ~ (20c)J. Thus, we conclude: Proposition 1 : Except for the Knudsen-layer correction, the velocity and the temper-ature fields of a slightly rarefied gas can be calculated correctly up to the second order in the Knudsen number by the slightly compressible Navier-Stokes equations with the slip boundary conditions. The effect of gas rarefaction comes in through the boundary condition. (N. B. In an infinite-domain problem where the pressure is specified at infinity, the pressure modified by Eq. (24) should be used. In most physical problems, however, o2rHI/ oxJ vanishes at infinity and no correction is necessary.) ill. Force and Its Moment on a Closed Body Take a closed body Bi in a gas. The gas may or may not be bounded, and other bodies may lie in the gas. We will investigate the force and its moment on B1• In the following analysis, oBi denotes the boundary of Bi; oBo a closed surface that encloses only Bi in the gas ; n, the unit normal of the surface of integration under consideration pointed to the region including infinity; dS its surface element. Theorem 1 : The Knudsen-layer part of the momentum flux does not contribute to the force acting on a closed body. Proof: Let Po (o,;+ W,;) be momentum flux tensor, where W,;=P,;+ 2 ( 1 +w)u,u;, and F, be the force, normalized by -p0L2, acting on a closed body Bi in the gas. Then, (25) Because oW,;/ ox1= O in the gas (App. 3 ), the surface of integration can be deformed arbitrarily in the gas. Taking a surface of integration oBo outside the Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body 245 Knudsen layer, we have F,= f 'IJf,;Hn;dS, 8B., (26) since 'IJf,;x vanishes there. Further, because o'IJf,iH/ ox;= O (App. 3 ), we can deform oBo in Eq. (26) arbitralily in the gas. oB0 may be in the Knudsen layer, especially on the body oB1• (QED) Theorem 2 : The Knudsen-layer part of the momentum flux does not contribute to the moment of force acting on a closed body. Proof: The moment of force M, around origin, normalized by P0L3, is expressed by (27) where e,;• is Eddington's e. The proof goes parallel to that of Theorem 1 if '1Jf0 a is replaced by Em.xh'IJfk; because -0 e,hkxh'IJf•;= 0 from o'IJf,;/ox;= 0 and W,;=W;,. X; (QED) Corollary: On a solid boundary, F, and M, are calculated correctly up to the k3-order only by P,;H on 0B1. Proof: From the Knudsen-layer analysis, uiH1n,=u"nn,= 0 on a solid boundary [cf. Eqs. (17a) and (18b)). (Incidentally, u,113n, is not necessarily zero.) (QED) As in Theorem 1 , we can prove the following theorem with the aid of the formulae in Appendix 3. (Proof omitted) Theorem 3 : The Knudsen-layer part of mass (energy) flux does not contribute to the mass (energy) flow to a closed body. We prepare a lemma for Theorem 4 : Lemma : Let f (x ,) be a function three times continuously differentiable in a domain containing a closed surf ace (say oB0). Then (28 a) (28 b) Proof: Extend f(x,) over the whole region inside oB0 keeping its smoothness and apply Gauss theorem. (QED) Theorem 4 : The non Navier--Stokes stress in Pm system contributes neither to the force nor to the moment of force on a closed body. 246 Yoshio SoNE and Kazuo AoKI The non N-S stress in Pm syst. means the thermal stress in P,;HJ modified by the replacement (24). Proof: Its contributions to F, and M, are, respectively, proportional to: (29 a) and (29 b) After deforming aB1 to 8B0, apply the lemma. (QED) Combining Theorems 1, 2, 4 with Proposition 1 we find: Proposition 2 : Under the condition of Proposition 1 , the force and the moment of force on a closed body can be computed correctly up to the k 3-order of F, and M, by the classical hydrodynamic procedure based on the Navier-Stokes solution and the N -S stress if the slip boundary condition is taken into account. The results of this section are the generalization of the senior author's work (Ref. 14) developed for the linearized Boltzmann equation. Appendix 1 . Numerical constants r, Let A ((2) and B(t2) be the solutions of the integral equations: L[t,A (t2)] = -t,(t2 - ~ ), with the subsidiary condition: where L [···] is the linearized collision operator (cf. Eq. ( 1 )J and t 2=tl C(t2), D(t2), and G(t2) are introduced by Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body 247 2] [1;2- ~, l;,A (/;2)] =/;,G(/;2). The r, are defined by the integrals of these functions : ra=MAB), rs=- 6r2+ 2Ia(A) + 2MAG), where with F=A, B, etc. For B-K-W equation r,= 1. and for the hard sphere model r, are1si: r1= 1.2700. r2= 1.9223, ra= 1.9479, 2 . Relations between u,, r, etc. and (jJ where w= f (j)Edt;, (1 +w)u,= f l;,(/JEdt;, ~ ( 1 +w) r= f (/;,2- ~ )(j)Edt; - ( 1 +w)ur, P.;= 2 f l;,(;(j)Edt;-2( 1 +w) u,u;, r4 = 0.63489, P=w+r+wr, and the integration is carried out over the whole space of /;,. 3. Conservation equations rs= 0.96070. Multiplying Eq. ( 1 ) by E, l;,E, or /;,2 E and integrating over the whole space of I;,, we have a ax; [( 1 +w)u;] = 0, a ,,-- [2( 1 +w)u,u;+ P,;] = 0, UX; Yoshio SoNi,; and Kazuo AOKI These relations also hold with subscript H since hydrodynamic part is a solution of Eq. ( l l. References I ) II. Grad, in "Handbuch der Physik," edited by S. Fli.igge, Springer-Verlag, Berlin ( 1958), Vol. 12. p. 205. 2 ) C. Cercignani, "Theory and Application of the Boltzmann Equation," Scottish Academ-ic. Edinburgh (1975). 3 ) Y. Sane, in "Rarefied Gas Dynamics," edited by H. Oguchi, University of Tokyo Press, Tokyo <1984), Vol. l, p. 71. 4) Y. Sane, in "Rarefied Gas Dynamics," edited by L. Trilling and H. Y. Wachman, Academic, New York (1969), Vol. 1. p. 243. 5) Y. Sane, in "Rarefied Gas Dynamics," edited by D. Dini, Editrice Tecnico Scientifica, Pisa (1971), Vol. 2, p. 737. 6 ) H. Grad, in "Transport Theory," edited by R. Bellman, G. Birkhoff, and I. Abu-Shumays, American Mathematical Society, Providence (1969), p. 269. 7 ) H. Grad, Phys. Fluids, 6, 147 ( 1963). 8 ) C. Cercignani, Phys. Fluids, 11, 303 (1968). 9) J. S. Darrozes, in "Rarefied Gas Dynamics," edited by L. Trilling and H. Y. Wachman. Academic, New York (1969), Vol. l, p.111. 10) W. Greenberg and C. V. M. van der Mee, Z. Angew. Math. Phys., 35, 156 (1984). 11) C. Bardos, R. E. Caflisch, and B. Nicolaenko, Comm. Pure Appl. Math., 39, 323 (1986). 12) C. Cercignani, in "Trends in Applications of Pure Mathematics to Mechanics," edited by E. Kroner and K. Kirchgassner, Springer-Verlag, Berlin (1986), p. 35. 13) Y. Onishi and Y. Sane, J. Phys. Soc. Japan, 47, 1676 (1979). 14) Y. Sane, in "Rarefied gas Dynamics," edited by H. Oguchi, University of Tokyo Press, Tokyo (1984), Vol. 1, p.117. 15) T. Ohwada, private communication. 

Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body

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Asymptotic Theory of Slightly Rarefied Gas Flow and Force on a Closed Body

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