Concise and accurate solutions to half-space binary-gas flow problems defined by the McCormack model and specular-diffuse wall conditions

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1 European Journal of Mechanics B/Fluids 23 (24) Concise accurate solutions to half-space binary-gas flow problems defined by the McCormack model specular-diffuse wall conditions C.E. Siewert a D. Valougeorgis b a Mathematics Department North Carolina State University Raleigh NC USA b Department of Mechanical Industrial Engineering University of Thessaly Volos Greece Received 1 August 23; received in revised form 6 December 23; accepted 9 December 23 Available online 15 January 24 Abstract An analytical version of the discrete-ordinates method (the ADO method) is used to establish concise particularly accurate solutions to the viscous-slip the half-space thermal-creep problems for a binary gas mixture. The kinetic equations used to describe the flow are based on the McCormack model for mixtures. In addition to a computation of the viscous-slip thermal-slip coefficients for the case of Maxwell boundary conditions for each of the two species the velocity heat-flow shear-stress profiles are established for both types of particles. Numerical results are reported for three binary mixtures (Ne Ar He Ar He Xe) with various molar concentrations. The complete solution requires only a (matrix) eigenvalue/eigenvector routine the solution of a system of linear algebraic equations thus the algorithm is considered especially easy to use. The developed (FORTRAN) code requires typically less than.1 seconds on a 1.2 GHz Pentium-based PC to solve both problems. 23 Elsevier SAS. All rights reserved. 1. Introduction The study of slip phenomena in gas flows over plane boundaries is of major importance in gas dynamics especially when the flow is in the transition or in the slip regimes [12]. In the transition regime the application of the Boltzmann equation (BE) or of kinetic model equations is necessary to describe the thermal creep the mechanocaloric effects. In addition in the slip regime the determination of the appropriate slip boundary conditions to be coupled with the hydrodynamic continuum equations should be obtained from the solution of kinetic type equations (BE or suitable models). The fundamental theoretical significance the great practical importance of the slip coefficients easily justify the interest in this area of research. Most work in this regard has been focused on the case of a single gas [3 6]; however the case of gas mixtures has also received some significant attention [7 12]. Efforts are now being made [13 18] to extend early work on gas mixtures in order to solve complicated binary gas problems in an efficient accurate manner. This is achieved by adapting well-developed techniques for single-component gases to gas mixtures. The renewed interest in these problems is justified by the basic need of a thorough understing of micro mesoscale transport phenomena in mixtures due to an increasing number of technological applications [192]. One of the major difficulties in dealing with gas mixtures is the large number of parameters (concentration ratios molecular masses diameters gas-surface accommodation coefficients intermolecular laws forces) which are involved in the * Corresponding author. address: siewert@ncsu.edu (C.E. Siewert) /$ see front matter 23 Elsevier SAS. All rights reserved. doi:1.116/j.euromechflu

2 71 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) calculations. Some of these parameters are deduced from modeling while others are obtained from experimental data. Thus it is important to develop methodologies able to hle all these parameters with a modest computational effort. To deal with this kind of problem Ivchenko Loyalka Tompson [1617] have developed general convenient expressions for the slip coefficients of various binary gases. These authors compared their results with experimental work [2122] concluded that there was good agreement when the theoretical results are obtained through a second-order Chapman-Enskog approximation. Sharipov Kalempa [1415] have recently reported solutions of the velocity-slip thermal-slip problems for three different gas mixtures based on the McCormack model equation [23]. Sharipov Kalempa implemented the discrete-velocity method to solve the coupled equations they reported that they found good agreement with results [1617] deduced from the Boltzmann equation. We consider that Sharipov Kalempa [13 15] have demonstrated the merits of the McCormack kinetic model for describing binary gas flows. During the last few years an analytical version of the discrete-ordinates (ADO) method has been developed [24] established as a simple efficient highly accurate methodology for solving problems in rarefied gas dynamics. A large number of a single-gas flow thermal problems has been solved in a unified manner [25 27] while the method has also been used to solve [2829] problems for mixtures described by the Hamel model [3]. In the present work the ADO method is used to solve the half-space viscous-slip (or the Kramers ) thermal-slip (or the half-space thermal-creep) binary-gas flow problems defined by the McCormack model with specular-diffuse boundary conditions. Our objective here is to provide concise accurate solutions (to the considered problems) that define what we consider to be a high stard of accuracy. In addition to defining good numerical results these new solutions are valid for wall conditions described by a general speculardiffuse scattering law the solutions can be implemented at a computational cost much less we believe than the cost of evaluating basic quantities of interest with strictly numerical solutions. Finally we note that our numerical results are reported on a species-specific basis so that various ways (that could depend on a specific application) of defining the velocity heat-flow shear-stress profiles for the binary mixture can be used. While we report in this work the viscous-slip thermal-slip coefficients we also establish velocity heat-flow shear-stress profiles. 2. The McCormack model In this work we base our analysis of a binary gas mixture on the McCormack model as introduced in an important paper [23] published in While we use this model as defined in [23] we use an explicit notation that is appropriate to the analysis computations we report here. We consider that the required functions h α (x v) for the two type of particles (α = 1 2) denote perturbations from Maxwellian distributions for each species i.e. f α (x v) = f α (v) [ 1 + h α (x v) ] (1) where ( ) λα 3/2 f α (v) = n α e λ αv 2 λ α = m α. (2) π 2kT Here k is the Boltzmann constant m α n α are the mass the equilibrium density of the α-th species x is the spatial variable (measured for example in cm) v with components v x v y v z magnitude v is the particle velocity T is a reference temperature. It follows from McCormack s work [23] that the perturbations satisfy (for the case of variations in only one spatial variable) the coupled equations c x x h α(x c) + ω α γ α h α (x c) = ω α γ α L α {h 1 h 2 }(x c) α = 1 2 (3) where the vector c with components c x c y c z magnitude c is a dimensionless variable [ ] mα 1/2 ω α = (4) 2kT the collision frequencies γ α are to be defined. Here we write the integral operators as L α {h 1 h 2 }(x c) = 1 π 3/2 2 β=1 e c 2 h β (x c )K βα (c c) dc x dc y dc z (5) where the kernels K βα (c c) are listed explicitly in Appendix A of this paper. We note that in obtaining Eq. (3) from the form given by McCormack [23] we have introduced the dimensionless vector c differently in the two equations i.e. for the case α = 1 we used the transformation c = ω 1 v whereas for the case α = 2 we used the transformation c = ω 2 v. It can be noted that

3 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) by changing the independent variable v differently in the two balance equations we arrive at the convenient form displayed in Eq. (3). As we wish to work with a dimensionless spatial variable we introduce τ = x l (6) where l = µv (7) P is the mean-free path (based on viscosity) introduced by Sharipov Kalempa [14]. Here following [14] we write ( ) 1/2 2kT v = (8) m where m = n 1m 1 + n 2 m 2. (9) n 1 + n 2 Continuing to follow [14] we express the viscosity of the mixture in terms of the partial pressures P α the collision frequencies γ α as µ = P 1 + P 2 γ 1 γ 2 where P α = n α P n 1 + n 2 (1) (11) γ 1 = [ Ψ 1 Ψ 2 ν (4) ][ 12 ν(4) 21 Ψ2 + ν 12] (4) 1 (12a) γ 2 = [ Ψ 1 Ψ 2 ν (4) ][ 12 ν(4) 21 Ψ1 + ν 21] (4) 1. (12b) Here definitions given in Appendix A have been used Ψ 1 = ν (3) 11 + ν(3) 12 ν(4) 11 (13a) Ψ 2 = ν (3) 22 + ν(3) 21 ν(4) 22. (13b) Finally to compact our notation we introduce σ α = γ α ω α l (14) or more explicitly σ α = γ α n 1 /γ 1 + n 2 /γ 2 n 1 + n 2 so we rewrite Eq. (3) in terms of the τ variable as ( ) mα 1/2 (15) m c x τ h α(τ c) + σ α h α (τ c) = σ α L α {h 1 h 2 }(τ c). (16) In this work we consider the viscous-slip thermal-slip problems so we will seek solutions of Eqs. (16) that are valid for all τ> we use Maxwell boundary conditions at the wall viz. h α (c x c y c z ) = (1 a α )h α ( c x c y c z ) + a α I{h α }() (17) for c x > allc y c z. Note that h α (τ c) h α (τ c x c y c z ) (18)

4 712 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) that we use a 1 a 2 to denote the two accommodation coefficients. In addition we have used I{h α }(τ) = 2 π e c 2 h α (τ c x c y c z )c x dc x dc y dc z (19) to denote the diffuse term in Eq. (17). In this work we consider half-space problems so we must in addition to the boundary condition listed as Eq. (17) specify some conditions on the desired solutions as τ tends to infinity. This point will be addressed later in this work when the two specific problems of interest are discussed in detail. Here we seek to compute the velocity profiles u α (τ) = 1 π 3/2 e c2 h α (τ c)c z dc x dc y dc z the shear-stress profiles p α (τ) = 1 π 3/2 e c2 h α (τ c)c x c z dc x dc y dc z the heat-flow profiles q α (τ) = 1 π 3/2 e c2 h α (τ c)(c 2 5/2)c z dc x dc y dc z so we can obtain these quantities from moments of Eq. (16). To this end we first multiply Eq. (16) by (2a) (2b) (2c) φ 1 (c y c z ) = 1 π e (c2 y +c2 z ) c z (21) integrate over all c y all c z. We then repeat this procedure using Defining φ 2 (c y c z ) = 1 π e (c2 y +c2 z )( c 2 y + c2 z 2) c z. (22) g 2α 1 (τ c x ) = φ 1 (c y c z )h α (τ c) dc y dc z g 2α (τ c x ) = φ 2 (c y c z )h α (τ c) dc y dc z we find from these projections four coupled balance equations which we write (in matrix notation) as ξ G(τ ξ) + ΣG(τ ξ) = Σ τ (23a) (23b) ψ(ξ )K(ξ ξ)g(τ ξ ) dξ (24) where the components of G(τ ξ) are g α (τ ξ) forα = where we now use ξ in place of c x where Σ = diag{σ 1 σ 1 σ 2 σ 2 } (25) ψ(ξ)= π 1/2 e ξ 2. (26) In addition the components k ij (ξ ξ)of the kernel K(ξ ξ) are as listed in Appendix B of this work. So if we can solve Eq. (24) we can compute the quantities of interest from

5 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) u α (τ) = ψ(ξ)g 2α 1 (τ ξ) dξ p α (τ) = ψ(ξ)g 2α 1 (τ ξ)ξ dξ q α (τ) = [( ψ(ξ) ξ 2 1 ) ] g 2 2α 1 (τ ξ) + g 2α (τ ξ) dξ. To complete this section we project Eq. (17) against φ 1 (c y c z ) φ 2 (c y c z ) to find the boundary condition (27a) (27b) (27c) G(ξ) SG( ξ)= ξ ( ) (28) subject to which we must solve Eq. (24). Here S = diag{1 a 1 1 a 1 1 a 2 1 a 2 }. (29) In addition to the boundary condition listed as Eq. (28) a condition for each of the two considered problems on G(τ ξ) as τ tends to infinity must also be specified. This will be done later in this work. 3. The elementary solutions As we intend to find a particular solution if an inhomogeneous source term S(ξ) is added to Eq. (24) we now proceed to establish (in terms of the ADO method) the elementary solutions of ξ G(τ ξ) + ΣG(τ ξ) = Σ τ We seek solutions of Eq. (3) of the form ψ(ξ )K(ξ ξ)g(τ ξ ) dξ. (3) G(τ ξ) = Φ(ν ξ) e τ/ν (31) where the separation constants ν the elementary solutions Φ(ν ξ) are to be determined. Substituting Eq. (31) into Eq. (3) we find (νσ ξi)φ(ν ξ) = νσ ψ(ξ ) [ K(ξ ξ)φ(ν ξ ) + K( ξ ξ)φ(ν ξ ) ] dξ (32a) (νσ + ξi)φ(ν ξ)= νσ ψ(ξ ) [ K(ξ ξ)φ(ν ξ ) + K( ξ ξ)φ(ν ξ ) ] dξ from which we conclude since (32b) that K(ξ ξ)= K( ξ ξ) Φ(ν ξ) = Φ( ν ξ). (33) (34) Now adding subtracting Eqs. (32) one from the other we find that [ ] 1 ξ 2 Σ 2 V (ν ξ) ψ(ξ )K(ξ ξ)v(ν ξ ) dξ = λv (ν ξ) (35a)

6 714 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) U(ν ξ) = ν [V ξ Σ (ν ξ) ψ(ξ )K (ξ ξ)v(ν ξ ) dξ ] (35b) where U(ν ξ) = Φ(ν ξ) + Φ(ν ξ) V (ν ξ) = Φ(ν ξ) Φ(ν ξ). Here λ = 1 ν 2 K + (ξ ξ)= K(ξ ξ)+ K( ξ ξ) K (ξ ξ)= K(ξ ξ) K( ξ ξ) K(ξ ξ)= ξ ξ ΣK +(ξ ξ)σ + Σ 2 K (ξ ξ) ψ(ξ ) ξ ξ ΣK +(ξ ξ)σk (ξ ξ ) dξ. (39) We now introduce a half-range quadrature scheme (with weights nodes w k ξ k ) rewrite Eqs. (35) evaluated at the quadrature points as [ ] 1 N ξi 2 Σ 2 V (ν j ξ i ) w k ψ(ξ k )K(ξ k ξ i )V (ν j ξ k ) = λ j V (ν j ξ i ) (4) k=1 U(ν j ξ i ) = ν [ ] j N Σ V (ν j ξ i ) w k ψ(ξ k )K (ξ k ξ i )V (ν j ξ k ) (41) ξ i k=1 for i = N. Eq. (4) defines our eigenvalue problem to which we have added the subscript j to label the eigenvalues eigenvectors. Once this eigenvalue problem is solved we have the elementary solutions from Φ(ν j ξ i ) = 1 2 [ U(νj ξ i ) + V (ν j ξ i ) ] (42a) Φ(ν j ξ i ) = 1 2 [ U(νj ξ i ) V (ν j ξ i ) ]. (42b) Note that the separation constants defined by ν j =±λ 1/2 j (43) occur in ± pairs. From this point we take ν j to be the positive root listed in Eq. (43). Once we have solved the eigenvalue problem defined by Eq. (4) we can write our general (discrete ordinates) solution to Eq. (3) as 4N [ G(τ ±ξ i ) = Aj Φ(ν j ±ξ i ) e τ/ν j + B j Φ(ν j ξ i ) e τ/ν ] j (44) j=1 for i = N. Here the arbitrary constants {A j } {B j } are to be determined from the boundary conditions of a specific problem. Before proceeding to develop our solutions to the viscous-slip the thermal-slip problems we make one modification to Eq. (44). While we could use the solution as given by Eq. (44) we can also improve it. We have found that the eigenvalue problem yields one separation constant say ν 1 that approximates the one expected unbounded separation constant. And so instead of using what we see as an approximate solution that corresponds to ν 1 in Eq. (44) we replace the contribution (36a) (36b) (37) (38a) (38b)

7 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) from ν 1 with the exact solution we would expect if the approximation parameter N were unbounded. In this way we replace one (the least accurate) approximate solution with an exact solution that allows us to capture the correct asymptotic form 4N [ G(τ ±ξ i ) = AG + + BG (τ ±ξ i ) + Aj Φ(ν j ±ξ i ) e τ/ν j + B j Φ(ν j ξ i ) e τ/ν ] j (45) for i = N.Here 1 G + = G s (τ ξ) = σ 1 τ ξ sσ 1 (τ ξ/σ (46ab) are two exact solutions of Eq. (3). In the work we use ( ) 1/2 ( ) 1/2 m1 m2 r = s = (47ab) m 2 m 1 in order to compact our notation. 4. The problems Having developed our elementary solutions of Eq. (3) we are now ready to use them to solve the two specific problems basic to our current study. First however we note that while Eqs. (2) define the species-specific quantities u α (τ) p α (τ) q α (τ) forα = 1 2 the way to define the basic elements for a binary gas mixture is not so clear. Here we follow other works [13 18] define these basic elements as u(τ) = ϕ u1 u 1 (τ) + ϕ u2 u 2 (τ) p(τ) = ϕ p1 p 1 (τ) + ϕ p2 p 2 (τ) q(τ) = ϕ q1 q 1 (τ) + ϕ q2 q 2 (τ) (48c) where the adaptation coefficients ϕ iα i= u p q α = 1 2 are to be specified. Since these factors have been defined in several ways in other works [13 18] since the choice of these factors could conveniently be made differently for different applications of the theory we develop our solution report our numerical results without specifying these factors The viscous-slip problem The viscous-slip problem (also known as Kramers problem) has no driving (inhomogeneous) term in Eq. (24) so our solution can be constructed from Eq. (45). Since there is no explicit driving term for this problem the solution is required to diverge as τ tends to infinity but at the same time the bulk velocity of the mixture should satisfy d lim τ dτ u(τ) = k p (49) where k p is a normalization constant that depends on the choice of adaptation factors to be used. The condition listed as Eq. (49) requires us to take B j = in Eq. (45) in order to accommodate all choices of adaptation factors to be used we first normalize our solution by taking (48a) (48b) B = 1 σ 1. (5) It follows that the constants A A j can be obtained from the system of linear algebraic equations we find when G(τ ±ξ i ) = AG N G (τ ±ξ i ) + A j Φ(ν j ±ξ i ) e τ/ν j (51) σ 1 is substituted into a discrete-ordinates version of Eq. (28) evaluated at the N quadrature points ξ i. Once these constants are established we can use Eqs. (27) express our final results as

8 716 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) N u 1 (τ) = A + τ + A j N u1 (ν j ) e τ/ν j 4N u 2 (τ) = s(a + τ)+ A j N u2 (ν j ) e τ/ν j p 1 (τ) = 1 4N 2 + A j N p1 (ν j ) e τ/ν j (52c) p 2 (τ) = 1 sσ 1 4N + A j N 2 σ p2 (ν j ) e τ/ν j 2 (52d) 4N q α (τ) = A j N qα (ν j ) e τ/ν j. (52e) If we use uasy(τ) to denote the asymptotic part (the part that excludes the exponential factors) of u(τ) then u 1asy (τ) = A + τ (53a) u 2asy (τ) = s(a + τ) (53b) where Eq. (47b) has been noted. And so using the definition ζ P = u asy() u asy() (54) we see that ζ P = A (55) is the viscous slip coefficient for the bulk velocity of the mixture defined by Eq. (48a) for any choice of the adaptation factors. To complete Eqs. (52) we find N N uα (ν j ) = F T α w k ψ(ξ k ) [ Φ(ν j ξ k ) + Φ(ν j ξ k ) ] (56a) k=1 N N pα (ν j ) = F T [ α w k ψ(ξ k )ξ k Φ(νj ξ k ) Φ(ν j ξ k ) ] (56b) k=1 N N qα (ν j ) = w k ψ(ξ k )F T qα (ξ k) [ Φ(ν j ξ k ) + Φ(ν j ξ k ) ] (56c) k=1 where we use the superscript T to denote the transpose operation where 1 ξ 2 1/2 1 F 1 = F 2 = F 1 q1 (ξ) = F q2 (ξ) = ξ 2 1/2 1. (52a) (52b) (57a d) While we have introduced the (arbitrary) normalization listed in Eq. (5) the bulk velocity ϕ u1 u 1 (τ) + ϕ u2 u 2 (τ) u(τ) = k P (58) ϕ u1 + sϕ u2 will satisfy Eq. (49) for any choice of the adaptation factors. In regard to our numerical work we report in a following section of this work the viscous-slip coefficient ζ P the speciesspecific profiles u α (τ) q α (τ) for selected sets of data. In this way results are available for any choice of the adaptation factors ϕ iα. Having completed our solution to the viscous-slip problem we look now at the second of the two problems solved in this work.

9 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) The half-space thermal creep problem In regard to the case of thermal creep the flow is caused by a constant temperature gradient in the direction (z: parallel to the wall) of the flow so it is helpful to linearize the particle velocity distribution functions about a local Maxwellian rather than an absolute Maxwellian as was done in Eq. (1). We thus write { [( f α (τ η c) = f α (c) 1 + η c 2 3 ] } )K η + R η + h α (τ c) (59) 2 where η = z/l K η R η are the (non-dimensional) temperature density gradients in the η direction ( ) 3/2 λα f α (c) = n α e c2 λ α = m α. (6) π 2kT For the problem of thermal creep we take R η = K η then let k T = K η so that Eq. (59) becomes [ ( f α (τ η c) = f α (c) 1 + η c 2 5 ) ] k T + h α (τ c). (61) 2 As a result of this linearizion an inhomogeneous source (1/2)(ξ 2 1/2) S(ξ) = k T 1 (1/2)(ξ 2 1/2) (62) 1 must now be added to Eq. (24) to yield S(ξ) + ξ G(τ ξ) + ΣG(τ ξ) = Σ ψ(ξ )K(ξ ξ)g(τ ξ ) dξ. (63) τ Since we have linearized about a local Maxwellian we now require our solution to Eq. (63) to be bounded as τ tends to infinity so we write our (discrete ordinates) solution as 4N G(τ ±ξ i ) = G p (τ ±ξ i ) + AG + + A j Φ(ν j ±ξ i ) e τ/ν j (64) where G p (τ ξ) is a particular solution that corresponds to the inhomogeneous driving term S(ξ). We impose the (arbitrary) normalization k T = 1 find that we can write E(ξ 2 1/2 sw) 2E G p (τ ξ) = F(ξ 2 (65) 1/2 rw) 2F where we have again used Eqs. (47). After we note definitions given in Appendix A we can write w = 5 4 r ν(2) 12 ν (1). (66) 12 In addition the two constants E F required in Eq. (65) are defined by a system of linear algebraic equations which we write as [ ] E N = 1 [ ] 1/σ1 (67) F 2 1/σ 2 where the elements of N are given by n 11 = Φ [η (2) 12 ]2 8 η (1) 12 n 12 = η (6) 12 5 [η(2) 12 ]2 r3 8 η (1) 12 (68a) (68b)

10 718 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) n 21 = η (6) 21 5 [η(2) 21 ]2 s3 8 η (1) (68c) 21 n 22 = Φ [η (2) 21 ]2 8 η (1) (68d) 21 with Φ 1 = η (5) 11 + η(5) 12 η(6) 11 (69a) Φ 2 = η (5) 22 + η(5) 21 η(6) 22. (69b) At this point we can substitute Eq. (64) into a discrete-ordinates version of the boundary condition listed as Eq. (28) to define a system of linear algebraic equations. After solving the set of algebraic equations for the constants A {A j } we can write our final results for the half-space thermal-creep problem as 4N u 1 (τ) = A swe + A j N u1 (ν j ) e τ/ν j 4N u 2 (τ) = sa rwf + A j N u2 (ν j ) e τ/ν j 4N p α (τ) = A j N pα (ν j ) e τ/ν j q 1 (τ) = 5 4N 2 E + A j N q1 (ν j ) e τ/ν j q 2 (τ) = 5 4N 2 F + A j N q2 (ν j ) e τ/ν j where we continue to use the definitions given by Eqs. (56). Considering that the thermal-slip coefficient for the mixture is given by ζ T = lim τ u(τ) (71) we can write ζ T = (ϕ u1 + sϕ ua w(sϕ u1 E + rϕ u2 F). (72) In contrast to the viscous-slip problem we see that here the slip coefficient depends on the factors ϕ u1 ϕ u2 so for this reason we intend to report ζ 1 = u 1 ( ) = A swe (73a) ζ 2 = u 2 ( ) = sa rwf (73b) as well as species-specific profiles u α (τ) q α (τ) for selected sets of data. It follows that once adaptation factors are specified the thermal-slip coefficient for the mixture can be evaluated from ζ T = ϕ u1 ζ 1 + ϕ u2 ζ 2. (74) As our solutions are complete we continue with an implementation of the algorithms defined here to establish numerical results with what we consider to be a very good stard of accuracy. (7a) (7b) (7c) (7d) (7e)

11 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) Numerical results The first thing to note in regard to our numerical work is the way we defined the quadrature scheme for the analytical discrete-ordinates method used in this work. To keep matters simple we used the transformation v(ξ) = e ξ (75) to map ξ [ ) onto v [ 1] we then used the Gauss Legendre scheme mapped (linearly) onto the interval [ 1]. In order to evaluate the merits of the solutions developed here for the half-space viscous-slip thermal-creep problems we have elected to use the three data cases defined by Sharipov Kalempa [14]. These three data cases refer to a mixture of the species: (i) Ne Ar (ii) He Ar (iii) He Xe. As we are reporting numerical work only for the case of rigid-sphere interactions we can see that the McCormack model requires for this case only three ratios: the mass ratio (m 1 /m the diameter ratio (d 1 /d the density ratio (n 1 /n. In addition by formulating the McCormack model in terms of a convenient mean-free path by observing the ratios of parameters that result we can see that the constant factor (πkt /32) 1/2 in Eq. (A.35) of Appendix A need not be specified. For the sake of our computations we consider that the data m 2 = m 1 = m 2 = m 1 = 4.26 d 2 = 1.46 d 1 d 2 = d 1 (Ne Ar mixture) (He Ar mixture) d m 2 = m 1 = = (He Xe mixture) d 1 are exact. We follow Sharipov Kalempa [14] tabulate our results for these three cases in terms of the molar concentration defined (in terms of the first particle) as C = n 1/n 2. (76) 1 + n 1 /n 2 In Tables 1 3 we list some typical results for the viscous-slip problem similar results for half-space thermal-creep problem are listed in Tables 4 7. Although we have computed all the basic quantities u α (τ) q α (τ) p α (τ) we have (for economy of space) omitted p α (τ) from our tabulations we report results only for two choices of the accommodation coefficients. While we have no definitive proof of the accuracy of our results we believe the results listed in our tables are correct (within the context of the kinetic model used) to all digits given. To establish some confidence in our numerical results we found stability in the results as we varied the only approximation parameter N from 2 to 1 we obtained identical results from two independently implemented numerical codes: one based on MATLAB the other on FORTRAN. In addition we have also used the identity n 1 n p 1 (τ) + n 2 n p 2 (τ) = (77) Table 1 The viscous-slip coefficient Ne Ar mixture He Ar mixture He Xe mixture a 1 = 1. a 1 =.3 a 1 = 1. a 1 =.3 a 1 = 1. a 1 =.3 a 2 = 1. a 2 =.6 a 2 = 1. a 2 =.6 a 2 = 1. a 2 =.6 C ζ P

12 72 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) Table 2 The viscous-slip problem: velocity profiles for each species for the case a 1 = 1. a 2 = 1. C =.5 Ne Ar mixture He Ar mixture He Xe mixture τ u 1 (τ) u 2 (τ) u 1 (τ) u 2 (τ) u 1 (τ) u 2 (τ) ( 1) ( 1) ( 1) ( 1) ( 1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) Table 3 The viscous-slip problem: heat-flow profiles for the case a 1 = 1. a 2 = 1. C =.5 Ne Ar mixture He Ar mixture He Xe mixture τ q 1 (τ) q 2 (τ) q 1 (τ) q 2 (τ) q 1 (τ) q 2 (τ) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 1.665( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 1.229( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 2) ( 2) 1.396( 1) ( 1) ( 1) ( 1) ( 2) ( 2) ( 2) ( 1) ( 1) ( 1) ( 2) ( 2) ( 2) ( 1) ( 1) ( 1) ( 2) ( 2) ( 2) ( 1) 1.15( 1) ( 1) ( 2) ( 2) ( 2) ( 1) ( 2) ( 1) ( 2) 5.654( 2) ( 2) ( 1) ( 2) ( 1) ( 2) ( 2) ( 2) 6.319( 2) ( 2) ( 1) ( 3) ( 3) ( 3) ( 3) ( 2) ( 2) ( 4) ( 4) ( 3) ( 3) 3.23( 3) ( 3) Table 4 The thermal-slip coefficients for the case a 1 = 1. a 2 = 1. Ne Ar mixture He Ar mixture He Xe mixture C ζ 1 ζ 2 ζ 1 ζ 2 ζ 1 ζ ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 6.493( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 4.858( 1) ( 1) ( 1) 6.571( 1) ( 1) ( 1) ( 1) ( 1) 5.199( 1) ( 1) ( 1) ( 1) 5.49( 1) ( 1) ( 1) ( 1) ( 1) 5.736( 1) ( 1) 6.221( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 6.39( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 8.141( 1) ( 1)

13 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) Table 5 The thermal-slip coefficients for the case a 1 =.3 a 2 =.6 Ne Ar mixture He Ar mixture He Xe mixture C ζ 1 ζ 2 ζ 1 ζ 2 ζ 1 ζ ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 4.762( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 4.575( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 5.52( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 4.729( 1) Table 6 The thermal-creep problem: velocity profiles for each species for the case a 1 = 1. a 2 = 1. C =.5 Ne Ar mixture He Ar mixture He Xe mixture τ u 1 (τ) u 2 (τ) u 1 (τ) u 2 (τ) u 1 (τ) u 2 (τ) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 2.667( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 3.289( 1) ( 1) ( 1) ( 1) 3.618( 1) ( 1) 3.489( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 3.535( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) Table 7 The thermal-creep problem: heat-flow profiles for the case a 1 = 1. a 2 = 1. C =.5 Ne Ar mixture He Ar mixture He Xe mixture τ q 1 (τ) q 2 (τ) q 1 (τ) q 2 (τ) q 1 (τ) q 2 (τ) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)

14 722 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) where n = n 1 + n 2 to see that our computations confirmed (to the same number of digits listed in our tables) that Π(τ) = n 1 n p 1(τ) + n 2 n p 2(τ) (79) is constant. We have also been able (after taking note of some differing definitions) to confirm for the case of rigid-sphere interactions the three four digit numerical results for the viscous-slip problem the half-space thermal-creep problem reported for the case of purely diffuse boundary conditions i.e. a 1 = 1a 2 = 1 by Sharipov Kalempa [1415] for the three mixtures defined in that work. Finally we have seen that our computations yield known results available from the S-model calculations [26] when we allow our data to collapse to the single-species gas. We have obtained this reduction to a single gas in three ways: (i) n 1 = for which the quantities with subscript 2 yield the single-gas results (ii) n 2 = for which the quantities with subscript 1 yield the single-gas results (iii) m 1 = m 2 d 1 = d 2. That we obtain identical results from these three limiting cases can be attributed we believe to the good way the mean-free path l used in this work is defined [14]. We find it especially interesting to see that the S model can be obtained from the McCormack model when the data for the gas mixture is reduced to that of a single species. In regard to adaptation factors we note that ( ) n α m α m 1/2 ϕ uα = α = 1 2 (8) n 1 m 1 + n 2 m 2 m α can be used with our solutions in the manner of Eq. (48a) to obtain the bulk velocity of the mixture the slip coefficients as they are defined by Sharipov Kalempa [1415]. On the other h Sharipov Kalempa [13] Naris Valougeorgis Sharipov Kalempa [18] have used the adaptation factors ϕ uα = n ( ) α m 1/2 α= 1 2 (81) n 1 + n 2 m α to define the velocity profile for a binary mixture. Note that the mean molecular mass m is defined in Eq. (9). (78) 6. A relationship between the two considered problems In a recent work [31] Sharipov used physical arguments to provide within the context of the S model for a single-species gas a relationship between the heat flow from Kramers problem the thermal-slip coefficient. In a following work [32] Siewert used the defining equations relevant to the linearized Boltzmann equation ( some kinetic models) a form of the boundary condition (that includes the Maxwell Cercignani-Lampis boundary conditions as special cases) to find evaluate an explicit relationship between the heat flow from Kramers problem the thermal-slip coefficient. These two results [3132] define for the case of a single-species gas a relationship between the viscous-slip problem the half-space thermal-creep problem that can be used for example to help evaluate numerical results obtained for the two problems. We have extended the mentioned results [3132] to the currently considered case of a binary-gas mixture described by the McCormack model. Since our derivation follows very closely the one given in [32] we list here only the final result. If we add subscripts K for the viscous-slip problem T for the half-space thermal-creep problem we find we can write n 1 n Q K1 + rn 2 n Q K2 = β ( n1 nσ 1 ζ T1 + n 2 nσ 2 ζ T2 ) (82) where ( n1 β = E + n ) 2 F. nσ 1 nσ 2 Here E F are the constants defined by Eq. (67) the thermal-slip coefficients ζ Tα are as defined by Eqs. (73) Q Kα = q Kα (τ) dτ where the heat flow profiles for Kramers problem are defined by Eq. (2c). We have confirmed to many significant figures Eq. (82) as part of our testing of the solutions reported in this work. (83) (84)

15 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) Concluding remarks To conclude this work we note that we believe our solutions to the considered problems of viscous slip thermal creep are especially concise easy to use. We have included a general form of the Maxwell boundary condition in our formulation we have reported what we believe to be highly accurate results for some test cases. It should be noted that our complete species-specific results listed for the two considered problems in Eqs. (52) (7) are continuous in the τ variable thus are valid for all τ. In this work we have considered only the case of rigid-sphere interactions but as pointed out by Sharipov Kalempa [14] the solutions can be used for other scattering laws such the one defined by the Lennard-Jones potential simply by using instead of Eqs. (A.34) (A.35) appropriate definitions of the omega integrals [12]. In addition it is clear that we now have essentially all we need to solve well the classical flow problems (Poiseuille thermal creep Couette) in a plane channel for a binary gas mixture that can be described by the McCormack model. In developing our solutions here for the McCormack model we have noted one aspect of the computation that made this work especially interesting when compared to our previous work [2829] for mixtures described by the Hamel model. This aspect of the solution can be seen in Eq. (39) where there appears an integral term that we have not seen in any of our previous work with the ADO method. While this integral term has required some attention the ensuing additional work is considered modest when we take into account the merits of the McCormack model. We also see that the way in which Eqs. (35) (39) were formulated can be utilized for other problems in rarefied gas dynamics. Since our solutions require only a matrix eigenvalue/eigenvector routine a solver of linear algebraic equations the algorithm is especially efficient fast easy to implement. In fact the developed (FORTRAN) code solves both problems for all quantities of interest with 5 or 6 figures of accuracy in less than.1 seconds on a 1.2 GHz mobile Pentium III machine which confirms we believe the merit of this work. Acknowledgement The authors takes this opportunity to thank L.B. Barichello Felix Sharipov J.R. Thomas Jr. for some helpful discussions regarding this work. Appendix A. Basic elements of the defining equations Here we list some basics results that are required to define certain elements of the main text of this paper. First of all in regard to Eq. (5) we note that K βα (c c) = K (1) βα (c c) + K (2) βα (c c) + K (3) βα (c c) + K (4) βα (c c) α β = 1 2 (A.1) where K (1) 11 (c c) = 1 + { 2 [ 1 η (1) ] (2) 12 η 12( c } 5/2 c c (A.2) K (2) 11 (c c) = (2/3) [ 1 2r η 12]( (1) c ( 3/2 c 3/2 (A.3) K (3) [ 11 (c c) = 2ϖ 1 (c c) 2 (1/3)c 2 c 2 ] (A.4) K (4) 11 (c c) = [ ( (4/5)β 1 c (2) 5/2 η 12]( c 5/2 c c (A.5) K (1) 21 (c c) = r { 2η (1) [ η(2) r 2 ( c 2 5/ + c 2 5/2 ]} c c (A.6) K (2) 21 (c c) = (4/3)r η 12( (1) c ( 3/2 c 3/2 (A.7) K (3) 21 (c c) = 2η 12[ (4) (c c) 2 (1/3)c 2 c 2 ] (A.8) K (4) 21 (c c) = (4/5)η 12( (6) c 2 5/( c 2 5/ c c (A.9) K (1) 22 (c c) = 1 + { 2 [ 1 η (1) ] (2) 21 η 21( c } 5/2 c c (A.1) K (2) 22 (c c) = (2/3) [ 1 2s η 21]( (1) c 2 3/( c 2 3/ (A.11) K (3) [ 22 (c c) = 2ϖ 2 (c c) 2 (1/3)c 2 c 2 ] (A.12)

16 724 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) K (4) 22 (c c) = [ ( (4/5)β 2 c (2) 5/2 η 21]( c 5/2 c c (A.13) K (1) 12 (c c) = s { 2η (1) [ η(2) s 2 ( c 2 5/ + c 2 5/2 ]} c c (A.14) K (2) 12 (c c) = (4/3)s η 21( (1) c ( 3/2 c 3/2 (A.15) K (3) 12 (c c) = 2η 21[ (4) (c c) 2 (1/3)c 2 c 2 ] (A.16) K (4) 12 (c c) = (4/5)η 21( (6) c ( 5/2 c 5/2 c c. (A.17) Here r = r 2 / ( 1 + r 2) (A.18) s = s 2 / ( 1 + s 2). (A.19) In addition ϖ 1 = 1 + η (4) 11 η(3) 11 η(3) 12 (A.2) ϖ 2 = 1 + η (4) 22 η(3) 22 η(3) 21 (A.21) β 1 = 1 + η (6) 11 η(5) 11 η(5) 12 (A.22) β 2 = 1 + η (6) 22 η(5) 22 η(5) 21 (A.23) where η (k) ij = ν(k) ij /γ i. (A.24) Following McCormack [23] we write ν (1) αβ = 16 3 ν (2) αβ = ν (3) αβ = 16 5 ν (4) αβ = 16 5 ν (5) αβ = m αβ n β Ω m αβ 11 α ( mαβ m α ν (6) αβ = 64 ( mαβ 15 m α ) 2 n β ( Ω 12 αβ 5 2 Ω11 αβ m α ( ) mαβ 2 ( mα 1 n β m α m β ( ) mαβ 2 mα n β m α m β ( ) mαβ 3 mα n β Γ (5) m αβ β ) 3 ( mα m β (A.25) ) (A.26) 3 Ω11 αβ + m β m α Ω 22 αβ ( 1 3 Ω11 αβ Ω22 αβ ) 3/2 n β Γ (6) αβ with ( Γ (5) 15mα αβ = Ω22 αβ m ) ( ) β Ω 11 mβ (5Ω 4m β 8m αβ 12 α 2m αβ ) Ω13 αβ α after a correction by Sharipov Kalempa [14] ) (A.27) ) (A.28) (A.29) (A.3) (A.31) Γ (6) αβ = Ω22 αβ Ω11 αβ 5 2 Ω12 αβ Ω13 αβ. In addition m αβ = m α m β /(m α + m β ) (A.32) (A.33)

17 C.E. Siewert D. Valougeorgis / European Journal of Mechanics B/Fluids 23 (24) the Ω functions are the Chapman Cowling integrals [12] which for the case of rigid-sphere interactions take the simple forms with Ω 12 αβ = 3Ω11 αβ Ω13 αβ = 12Ω11 αβ Ω 22 αβ = 2Ω11 αβ (A.34a c) Ωαβ 11 = 1 ( ) 1/2 πkt (d α + d β ) 2. (A.35) 4 2m αβ Appendix B. The basic kernels for flow problems The components of the kernel K(ξ ξ) required in Eq. (24) are as follows: k 11 (ξ ξ)= 2ϖ 1 ξ ξ + 1 η (1) 12 12( η(2) ξ 2 + ξ 2 ) ( 1 /2 + 2β1 ξ 2 )( 1/2 ξ 2 ) 1/2 /5 (B.1) k 12 (ξ ξ)= (1/2)η (2) β 1( ξ 1/2 /5 (B.2) k 13 (ξ ξ)= 2η (4) 12 ξ ξ + r { η (1) 12 + [ η(2) 12 r 2 ( ξ 2 1/ + ξ 2 1/2 ] /2 } + 2η 12( (6) ξ ( 1/2 ξ 1/2 /5 (B.3) k 14 (ξ ξ)= (1/2)r 3 η (2) ( 2η(6) ξ 1/2 /5 (B.4) k 21 (ξ ξ)= η (2) β 1( ξ 1/2 /5 (B.5) k 22 (ξ ξ)= (4/5)β 1 (B.6) k 23 (ξ ξ)= rη (2) ( 4η(6) ξ 1/2 /5 (B.7) k 24 (ξ ξ)= (4/5)η (6) 12 k 31 (ξ ξ)= 2η (4) 21 ξ ξ + s { η (1) 21 + [ η(2) 21 s 2 ( ξ 2 1/ + ξ 2 1/2 ] /2 } + 2η 21( (6) ξ ( 1/2 ξ 1/2 /5 (B.9) k 32 (ξ ξ)= (1/2)s 3 η (2) ( 2η(6) ξ 2 ) 1/2 /5 (B.1) k 33 (ξ ξ)= 2ϖ 2 ξ ξ + 1 η (1) 21 21( η(2) ξ 2 + ξ ( 1 /2 + 2β2 ξ ( 1/2 ξ 1/2 /5 (B.11) k 34 (ξ ξ)= (1/2)η (2) β 2( ξ 1/2 /5 (B.12) k 41 (ξ ξ)= sη (2) ( 4η(6) ξ 1/2 /5 (B.13) k 42 (ξ ξ)= (4/5)η (6) 21 (B.14) k 43 (ξ ξ)= η (2) β ( 2 ξ 2 1/ /5 (B.15) k 44 (ξ ξ)= (4/5)β 2. (B.16) References [1] S. Chapman T.G. Cowling The Mathematical Theory of Non-Uniform Gases Cambridge University Press Cambridge [2] J.H. Ferziger H.G. Kaper Mathematical Theory of Transport Processes in Gases North-Holl Amsterdam [3] C. Cercignani Rarefied Gas Dynamics: from Basic Concepts to Actual Calculations Cambridge University Press Cambridge 2. [4] E.P. Muntz Rarefied gas dynamics Ann. Rev. Fluid Mech. 21 (1989) [5] G.A. Bird Molecular Gas Dynamics the Direct Simulation of Gas Flows Oxford University Press New York [6] F. Sharipov V. Seleznev Data on internal rarefied gas flows J. Phys. Chem. Ref. Data 27 (1998) [7] H. Lang S.K. Loyalka An exact expression for the diffusion slip velocity in a binary gas mixture Phys. Fluids 13 (197) [8] S.K. Loyalka Velocity slip coefficient the diffusion slip velocity for a multicomponent gas mixture Phys. Fluids 14 (1971) [9] V.G. Chernyak V.V. Kalinin P.E. Suetin The kinetic phenomena in nonisothermal motion of a binary gas mixture through a plane channel Int. J. Heat Mass Transfer 27 (1984) [1] Y. Onishi On the behavior of a slightly rarefied gas mixture over plane boundaries J. Appl. Math. Phys. (ZAMP) 37 (1986) [11] D. Valougeorgis Couette flow of a binary gas mixture Phys. Fluids 31 (1988)

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