Mass transport in a strongly sheared binary mixture of Maxwell molecules
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Mass transport in a strongly sheared binary mixture of Maxwell molecules
Vicente Garz´o ∗ Departamento de F´ısica, Universidad de Extremadura, E-06071 Badajoz, Spain (Dated: November 4, 2018)Transport coefficients associated with the mass flux of a binary mixture of Maxwell moleculesunder uniform shear flow are exactly determined from the Boltzmann kinetic equation. A normalsolution is obtained via a Chapman–Enskog-like expansion around a local shear flow distributionthat retains all the hydrodynamics orders in the shear rate. In the first order of the expansionthe mass flux is proportional to the gradients of mole fraction, pressure, and temperature but,due to the anisotropy induced in the system by the shear flow, mutual diffusion, pressure diffusionand thermal diffusion tensors are identified instead of the conventional scalar coefficients. Thesetensors are obtained in terms of the shear rate and the parameters of the mixture (particle masses,concentrations, and force constants). The description is made both in the absence and in thepresence of an external thermostat introduced in computer simulations to compensate for the viscousheating. As expected, the analysis shows that there is not a simple relationship between the resultswith and without the thermostat. The dependence of the three diffusion tensors on the shear rate isillustrated in the tracer limit case, the results showing that the deviation of the generalized transportcoefficients from their equilibrium forms is in general quite important. Finally, the generalizedtransport coefficients associated with the momentum and heat transport are evaluated from a modelkinetic equation of the Boltzmann equation.
PACS numbers: 51.10.+y, 05.20.Dd, 05.60.-k, 47.50.-d
I. INTRODUCTION
The description of transport properties for states close to equilibrium in gaseous binary mixtures is well established.In these situations, the Curie principle states that the presence of a velocity gradient (second-rank tensorial quantity)cannot modify a vectorial quantity such as the mass flux j , which is generated by gradients of mole fraction x ,pressure p , and temperature T . As a consequence, the mutual diffusion coefficient D (which couples the mass currentwith ∇ x ), the pressure diffusion coefficient D p (which couples the mass current with ∇ p ) and the thermal diffusioncoefficient D T (which couples the mass current with ∇ T ) do not depend on the velocity gradient. However, when theshear rate applied is large, non-Newtonian effects are important so that the Curie principle does not hold and thecoefficients associated with the mass transport are affected by the presence of shear flow. In particular, if the spatialgradients ∇ x , ∇ p , and ∇ T are weak, one expects that the flux j is still linear in these gradients but the standardscalar coefficients { D, D p , D T } must be replaced by the shear-rate dependent second-rank tensors { D ij , D p,ij , D T,ij } .The aim of this paper is to determine the above tensors in the framework of the Boltzmann equation.We are interested in a situation where weak spatial gradients of mole fraction, pressure, and temperature coexistwith a strong shear rate. Under these conditions, the application of the conventional Chapman-Enskog expansion around the local equilibrium state to get higher order hydrodynamic effects (Burnett, super-Burnett, . . . ) to the massflux turns out to be extremely difficult. This gives rise to look for alternative approaches. A possibility is to expandaround a more relevant reference state than local equilibrium. Since we want to compute the mass transport in astrongly sheared mixture, the so-called uniform shear flow (USF) state can be chosen as the reference state. The USFstate is characterized by constant mole fractions, a uniform temperature, and a linear velocity profile u x = ay , where a is the constant shear rate. Due to its simplicity, this state has been widely used in the past to shed light on thecomplexities associated with the nonlinear response of the system to the action of strong shearing. In addition, theUSF state is one of the rare exceptions for which the hierarchy of moments of the Boltzmann equation admits anexact solution for single and multicomponent gases of Maxwell molecules (repulsive potential of the form r − ). Inthis case, explicit expressions of the pressure tensor (which is the relevant irreversible flux of the problem) have beenobtained for arbitrary values of the shear rate and the parameters of the system (masses, concentrations and forceconstants).As said before, here we want to compute the mass transport under USF for Maxwell molecules. Since the mixtureis slightly perturbed from the USF, the Boltzmann equation can be solved by an expansion in small gradients aroundthe (local) shear flow distribution instead of the (local) equilibrium. This is the main feature of the expansion sincethe reference state is not restricted to small values of the shear rate. In the first order of the expansion, the setof generalized transport coefficients { D ij , D p,ij , D T,ij } are identified from the mass flux j as nonlinear functions ofthe shear rate and the parameters of the mixture. This Chapman-Enskog-like expansion has been used to analyzetransport properties in spatially inhomogeneous states near USF in the case of ordinary gases and more recentlyin the context of granular gases. Some previous attempts have been carried out earlier by the author and coworkers in the case of thediffusion tensor D ij . However, all these studies have been restricted to perturbed steady states with the constraints p = const and T = const. Although steady states are in general desirable for practical purposes, especially in computersimulations, here we extend the above studies to a general time and space dependence of the hydrodynamic fields.This allows us to evaluate new contributions to the mass flux (those proportional to ∇ p and ∇ T ), which where nottaken into account in the previous studies. The plan of the paper is as follows. First, a brief summary of the results obtained from the Boltzmann equationfor a binary mixture of Maxwell molecules under USF is presented in Sec. II. Section III deals with the perturbationscheme used to solve the Boltzmann equation for the mixture to first order in the deviations of the hydrodynamicfield gradients from their values in the reference shear flow state. The generalized transport coefficients characterizingthe mass transport around USF are also defined in Sec. III. These coefficients are explicitly obtained in Sec. IV withand without the presence of an external thermostat introduced usually in computer simulations to compensate forthe viscous heating. The dependence of some of these coefficients on the shear rate is illustrated with detail in thetracer limit case, showing that the influence of shear flow on mass transport is quite significant. The paper is closedby a brief discussion of the results in Sec. V, the generalized transport coefficients associated with the momentumand heat transport are evaluated in Appendix D from a simple model kinetic equation of the Boltzmann equation.
II. A BINARY MIXTURE UNDER UNIFORM SHEAR FLOW
We consider a dilute binary mixture where f s ( r , v ; t ) is the one-particle velocity distribution function of species s ( s = 1 , f s is given by the set of two coupled nonlinear Boltzmann equations: (cid:18) ∂ t + v · ∇ + ∂∂ v · F s m s (cid:19) f s ( r , v , t ) = X r =1 J sr [ v | f s ( t ) , f r ( t )] , (2.1)where m s is the mass of a particle of species s , F s is a possible external force acting on particles of species s , and J sr [ v | f s , f r ] is the Boltzmann collision operator, which in standard notation reads J sr [ f s , f r ] = Z d v Z d Ω | v − v | σ sr ( v − v , θ )[ f s ( v ′ ) f r ( v ′ ) − f s ( v ) f r ( v )] . (2.2)The basic moments of f s are the species number densities n s = Z d v f s , (2.3)and the mean velocity of species s u s = 1 n s Z d vv f s . (2.4)These quantities define the total number density n = P s n s and the flow velocity u = P s ρ s u s /ρ , where ρ s = m s n s is the mass density of species s and ρ = P s ρ s is the total mass density. The temperature T is defined as nk B T = X s n s k B T s = X s m s Z d vV f s , (2.5)where k B is the Boltzmann constant and V = v − u is the peculiar velocity. The second identity in (2.5) defines thepartial kinetic temperatures T s of species s . They measure the mean kinetic energy of particles of species s . Moreover,in a dilute gas the hydrostatic pressure p is given by p = nk B T . The quantities n s , u , and T are associated withthe densities of conserved quantities (mass of each species, total momentum, and total energy). The correspondingbalance equations define the dissipative fluxes of mass j s = m s Z d v V f s , (2.6)momentum (pressure tensor), P = X s P s = X s m s Z d v VV f s , (2.7)and energy (heat flux) q = X s q s = X s m s Z d v V V f s . (2.8)The second equalities in Eqs. (2.7) and (2.8) define the partial contributions P s and q s to the pressure tensor andheat flux, respectively. The fact that the mass flux j s is defined with respect to the local center-of-mass velocity u implies that X s j s = . (2.9)The USF state is macroscopically defined by constant densities n s , a spatially uniform temperature T ( t ) and alinear velocity profile u ( y ) = u ( y ) = u ( y ) = ay b x , where a is the constant shear rate. Since n s and T are uniform,then j s = q = , and the transport of momentum (measured by the pressure tensor) is the relevant phenomenon.In the USF problem, the temperature tends to increase in time due to viscous heating. Usually, an external force(thermostat) is introduced in computer simulations to remove this heating effect and keep the temperature constant. The simplest choice is a Gaussian isokinetic thermostat given by F s = − m s α V , (2.10)where the thermostat parameter α is a function of the shear rate adjusted as to keep the temperature constant.The implicit assumption behind the introduction of these forces is that they play a neutral role in the transportproperties, so that the latter are the same with and without a thermostat, when conveniently scaled with the thermalspeed. Nevertheless, this expectation is not in general true, except for some specific situations and/or interactionpotentials. At a microscopic level, the USF is characterized by a velocity distribution function that becomes uniform in thelocal Lagrangian frame, i.e., f s ( r , v ; t ) = f s ( V , t ). In that case, Eq. (2.1) with the choice (2.10) reduces to ∂∂t f − ∂∂V i ( a ij V j + αV i ) f = J [ f , f ] + J [ f , f ] (2.11)and a similar equation for f . Here, a ij = aδ ix δ jy . The hierarchy of velocity moments associated with the Boltzmannequation (2.11) can be recursively solved in the particular case of Maxwell molecules, i.e., when particles of species r and s interact through a potential of the form V rs ( r ) = κ rs r − . The key point is that for this interaction the collisionrate gσ rs ( g, θ ) is independent of the relative velocity g and so the collisional moments of order k only involve momentsof degree smaller than or equal to k . In particular, the first- and second-degree collisional moments are given by m s Z d vV J sr [ f s , f r ] = − λ sr m s m r ( ρ s j r − ρ r j s ) , (2.12) m s Z d vVV J sr [ f s , f r ] = λ ′ sr ( m s + m r ) m s (cid:20)(cid:18) ρ s p r + ρ r p s − j s · j r (cid:19) − ρ s P r − ρ r P s + j s j r + j r j s ] − λ sr ( m s + m r ) m s (cid:20) (cid:18) m s m r ρ r P s − ρ s P r (cid:19) + (cid:18) − m s m r (cid:19) ( j s j r + j r j s ) (cid:21) , (2.13)where p s = tr P s = n s k B T s is the partial hydrostatic pressure and λ sr = 1 . π (cid:18) κ sr m s m r m s + m r (cid:19) / , λ ′ sr = 2 . π (cid:18) κ sr m s m r m s + m r (cid:19) / . (2.14)Thanks to the above property, exact expressions of the pressure tensor P for a binary mixture of Maxwell moleculesunder USF were obtained some time ago. The nonzero elements of P are related to the rheological properties of themixture, namely, the nonlinear shear viscosity and the viscometric functions. In reduced units, they turn out to benonlinear functions of the (reduced) shear rate a ∗ = a/ζ (where ζ is a convenient time unit defined below) and theparameters of the mixture: the mass ratio µ = m /m , the mole fraction x = n /n and the force constant ratios κ /κ and κ /κ . It must be noted that in the particular case of Maxwell molecules there is an exact equivalencebetween the USF results with and without the external forces (2.10). As will be shown below, beyond the USFproblem, the presence of the thermostat does not play a neutral role in the results and a certain influence may exist. III. CHAPMAN–ENSKOG-LIKE EXPANSION AROUND USF
As said in the Introduction, the main aim of this work is to analyze mass transport of a dilute binary mixturesubjected to USF. In that case, let us assume that the USF state is disturbed by small spatial perturbations. Theresponse of the system to those perturbations gives rise to contributions to the mass flux that can be characterizedby generalized transport coefficients. This Section is devoted to the evaluation of those coefficients.In order to analyze this problem we have to start from the set of Boltzmann equations (2.1) with a general time andspace dependence. Let u = a · r be the flow velocity of the undisturbed USF state, where the elements of the tensor a are a ij = aδ ix δ jy . In the disturbed state, however the true velocity u is in general different from u , i.e., u = u + δ u , δ u being a small perturbation to u . As a consequence, the true peculiar velocity is now c ≡ v − u = V − δ u , where V = v − u . In the Lagrangian frame moving with u , the Boltzmann equations (2.1) can be written as ∂∂t f − ∂∂V i ( a ij V j + αV i ) f + ( V + u ) · ∇ f + αδ u · ∂f ∂ V = J [ f , f ] + J [ f , f ] , (3.1a) ∂∂t f − ∂∂V i ( a ij V j + αV i ) f + ( V + u ) · ∇ f + αδ u · ∂f ∂ V = J [ f , f ] + J [ f , f ] , (3.1b)where here the derivative ∇ f s is taken at constant V . In addition, in Eqs. (3.1a) and (3.1b) the thermostat force hasbeen assumed to be proportional to the actual peculiar velocity, F s = − m s α ( V − δ u ) where now the parameter α is in general a function of r and t through their functional dependence on the hydrodynamic fields n s and T . Thegeneralization of α to the inhomogeneous case is essentially a matter of choice. Here, for the sake of simplicity, wewill take two different choices for α : (i) α = 0, so that the temperature grows in time, and (ii) the same expressionobtained in the (pure) USF problem, except that the densities and temperature are replaced by those of the generalinhomogeneous state.The macroscopic balance equations associated with this disturbed USF state are obtained by taking moments inEqs. (3.1a) and (3.1b) with the result ∂ t n s + u · ∇ n s + ∇ · ( n s δ u ) = − ∇ · j s m s , (3.2) ∂ t δu i + a ij δu j + ( u + δ u ) · ∇ δu i = − ρ − ∇ j P ij , (3.3)32 n∂ t T + 32 n ( u + δ u ) · ∇ T = − aP xy + 32 T X s =1 ∇ · j s m s − ( ∇ · q + P : ∇ δ u + 3 pα ) , (3.4)where the mass flux j s , the pressure tensor P , and the heat flux q are defined by Eqs. (2.6), (2.7), and (2.8), respectively,with the replacement V → c . The corresponding balance equations for the mole fraction x = n /n and the pressure p = nk B T can be obtained from Eqs. (3.2) and (3.4). They are given by ∂ t x + ( u + δ u ) · ∇ x = − ρn m m ∇ · j , (3.5) ∂ t p + ( u + δ u ) · ∇ p + p ∇ · δ u = −
23 ( aP xy + ∇ · q + P : ∇ δ u + 3 pα ) . (3.6)We assume that the deviations from the USF state are small. This means that the spatial gradients of the hy-drodynamic fields are small. For systems near equilibrium, the specific set of gradients contributing to each flux isrestricted by fluid symmetry, Onsager relations, and the form of entropy production. However, in far from equilibriumsituations (such as the one considered in this paper), only fluid symmetry applies and so there is more flexibility inthe representation of the heat and mass fluxes since they can be defined in a variety of equivalent ways dependingon the choice of hydrodynamic gradients used. In fact, some care is required in comparing transport coefficients indifferent representations using different independent gradients for the driving forces. Here, as in previous works, the mole fraction x , the pressure p , the temperature T , and the local flow velocity δ u are chosen as hydrodynamicfields.Since the system is strongly sheared, a solution to the set of Boltzmann equations (3.1a) and (3.1b) can be obtainedby means of a generalization of the conventional Chapman-Enskog method in which the velocity distribution functionis expanded around a local shear flow reference state in terms of the small spatial gradients of the hydrodynamicfields relative to those of USF. This is the main new ingredient of the expansion. This type of Chapman-Enskog-like expansion has been already considered to get the set of shear-rate dependent transport coefficients inthermostatted shear flow problems and it has also been recently used for inelastic gases. In the context of the Chapman–Enskog method, we look for a normal solution of the form f s ( r , V , t ) ≡ f s [ A ( r , t ) , V ] , (3.7)where A ( r , t ) ≡ { x ( r , t ) , p ( r , t ) , T ( r , t ) , δ u ( r , t ) } . (3.8)This special solution expresses the fact that the space dependence of the reference shear flow is completely absorbedin the relative velocity V and all other space and time dependence occurs entirely through a functional dependenceon the fields A ( r , t ). The functional dependence (3.4) can be made local by an expansion of the distribution functionin powers of the hydrodynamic gradients: f s [ A ( r , t, V ] = f (0) s ( V ) + f (1) s ( V ) + · · · , (3.9)where the reference zeroth-order distribution function corresponds to the USF distribution function but taking intoaccount the local dependence of the concentration, pressure and temperature and the change V → V − δ u ( r , t ) = c .The successive approximations f ( k ) s are of order k in the gradients of x , p , T , and δ u but retain all the orders in theshear rate a . Here, only the first-order approximation will be analyzed.When the expansion (3.9) is substituted into the definitions (2.6), (2.7), and (2.8), one gets the correspondingexpansions for the fluxes: j s = j (0) s + j (1) s + · · · , (3.10a) P = P (0) + P (1) + · · · , q = q (0) + q (1) + · · · . (3.10b)Finally, as in the usual Chapman-Enskog method, the time derivative is also expanded as ∂ t = ∂ (0) t + ∂ (1) t + ∂ (2) t + · · · , (3.11)where the action of each operator ∂ ( k ) t is obtained from the hydrodynamic equations (3.2)–(3.4). These results providethe basis for generating the Chapman-Enskog solution to the Boltzmann equations (3.1a) and (3.1b). A. Zeroth-order approximation
Substituting the expansions (3.10a)–(3.11) into Eq. (3.1a), the kinetic equation for f (0)1 is given by ∂∂t f (0)1 − ∂∂V i ( a ij V j + αV i ) f (0)1 + ( V + u ) · ∇ f (0)1 + αδ u · ∂f (0)1 ∂ V = J [ f (0)1 , f (0)1 ] + J [ f (0)1 , f (0)2 ] . (3.12)To lowest order in the expansion the conservation laws give ∂ (0) t x = 0 , T − ∂ (0) t T = p − ∂ (0) t p = − p aP (0) xy − α, (3.13) ∂ (0) t δu i + a ij δu j = 0 . (3.14)If α = 0, then T − ∂ (0) t T = p − ∂ (0) t p = − aP (0) xy / p while if α = − aP (0) xy / p then ∂ (0) t T = ∂ (0) t p = 0.Since f (0)1 is a normal solution, the time derivative in Eq. (3.12) can be represented more usefully as ∂ (0) t f (0)1 = ∂f (0)1 ∂x ∂ (0) t x + ∂f (0)1 ∂p ∂ (0) t p + ∂f (0)1 ∂T ∂ (0) t T + ∂f (0)1 ∂δu i ∂ (0) t δu i = − (cid:18) p aP (0) xy + 2 α (cid:19) (cid:18) p ∂∂p + T ∂∂T (cid:19) f (0)1 − a ij δu j ∂∂δu i f (0)1 = − (cid:18) p aP (0) xy + 2 α (cid:19) (cid:18) p ∂∂p + T ∂∂T (cid:19) f (0)1 + a ij δu j ∂∂c i f (0)1 , (3.15)where in the last step we have taken into account that f (0)1 depends on δ u only through the peculiar velocity c .Substituting Eq. (3.15) into Eq. (3.12) yields the following kinetic equation for f (0)1 : − (cid:18) p aP (0) xy + 2 α (cid:19) (cid:18) p ∂∂p + T ∂∂T (cid:19) f (0)1 − ac y ∂∂c x f (0)1 − α ∂∂ c · (cid:16) c f (0)1 (cid:17) = J [ f (0)1 , f (0)1 ] + J [ f (0)1 , f (0)2 ] . (3.16)A similar equation holds for f (0)2 . The partial pressure tensors P (0)1 and P (0)2 can be obtained from Eq. (3.16) and itscounterpart for f (0)2 when one multiplies both equations by m s cc and integrate over c . Their explicit forms can befound in the Appendix of Ref. 4. B. First-order approximation
The analysis to first order in the gradients is worked out in Appendix A. The distribution function f (1)1 is of theform f (1)1 = A · ∇ x + B · ∇ p + C · ∇ T + D : ∇ δ u , (3.17)where the vectors { A , B , C } , and the tensor D are functions of the true peculiar velocity c . They are the solutionsof the following set of linear integral equations: − (cid:18) p aP (0) xy + 2 α (cid:19) ( p∂ p + T ∂ T ) A − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) A + L A + M A = A + (cid:18) a p ∂ x P (0) xy + 2 ∂ x α (cid:19) ( p B + T C ) , (3.18) − (cid:18) p aP (0) xy + 2 α (cid:19) ( p∂ p + T ∂ T ) B − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) B + L B + M B − (cid:20) a ∂ p P (0) xy + 2(1 + p∂ p ) α (cid:21) B = B − (cid:20) aT p (1 − p∂ p ) P (0) xy − T ∂ p α (cid:21) C , (3.19) − (cid:18) p aP (0) xy + 2 α (cid:19) ( p∂ p + T ∂ T ) C − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) C + L C + M C − (cid:20) a p (1 + T ∂ T ) P (0) xy + 2(1 + T ∂ T ) α (cid:21) C = C + (cid:18) a ∂ T P (0) xy + 2 p∂ T α (cid:19) B , (3.20) − (cid:18) p aP (0) xy + 2 α (cid:19) ( p∂ p + T ∂ T ) D ,ℓj − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) D ,ℓj − aδ ℓy D ,xj + L D ,ℓj + M D ,ℓj = D ,ℓj , (3.21)where A ( c ), B ( c ), C ( c ), and D ( c ) are defined by Eqs. (A8)–(A11), respectively. In addition, L and M are thelinearized Boltzmann collision operators around the reference USF state: L X = − (cid:16) J [ f (0)1 , X ] + J [ X, f (0)1 ] + J [ X, f (0)2 ] (cid:17) , (3.22a) M X = − J [ f (0)2 , X ] . (3.22b)In this paper we are mainly interested in evaluating the first-order contribution to the mass flux j (1)1 . It is definedas j (1)1 = m Z d c c f (1)1 , j (1)2 = − j (1)1 . (3.23)Use of Eq. (3.17) into Eq. (3.23) gives the expression j (1)1 ,i = − m m nρ D ij ∂x ∂r j − ρp D p,ij ∂p∂r j − ρT D T,ij ∂T∂r j , (3.24)where D ij = − ρm n Z d c c i A ,j ( c ) , (3.25) D p,ij = − m pρ Z d c c i B ,j ( c ) , (3.26) D T,ij = − m Tρ Z d c c i C ,j ( c ) . (3.27)Upon writing Eqs. (3.25)–(3.27) use has been made of the symmetry properties of A , B , and C . In general, the setof generalized transport coefficients D ij , D p,ij , and D T,ij are nonlinear functions of the shear rate and the parametersof the mixture. It is apparent that the anisotropy induced by the presence of shear flow gives rise to new transportcoefficients for the mass flux, reflecting broken symmetry. According to Eq. (3.24), the mass flux is expressed in termsof a diffusion tensor D ij , a pressure diffusion tensor D p,ij , and a thermal diffusion tensor D T,ij .To get the explicit dependence of the above transport coefficients on the parameter space of the problem, the formof α must be chosen. As said before, two choices will be considered here: (i) the unthermostatted case α = 0, and (ii)the thermostatted case α = − aP (0) xy / p . Both cases will be separately studied in the next Section. IV. MASS TRANSPORT UNDER SHEAR FLOW
This Section is devoted to the determination of the generalized transport coefficients D ij , D p,ij , and D T,ij associatedwith the mass transport for the two choices of the thermostat parameter. These coefficients are given in terms of thesolutions to the integral equations (3.18)–(3.20).
A. Unthermostatted USF state
In the absence of an external thermostat ( α = 0), the integral equations (3.18)–(3.20) become − paP (0) xy ( p∂ p + T ∂ T ) A − ac y ∂∂c x A + L A + M A = A + 2 a p ( p B + T C ) ( ∂ x P (0) xy ) , (4.1) − p aP (0) xy ( p∂ p + T ∂ T ) B − (cid:18) a ∂ p P (0) xy + ac y ∂∂c x (cid:19) B + L B + M B = B − aT p C (1 − p∂ p ) P (0) xy , (4.2) − paP (0) xy ( p∂ p + T ∂ T ) C − (cid:20) a p (1 + T ∂ T ) P (0) xy + ac y ∂∂c x (cid:21) C + L C + M C = C + 2 a B ( ∂ T P (0) xy ) . (4.3)The dependence of P (0) ij on the pressure p and temperature T occurs explicitly and through its dependence on thereduced shear rate a ∗ = a/ζ . Here, the effective collision frequency ζ is given by ζ = 2 n λ ′ m + m = 2 pk B T λ ′ m + m , (4.4)where λ ′ is defined in Eq. (2.14). Consequently, ∂ p P (0) ij = ∂ p pP ∗ ij ( a ∗ ) = (cid:18) − a ∗ ∂∂a ∗ (cid:19) P ∗ ij ( a ∗ ) , (4.5) ∂ T P (0) ij = ∂ T pP ∗ ij ( a ∗ ) = pT a ∗ ∂∂a ∗ P ∗ ij ( a ∗ ) , (4.6)where P ∗ ij = P (0) ij /p . In addition, the dependence of P (0) ij on the mole fraction x is also rather intricate and so thederivatives with respect to x must be carried out with care. The generalized coefficients D ij , D p,ij , and D T,ij canbe obtained from Eqs. (4.1)–(4.3) when one multiplies those equations by m c i and integrates over c . After somealgebra (some technical details are provided in Appendix B), one arrives at the following set of coupled algebraicequations: (cid:20)(cid:18) ρλ m m − aP ∗ xy (cid:19) δ ik + a ik (cid:21) D kj = ρk B Tm m (cid:18) ∂ x P ∗ ,ij − ρ ρ ∂ x P ∗ ij (cid:19) + 2 aρ m m n ( ∂ x P ∗ xy ) ( D p,ij + D T,ij ) , (4.7) (cid:20)(cid:18) ρλ m m − a p (1 − a ∗ ∂ a ∗ ) P ∗ xy (cid:19) δ ik + a ik (cid:21) D p,kj = pρ (1 − a ∗ ∂ a ∗ ) (cid:18) P ∗ ,ij − ρ ρ P ∗ ij (cid:19) − a a ∗ D T,ij ( ∂ a ∗ P ∗ xy ) , (4.8) (cid:20)(cid:18) ρλ m m − a a ∗ ∂ a ∗ ) P ∗ xy (cid:19) δ ik + a ik (cid:21) D T,kj = pρ a ∗ ∂ a ∗ (cid:18) P ∗ ,ij − ρ ρ P ∗ ij (cid:19) + 2 a a ∗ D p,ij ( ∂ a ∗ P ∗ xy ) , (4.9)where P ∗ s = P (0) s /p and use has been made of the relations (4.5) and (4.6).In the absence of shear field ( a = 0), then P ∗ s,ij = x s δ ij , and P ∗ ij = δ ij , so that Eqs. (4.7)–(4.9) have the solutions D ij = D δ ij , D p,ij = D p, δ ij , and D T,ij = 0, where D and D p, are the conventional Navier-Stokes transportcoefficients for Maxwell molecules. Their expressions are D = k B Tλ , D p, = ρ ρ ρ ( m − m ) D . (4.10)In this case the mass flux j (1) can be written as j (1)1 = − m m ρ ρ k B ρ D ( ∇ φ ) T − ( ∇ φ ) T T , (4.11)where (cid:18) ∇ φ s T (cid:19) T = 1 m s ∇ ln( x s p ) , (4.12) φ s being the chemical potential per unit mass. The fact that the thermal diffusion coefficient vanishes when a ∗ = 0is due to the interaction potential considered (Maxwell molecules) since this coefficient is different from zero for moregeneral interaction potentials. However, when the mixture is strongly sheared, the Boltzmann equation leads tocontributions to the mass flux proportional to the thermal gradient, even for Maxwell molecules.In the case of mechanically equivalent particles ( µ = 1, κ = κ = κ ), P ∗ ,ij = x P ∗ ij , ∂ x P (0)1 ,ij = P (0)1 ,ij /x = P (0) xy ,and so D p,ij = D T,ij = 0. Moreover, Eq. (4.7) reduces to D ij = m − nλ /m − aP ∗ xy δ ik − a ik nλ /m − aP ∗ xy ! P (0) kj . (4.13)Equation (4.13) is consistent with previous results derived for the self-diffusion tensor. Furthermore, known resultsfor the diffusion tensor are also recovered in the tracer limit ( x → =10=4=0.2 D * kk / a* FIG. 1: Shear-rate dependence of the trace D ∗ kk of the mutual diffusion tensor for x = 0, κ = κ and several values of themass ratio µ = m /m . =10 =4=0.2 D * p kk / a* FIG. 2: Shear-rate dependence of the trace D ∗ p,kk of the pressure diffusion tensor for x = 0, κ = κ and several values ofthe mass ratio µ = m /m . B. Thermostatted USF state
Let us assume now that an external thermostat is introduced to compensate for the viscous heating effect. In thiscase, α = − aP (0) xy / p , and the integral equations (3.18)–(3.20) become − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) A + L A + M A = A , (4.14) − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) B + L B + M B = B , (4.15) − (cid:18) ac y ∂∂c x + α ∂∂ c · c (cid:19) C + L C + M C = C . (4.16)In contrast to what happens in the unthermostatted case, the different integral equations are now decoupled andhence the generalized coefficients of the mass transport can be obtained more easily. The mathematical steps to getthem are similar to those made before when α = 0 and so only the final results are presented. The explicit expressionsfor D ij , D p,ij , and D T,ij are given by D ij = ρk B Tm m α + ρλ m m δ ik − a ik α + ρλ m m ! (cid:18) ∂ x P ∗ ,kj − ρ ρ ∂ x P ∗ kj (cid:19) , (4.17)0 =10 =4=0.2 D * T kk / a* FIG. 3: Shear-rate dependence of the trace D ∗ T,kk of the thermal diffusion tensor for x = 0, κ = κ and several values ofthe mass ratio µ = m /m . =10=4=0.2 - D * xy a* FIG. 4: Shear-rate dependence of the off-diagonal element − D ∗ xy of the mutual diffusion tensor for x = 0, κ = κ andseveral values of the mass ratio µ = m /m . =10=4=0.2 - D * p xy a* FIG. 5: Shear-rate dependence of the off-diagonal element − D ∗ p,xy of the pressure diffusion tensor for x = 0, κ = κ andseveral values of the mass ratio µ = m /m . =10=4=0.2 D * T xy a* FIG. 6: Shear-rate dependence of the off-diagonal element D ∗ T,xy of the thermal diffusion tensor for x = 0, κ = κ andseveral values of the mass ratio µ = m /m . D p,ij = pρ α + ρλ m m δ ik − a ik α + ρλ m m ! (1 − a ∗ ∂ a ∗ ) (cid:18) P ∗ ,kj − ρ ρ P ∗ kj (cid:19) , (4.18) D T,ij = pρ a ∗ α + ρλ m m δ ik − a ik α + ρλ m m ! ∂ a ∗ (cid:18) P ∗ ,kj − ρ ρ P ∗ kj (cid:19) . (4.19)In order to get these expressions use has been made of the identity( b a ) − = b − − b − a , (4.20)where b is an arbitrary constant and a is the tensor with elements a ij = aδ ix δ jy .In the case of mechanically equivalent particles, D p,ij = D T,ij = 0 and Eq. (4.17) reduces to D ij = m − α + nλ /m (cid:18) δ ik − a ik α + nλ /m (cid:19) P (0) kj . (4.21)Equation (4.21) gives the self-diffusion tensor of tagged particles under thermostatted USF. For a general binarymixture, the expression (4.17) for the diffusion tensor D ij coincides with the one derived before in a stationarystate with the constraints p = const and T = const. Finally, it is also apparent that, except for vanishing shear rates,the expressions of the generalized transport coefficients (4.17)–(4.19) in the thermostatted state differ from the onesderived in the absence of a thermostat, Eqs. (4.7)–(4.9). This shows again that the presence of the thermostat affectsthe transport properties of the system. V. ILLUSTRATIVE EXAMPLES IN THE TRACER LIMIT
The results obtained in the preceding Section give all the relevant information on the influence of shear flow onthe mass transport. In general, the elements D ij , D p,ij and D T,ij present a complex dependence on the shear rateand the parameters of the mixture without any restriction on their values. However, although the solution to Eqs.(4.7)–(4.9) (in the unthermostatted case) and Eqs. (4.17)–(4.19) (in the thermostatted case) is simple, it involvesquite a tedious algebra due to the complex dependence of the partial pressure tensors P (0) s,ij and the thermostatparameter α on the mole fraction x and the reduced shear rate a ∗ . To show the shear-rate dependence of the tensors T ij ≡ { D ij , D p,ij , D T,ij } in a clearer way, the tracer limit ( x →
0) will be considered here in detail. In addition,to make some contact with computer simulation results, the thermostatted case will be studied. In the tracer limitcase, P ≃ P and the partial pressure tensors P and P have a more simplified forms. In particular, ∂ x P (0) ij = 0and ∂ x P (0)1 ,ij = P (0)1 ,ij /x . The explicit expressions of the partial pressure tensors in the tracer limit are provided inAppendix C.2As expected, T xz = T zx = T yz = T zy = 0, in agreement with the symmetry of the problem. As a consequence,there are five relevant elements: the three diagonal ( T xx , T yy , and T zz ) and two off-diagonal elements ( T xy and T yx ).In addition, T xx = T yy = T zz and T xy = T yx . The equality P s,yy = P s,zz implies T yy = T zz . This property is aconsequence of the interaction model considered since for non-Maxwell molecules computer simulations show that the yy and zz elements of the pressure tensor are different. In Figs. 1–6, the relevant elements of tensors D ∗ ij , D ∗ p,ij and D ∗ T,ij are plotted as functions of the reduced shear rate a ∗ for κ = κ and several values of the mass ratio µ .Here, the tensors have been reduced with respect to their Navier-Stokes values (except D T,ij ), namely, D ∗ ij = D ij /D , D ∗ p,ij = D p,ij /D p, and D ∗ T,ij = D T,ij /x D . One third of the trace of these tensors is plotted in Figs. 1–3, whilethe xy element is plotted in Figs. 4–6. We observe that in general the influence of shear flow on the mass transportis quite important. It is also apparent that the anisotropy of the system, as measured by the traces D ∗ kk , D ∗ p,kk ,and D ∗ pT kk , grows with the shear rate. This anisotropy is more significant when the impurity is heavier than theparticles of the gas. Moreover, the shear field induces cross effects in the diffusion of particles. This is measured bythe (reduced) off-diagonal elements D ∗ xy , D ∗ p,xy and D ∗ T,xy . These coefficients give the transport of mass along the x axis due to gradients parallel to the y axis. While D ∗ xy and D ∗ p,xy are negative, the coefficient D T,xy can be positivein the region of small shear rates. We observe that, regardless of the mass ratio, the shapes of D ∗ xy and D ∗ p,xy arequite similar: there is a region of values of a ∗ for which − D ∗ xy and − D ∗ p,xy increase with increasing shear rate, whilethe opposite happens for larger shear rates. The magnitude of D ∗ T,xy is smaller than that of the elements − D ∗ xy and − D ∗ p,xy , especially when the tracer particles are lighter than the particles of the gas. In this latter case, D ∗ T,xy ispractically negligible.
VI. DISCUSSION
Diffusion of particles in a binary mixture in non-Newtonian regimes is a subject of great interest from a fundamentaland practical points of view. If the mixture is strongly sheared, the mass flux j can be significantly affected by thepresence of shear flow so that the corresponding transport coefficients may differ significantly from their equilibriumvalues. In addition, the resulting mass transport is anisotropic and thus it cannot be described by scalar transportcoefficients but by shear-rate dependent tensorial quantities whose explicit determination has been the main objectiveof this paper.In order to gain some insight into this complex problem, a dilute binary mixture of Maxwell molecules underUSF has been considered. This is perhaps the only interaction potential for which the Boltzmann equation can beexactly solved in some specific non-homogenous situations, such as in the case of the USF problem. In particular, thecorresponding rheological properties of the mixture (nonlinear shear viscosity and viscometric functions) have beenobtained for arbitrary values of the shear rate and without any restriction on the parameters of the mixture (masses,concentrations, and force constants). This exact solution is of great significance in providing insight into the typeof phenomena that can occur in conditions far away from equilibrium. In this paper, the interest has been focusedon situations that slightly deviate from the USF by small spatial gradients. Under these conditions, a generalizedChapman-Enskog method around the shear flow distribution has been used to determine mass transport in thefirst order of the deviations of the hydrodynamic field gradients from their values in the reference shear flow state f (0) s . In this case, the mass flux j (1)1 is given by Eq. (3.24), where the corresponding set of generalized transportcoefficients { D ij , D p,ij , D T,ij } are the solutions of the coupled algebraic equations (4.7)–(4.9) in the unthermostattedcase, while they are explicitly given by Eqs. (4.17)–(4.19) in the presence of a Gaussian thermostat. This type ofexternal forces are usually employed in nonequilibrium molecular dynamics simulations to compensate exactly for theviscous increase of temperature.As expected, the results show that the coefficients { D ij , D p,ij , D T,ij } present a complex dependence on the shearrate and on the masses, mole fractions, and force constants. This is clearly illustrated in Figs. 1–6 for the tracerlimit case ( x → { D ij , D p,ij , D T,ij } from their equilibrium values are basically due to threedifferent reasons. First, the presence of shear flow modifies the collision frequency of the conventional diffusion problem( ρλ /m m ) by a shear-rate dependent term. Second, given that the binary mixture is in general constituted byparticles mechanically different, the reference shear flow states f (0)1 and f (0)2 are completely different. This effect givesrise to terms proportional to P ∗ ,ij − ( ρ /ρ ) P ∗ ij . Third, in the unthermostatted case, the generalized coefficients arecoupled due to the inherent non-Newtonian features of the USF state. Each one of the three effects is a differentreflection of the extreme nonequilibrium conditions present in the mixture.It is apparent that the results presented here in the particular case of Maxwell molecules may be relevant forinterpreting computer simulation results. Sarman, Evans, and Baranyai carried out time ago molecular dynamicssimulations in a strongly sheared Lennard-Jones binary mixture to evaluate the self- and mutual-diffusion tensor bymeans of Green-Kubo formulae. To the best of my knowledge, this is the only computer experiment in which the3shear-rate dependence of the diffusion tensor D ij has been measured. They considered an equimolar Lennard-Jonesmixture at two different densities and the parameters in the potential were adjusted to model an argon-kryptonmixture, which means that the two components are fairly similar. As already said in Ref. 13, when one considersthis type of mixture ( x = 0 . m /m = 0 . κ = κ = κ ) in the thermostatted case, the general qualitativedependence of the (reduced) mutual diffusion tensor D ij ( a ∗ ) /D on the (reduced) shear rate agrees quite well withcomputer simulations. Thus, theory and simulation predict that in general, the xx element increases to a maximumand then it decreases again, while the yy element decreases with increasing shear rate. The off-diagonal elements xy and yx are negative and their magnitude increases with a ∗ for not very large values of the shear rate. However, kinetictheory predicts that | D xy | > | D yx | , while the opposite happens in computer simulations. On the other hand, at aquantitative level, the influence of shear flow on diffusion is much more modest in the molecular dynamics simulationsthan the one found theoretically for dilute gases. This is probably due to the fact that the shear rates (in reducedunits) applied in the simulations are not large enough to observe significant changes of the diffusion tensor relativeto its equilibrium value. An alternative to overcome the difficulties for reaching large shear rates in nonequilibriummolecular dynamics at low-density is the direct simulation Monte Carlo method. I hope that the results derived herefor Maxwell molecules for D ij , D p,ij and D T,ij stimulate the performance of Monte Carlo simulations to assess thereliability of the Maxwell results to describe mass transport in strongly sheared mixtures for more realistic interactionpotentials.As said before, it must noted that, in order to observe large effects of shear flow on the tensors { D ij , D p,ij , D T,ij } ,the (reduced) shear rate must be at least of the order of 1. This means that for the inert gas fluids considered inthis paper, non-Newtonian effects on mass transport could be observable for shear rates practically unattainable inthe laboratory. In this sense, one should look at fluids that are observed to be non-Newtonian, such as colloidalsuspensions, polymeric liquids, gels, · · · . Although the results derived in this paper have been focused on the mass transport, the remaining transportcoefficients associated with the pressure tensor P (1) ij and the heat flux vector q (1) could be determined from theintegral equations (3.18)–(3.21). Nevertheless, in practice this calculation cannot be carried out analytically by usingthe Boltzmann equation since the fourth-degree moments of USF (whose explicit expressions are not known in theBoltzmann equation, except for a single gas ) are needed to get the heat flux. In order to overcome such a difficultyone can use a model kinetic equation that preserves the essential features of the true Boltzmann equation but admitsa more practical analysis. Perhaps the most well-known model for gas mixtures is the Gross-Krook (GK) kineticmodel. In this model the Boltzmann operator J rs [ f r , f s ] is replaced by the relaxation term J rs [ f r , f s ] → − ν rs ( f r − f rs ) , (6.1)where f rs = n r (cid:18) m r πk B T rs (cid:19) / exp (cid:20) − m r k B T rs ( v − u rs ) (cid:21) (6.2)and u rs = m r u r + m s u s m r + m s , (6.3) T rs = T r + 2 m r m s ( m r + m s ) (cid:20) ( T s − T r ) + m s k B ( u r − u s ) (cid:21) . (6.4)The partial temperatures T r are defined by Eq. (2.5). For Maxwell molecules, the effective collision frequency ν rs isgiven by ν rs = An s (cid:18) κ rs m r + m s m r m s (cid:19) / , (6.5)where A is a constant to be fixed by requiring that the model reproduces some transport coefficient of the Boltzmannequation. An exact solution to the GK kinetic model for a binary mixture in USF has been found. The comparisonof the GK results with those from the Boltzmann equation at the level of the rheological properties shows goodagreement, confirming the reliability of the GK model in computing transport properties in far from equilibriumsituations as well. Starting from the USF solution of the GK model, the fluxes P (1) ij and q (1) are obtained inAppendix D in the thermostatted case. With all the transport coefficients known, the constitutive equations for the4mass, momentum and heat fluxes are completed and the corresponding set of closed hydrodynamic equations for themixture can be derived. This allows one to perform a linear stability analysis of the hydrodynamic equations withrespect to the USF and determine the conditions for instabilities at long wavelengths. Previous results for a singlegas have shown that USF is unstable when the perturbations are along the velocity gradient ( y direction). Theproblem now is to extend this analysis to the case of multicomponent systems. Work along this line will be reportedin the near future. Acknowledgments
Partial support from the Ministerio de Ciencia y Tecnolog´ıa (Spain) through Grant No. FIS2007–60977 and fromthe Junta de Extremadura through Grant No. GRU07046 is acknowledged.
APPENDIX A: CHAPMAN–ENSKOG-LIKE EXPANSION
In this Appendix, some technical details on the determination of the first-order approximation f (1)1 by means of theChapman–Enskog-like expansion are provided. Inserting the expansions (3.9) and (3.11) into Eq. (3.1a), one gets thekinetic equation for f (1)1 : ∂ (0) t f (1)1 − ∂∂V i ( a ij V j + αV i ) f (1)1 + αδ u · ∂f (1)1 ∂ V + L f (1)1 + M f (1)2 = − h ∂ (1) t + ( V + u ) · ∇ i f (1)1 . (A1)The velocity dependence on the right-hand side of Eq. (A1) can be obtained from the macroscopic balance equations(3.2)–(3.4) to first order in the gradients. They are given by ∂ (1) t x = − ( u + δ u ) · ∇ x , (A2) ∂ (1) t δ u = − ( u + δ u ) · ∇ δ u − ρ − ∇ · P (0) , (A3) ∂ (1) t p = − ( u + δ u ) · ∇ p − p ∇ · δ u − (cid:16) aP (1) xy + P (0) : ∇ δ u (cid:17) , (A4) ∂ (1) t T = − ( u + δ u ) · ∇ T − n (cid:16) aP (1) xy + P (0) : ∇ δ u (cid:17) , (A5)where use has been made of the result j (0)1 = q (0) = . In addition, P (1) ij = X s m s Z d c c i c j f (1) s ( c ) . (A6)Use of Eqs. (A2)–(A5) in Eq. (A1) yields ∂ (0) t f (1)1 − ∂∂V i ( a ij V j + αV i ) f (1)1 + αδ u · ∂f (1)1 ∂ V + L f (1)1 + M f (1)2 = A · ∇ x + B · ∇ p + C · ∇ T + D : ∇ δ u , (A7)where A ,i ( c ) = − ∂f (0)1 ∂x c i + 1 ρ ∂f (0)1 ∂δu j ∂P (0) ij ∂x , (A8) B ,i ( c ) = − ∂f (0)1 ∂p c i + 1 ρ ∂f (0)1 ∂δu j ∂P (0) ij ∂p , (A9)5 C ,i ( c ) = − ∂f (0)1 ∂T c i + 1 ρ ∂f (0)1 ∂δu j ∂P (0) ij ∂T , (A10) D ,ij ( c ) = p ∂f (0)1 ∂p δ ij − ∂f (0)1 ∂δu i c j + 23 p (cid:16) P (0) ij − aη xyij (cid:17) (cid:18) p ∂∂p + T ∂∂T (cid:19) f (0)1 . (A11)Upon writing Eq. (A11) use has been made of the expression of the total pressure tensor P (1) ij of the mixture P (1) ij = − η ijkℓ ∂δu k ∂r ℓ , (A12)where η ijkℓ is the viscosity tensor.The solution to Eq. (A7) has the form given by Eq. (3.17), where the coefficients A , B , C , and D are functionsof the peculiar velocity and the hydrodynamic fields x , p , T , and δ u . The time derivative acting on these quantitiescan be evaluated with the replacement ∂ (0) t → − (cid:18) p aP (0) xy + 2 α (cid:19) ( p∂ p + T ∂ T ) . (A13)Moreover, there are contributions from ∂ (0) t acting on the pressure, temperature, and velocity gradients given by ∂ (0) t ∇ p = −∇ (cid:18) aP (0) xy + 2 pα (cid:19) = − a ∂P (0) xy ∂x + 2 p ∂α∂x ! ∇ x − a ∂P (0) xy ∂p + 2 α + 2 ∂α∂p ! ∇ p − a ∂P (0) xy ∂T + 2 p ∂α∂T ! ∇ T, (A14) ∂ (0) t ∇ T = −∇ (cid:18) T p aP (0) xy + 2 T α (cid:19) = − aT p ∂P (0) xy ∂x + 2 T ∂α∂x ! ∇ x + aT p P (0) xy − aT p ∂P (0) xy ∂p − T ∂α∂p ! ∇ p − a p P (0) xy + 2 aT p ∂P (0) xy ∂T + 2 α + 2 T ∂α∂T ! ∇ T, (A15) ∂ (0) t ∇ i δu j = ∇ i ∂ (0) t δu j = − a jk ∇ i δu k . (A16)The corresponding integral equations (3.18)–(3.20) can be obtained when one identifies coefficients of independentgradients in Eq. (A7) and takes into account Eqs. (A14)–(A16) and the mathematical property ∂ (0) t X = ∂X∂p ∂ (0) t p + ∂X∂T ∂ (0) t T + ∂X∂δu i ∂ (0) t δu i = − (cid:18) p aP (0) xy + 2 α (cid:19) (cid:18) p ∂∂p + T ∂∂T (cid:19) X + a ij δu j ∂X∂c i , (A17)where in the last step it has been taken into account that X depends on δ u through c = V − δ u . APPENDIX B: GENERALIZED TRANSPORT COEFFICIENTS ASSOCIATED WITH THE MASSTRANSPORT
In the unthermostatted case ( α = 0), the integral equations defining the generalized transport coefficients D ij , D p,ij and D T,ij are given by Eqs. (4.1)–(4.3). To get these coefficients, one multiplies (4.1)–(4.3) by m c i and integrates6over velocity. The result is23 p aP (0) xy ( p∂ p + T ∂ T ) (cid:18) m m nρ D ij (cid:19) − m m nρ (cid:18) a ik D kj + ρλ m m D ij (cid:19) = m Z d c c i A ,j − aρ p ( ∂ x P (0) xy ) ( D p,ij + D T,ij ) , (B1)23 p aP (0) xy ( p∂ p + T ∂ T ) (cid:18) ρp D p,ij (cid:19) − ρp (cid:20) a ik D p,kj + (cid:18) ρλ m m − a p ∂ p P (0) xy (cid:19) D p,ij (cid:21) = m Z d c c i B ,j + 2 aρ p D T,ij (1 − p∂ p ) P (0) xy , (B2)23 p aP (0) xy ( p∂ p + T ∂ T ) (cid:16) ρT D T,ij (cid:17) − ρT (cid:20) a ik D T,kj + (cid:18) ρλ m m − a p (1 + T ∂ T ) P (0) xy (cid:19) D T,ij (cid:21) = m Z d c c i C ,j − aρ p D p,ij ( ∂ T P (0) xy ) , (B3)where P (0) s,ij = m s Z d c c i c j f (0) s . (B4)Upon writing Eqs. (B1)–(B3), use has been made of the relation (2.12), which yields the results m Z d c c i ( L A + M A ) = − nλ D ij , (B5a) m Z d c c i ( L B + M B ) = − ρ λ m m p D p,ij , (B5b) m Z d c c i ( L C + M C ) = − ρ λ m m T D
T,ij . (B5c)The velocity integrals appearing in Eqs. (B1)–(B3) can be performed by using Eqs. (A8)–(A10), m Z d c c i A ,j = − (cid:18) ∂ x P (0)1 ,ij − ρ ρ ∂ x P (0) ij (cid:19) , (B6) m Z d c c i B ,j = − ∂ p (cid:18) P (0)1 ,ij − ρ ρ P (0) ij (cid:19) , (B7) m Z d c c i C ,j = − ∂ T (cid:18) P (0)1 ,ij − ρ ρ P (0) ij (cid:19) . (B8)The generalized transport coefficients D ij , D p,ij , and D T,ij can be written as D ij = D D ∗ ij ( a ∗ ), D p,ij = D p, D ∗ p,ij ( a ∗ ),and D T,ij = D T, D ∗ T,ij ( a ∗ ) where D ∗ ij , D ∗ p,ij , and D ∗ T,ij are dimensionless functions of the shear rate. Moreover, fromdimensional analysis, D ∼ T , D p, ∼ T /p , and D T, ∼ T /p . Therefore,( p∂ p + T ∂ T ) (cid:18) m m nρ D ij (cid:19) = ( p∂ p + T ∂ T ) (cid:18) m m nρ D D ∗ ij (cid:19) = m m nρ D ij , (B9)( p∂ p + T ∂ T ) (cid:18) ρp D p,ij (cid:19) = ( p∂ p + T ∂ T ) (cid:18) ρp D p, D ∗ p,ij (cid:19) = 0 , (B10)( p∂ p + T ∂ T ) (cid:16) ρT D T,ij (cid:17) = ( p∂ p + T ∂ T ) (cid:16) ρT D T, D ∗ T,ij (cid:17) = 0 , (B11)where use has been made of the identity( p∂ p + T ∂ T ) X ( a ∗ ) = ( ∂ a ∗ X ) ( p∂ p a ∗ + T ∂ T a ∗ ) = 0 , (B12)with a ∗ = a/ζ ∼ T /p . Taking into account the above results one arrives at the set of algebraic equations (4.7)–(4.9).7
APPENDIX C: RHEOLOGICAL PROPERTIES IN THE TRACER LIMIT
The explicit expressions for the pressure tensors P ∗ s,ij ≡ P s,ij /p in the USF are provided in this Appendix for thespecial case of tracer limit ( x → P ∗ ,ij are given by P ∗ ,yy = P ∗ ,zz = 11 + 2 ωα ∗ , (C1) P ∗ ,xx = 1 + 6 ωα ∗ ωα ∗ , (C2) P ∗ ,yxy = − α ∗ a ∗ = − ωa ∗ (1 + 2 ωα ∗ ) , (C3)where a ∗ = a/ζ , α ∗ = α/ζ , ζ being defined by Eq. (4.4). Moreover, ω = 2 γ (1 + µ ) , γ = r κ κ µµ , (C4)where µ = m /m is the mass ratio. The (reduced) thermostat parameter is given by α ∗ = max( α , α ′ ) where α = 12 ω G ( ωa ∗ ) , α ′ = 14 µ G (2 µa ∗ ) − γ , (C5)where G ( z ) = sinh [ cosh − (1 + 9 z )] and γ = λ /λ ′ = 0 . α > α ′ except for very large shearrates and/or very disparate mass binary mixtures. The nonzero elements of P ∗ ,ij are given by P ∗ ,yy = P ∗ ,zz = x ∆(1 + 2 ωα ∗ ) (cid:26) ( γ −
12 )(2 α ∗ + ǫ ) + (2 γ − α ∗ β (1 + ωǫ + 4 ωα ∗ )+ 12 (1 + 2 ωα ∗ )(2 α ∗ + ǫ ) (cid:27) , (C6) P ∗ ,xx = x ∆(1 + 2 ωα ∗ ) (cid:26) ( γ −
12 )(2 α ∗ + ǫ ) + 3(2 γ − α ∗ (1 + ωǫ + 4 ωα ∗ ) (cid:18) α ∗ + ǫ − β (cid:19) + 12 (1 + 2 ωα ∗ ) (cid:2) (2 α ∗ + ǫ ) + 2 a ∗ ) + (2 γ − a ∗ (cid:3)(cid:27) , (C7) P ∗ ,xy = − x ωa ∗ ∆(1 + 2 ωα ∗ ) (cid:26) ( γ −
12 )(2 α ∗ + γ )(2 α ∗ + ǫ ) + ( γ −
12 ) 1 + 2 ωα ∗ ω × (2 α ∗ + ǫ + 2 ωα ∗ β ) + 12 ω (1 + 2 ωα ∗ ) (2 α ∗ + ǫ ) (cid:27) , (C8)where ǫ = γ + β , β = 1 / µ , and ∆ = (2 α ∗ + ǫ ) (2 α ∗ + γ ) − βa ∗ . (C9) APPENDIX D: MOMENTUM AND HEAT TRANSPORT AROUND USF FROM THE GK MODEL
This Appendix addresses the evaluation of the fluxes P (1) ij and q (1) in the thermosttated case from the GK kineticmodel (6.1). The first order corrections to the fluxes are P (1) ij = − X s η s,ijkℓ ∂δu k ∂r ℓ , (D1)8 q (1) i = − X s D ′′ s,ij ∂x ∂r j − X s L s,ij ∂p∂r j − X s λ s,ij ∂T∂r j , (D2)where the partial contributions to the transport coefficients are defined as η s,ijkℓ = − m s Z d c c i c j D s,kℓ ( c ) , (D3) D ′′ s,ij = − m s Z d c c c i A s,j ( c ) , (D4) L s,ij = − m s Z d c c c i B s,j ( c ) , (D5) λ s,ij = − m s Z d c c c i C s,j ( c ) . (D6)From the above partial contributions one can get the generalized shear viscosity η ijkℓ = η ,ijkℓ + η ,ijkℓ , the generalizedDuffour coefficient D ′′ ij = D ′′ ,ij + D ′′ ,ij , the generalized pressure energy coefficient L ij = L ,ij + L ,ij , and thegeneralized thermal conductivity λ ij = λ ,ij + λ ,ij . The quantities {A s,i , B s,i , C s,i , D s,ij } still verify the integralequations (3.18)–(3.21) (with α = − aP (0) xy / p ) with the only replacement L f (1)1 + M f (1)2 → ν f (1)1 − ν f (1)11 − ν f (1)12 , (D7)where ν = ν + ν , f (1)11 = f (0)11 n k B T c · j (1)1 , f (1)12 = f (0)12 n n k B T µ ( n − n ) c · j (1)1 , (D8)and f (0) rs = n r (cid:18) m r πk B T rs (cid:19) / exp (cid:18) − m r c k B T rs (cid:19) . (D9)In Eq. (D8), µ rs = m r / ( m r + m s ).In order to get the coefficients η s,ijkℓ , D ′′ s,ij , L s,ij and λ s,ij , it is convenient to introduce the velocity moments X ( i ) k,ℓ,m = Z d c c kx c ℓy c mz A ,i , (D10) Y ( i ) k,ℓ,m = Z d c c kx c ℓy c mz B ,i , (D11) Z ( i ) k,ℓ,m = Z d c c kx c ℓy c mz C ,i , (D12) W ( ij ) k,ℓ,m = Z d c c kx c ℓy c mz D ,ij , (D13)and similar definitions for the species 2. The knowledge of the above moments allows one to get the expressions ofthe coefficients η ,ijkℓ , D ′′ ,ij , L ,ij and λ ,ij . The method to evaluate the moments X ( i ) k,ℓ,m , Y ( i ) k,ℓ,m , Z ( i ) k,ℓ,m , and W ( ij ) k,ℓ,m is quite similar. Here, as an example, the mathematical steps to determine the moments X ( i ) k,ℓ,m associated with thetransport coefficients D ′′ ,ij will be analyzed in detail. First, in the thermosttated case, Eq. (3.18) with the change(D7) becomes − (cid:18) ac y ∂∂c x − ν + α ∂∂ c · c (cid:19) A ,i + m m nρ (cid:18) ν n k B T f (0)11 + ν n n k B T µ ( n − n ) f (0)12 (cid:19) c j D ji = A ,i , (D14)9where A is given by Eq. (A8). Upon writing (D14) use has been made of the constitutive form (3.24) for the massflux. Now, we multiply Eq. (D14) by c kx c ℓy c mz and integrate over velocity. After some algebra, we get akX ( i ) k − ,ℓ +1 ,m + [ ν + ( k + ℓ + m ) α ] X ( i ) k,ℓ,m = R ( i ) k,ℓ,m , (D15)where R ( i ) k,ℓ,m = A ,i − m m nρ (cid:18) k B Tm (cid:19) ( k + ℓ + m +1) / (cid:20) ν k B T χ ( k + ℓ + m − / + ν n k B T µ ( n − n ) χ ( k + ℓ + m − / (cid:21) × Λ k + δ jx ,ℓ + δ jy ,m + δ jz D ji , (D16) A ,i ≡ Z d c c kx c ℓy c mz A ,i = − ∂∂x M k + δ ix ,ℓ + δ iy ,m + δ iz + 1 ρ ∂P (0) ij ∂x ( δ jx kM k − ,ℓ,m + δ jy ℓM k,ℓ − ,m + δ jz mM k,ℓ,m − ) . (D17)In Eqs. (D16) and (D17), we have introduced the temperature ratios χ = T /T and χ = T /T and the (unper-turbed) moments of the USF M k,ℓ,m = Z d c c kx c ℓy c mz f (0)1 ( c ) . (D18)The explicit shear-rate dependence of χ , χ and M k,ℓ,m can be found in Ref. 31. Moreover,Λ k,ℓ,m = π − / Γ( k + 12 )Γ( ℓ + 12 )Γ( m + 12 ) (D19)if ( k, ℓ, m ) are even, being zero otherwise. The solution to Eq. (D15) can be written as X ( i ) k,ℓ,m = k X q =0 k !( k − q )! ( − a ) q [ ν + ( k + ℓ + m ) α ] − (1+ q ) R ( i ) k − q,ℓ + q,m . (D20)Note that Eq. (D20) is still formal since one needs to know the coefficients D ij . They can be consistently determinedfrom their definitions (3.25). Once these coefficients are known, Eq. (D20) allows one to get the coefficients D ′′ ,ij .The same method can be applied to evaluate the remaining moments. The moments Y ( i ) k,ℓ,m and Z ( i ) k,ℓ,m are given by Y ( i ) k,ℓ,m = k X q =0 k !( k − q )! ( − a ) q [ ν + ( k + ℓ + m ) α ] − (1+ q ) S ( i ) k − q,ℓ + q,m , (D21) Z ( i ) k,ℓ,m = k X q =0 k !( k − q )! ( − a ) q [ ν + ( k + ℓ + m ) α ] − (1+ q ) T ( i ) k − q,ℓ + q,m , (D22)where S ( i ) k,ℓ,m = B ,i − ρp (cid:18) k B Tm (cid:19) ( k + ℓ + m +1) / (cid:20) ν k B T χ ( k + ℓ + m − / + ν n k B T µ ( n − n ) χ ( k + ℓ + m − / (cid:21) × Λ k + δ jx ,ℓ + δ jy ,m + δ jz D p,ji , (D23) T ( i ) k,ℓ,m = C ,i − ρT (cid:18) k B Tm (cid:19) ( k + ℓ + m +1) / (cid:20) ν k B T χ ( k + ℓ + m − / + ν n k B T µ ( n − n ) χ ( k + ℓ + m − / (cid:21) × Λ k + δ jx ,ℓ + δ jy ,m + δ jz D T,ji . (D24)0The expressions of B ,i and C ,i are formally identical to that of A ,i , except that the operator ∂ x appearing in(D17) must be replaced by the operators ∂ p and ∂ T in the cases of B ,i and C ,i , respectively. Finally, the expressionof W ( ij ) k,ℓ,m is W ( ij ) k,ℓ,m = k X q =0 k !( k − q )! ( − a ) q [ ν + ( k + ℓ + m ) α ] − (1+ q ) h U ( ij ) k − q,ℓ + q,m + aδ iy W ( xj ) k,ℓ,m i , (D25)where U ( ij ) k,ℓ,m = − δ ij (cid:18) − p ∂∂p (cid:19) M k,ℓ,m + 23 p (cid:16) P (0) ij − aη xyij (cid:17) (cid:18) p ∂∂p + T ∂∂T (cid:19) M k,ℓ,m − M k,ℓ,m ( δ ix δ jx k + δ iy δ jy ℓ + δ iz δ jz m ) − kδ ix ( δ jy M k − ,ℓ +1 ,m + δ jz M k − ,ℓ,m +1 ) − ℓδ iy ( δ jx M k +1 ,ℓ − ,m + δ jz M k,ℓ − ,m +1 ) − mδ iz ( δ jx M k +1 ,ℓ,m − + δ jy M k,ℓ +1 ,m − ) . (D26)The transport coefficients D ′′ ,ij , L ,ij , λ ,ij , and η ,ijkℓ can be obtained from Eqs. (D20), (D21), (D22) and (D25),respectively, in terms of the shear rate and the parameters of the mixture. Their respective counterparts for species2 can be easily determined from them by making the changes: m → m , n → n , and κ → κ . The expressionof the Duffour tensor D ′′ ij coincides with the one obtained before in a stationary state with ∇ p = ∇ T = 0. 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