KineticandRelatedModels doi:10.3934/krm.2011.4.361 (cid:13)cAmericanInstituteofMathematicalSciences Volume4,Number1,March2011 pp. 361–384 1 NON-NEWTONIAN COUETTE–POISEUILLE FLOW OF 1 0 A DILUTE GAS 2 n a Mohamed Tij J D´epartement dePhysique,Universit´eMoulayIsma¨ıl 0 Mekn`es,Morocco 2 Andr´es Santos ] h DepartamentodeF´ısica,UniversidaddeExtremadura c E-06071Badajoz,Spain e m - Abstract. Thesteadystateofadilutegasenclosedbetweentwoinfinitepar- t allel plates in relative motion and under the action of a uniform body force a parallel to the plates is considered. The Bhatnagar–Gross–Krook model ki- t s netic equation is analytically solved for this Couette–Poiseuille flow to first t. order inthe forceand for arbitraryvalues of the Knudsen number associated a with the shear rate. This allows us to investigate the influence of the exter- m nalforceonthenon-NewtonianpropertiesoftheCouette flow. Moreover,the - Couette–Poiseuille flowisanalyzedwhentheshear-rateKnudsennumberand d thescaledforceareofthesameorderandtermsuptosecondorderareretained. n Inthisway,thetransitionfromthebimodaltemperatureprofilecharacteristic o of the pure force-driven Poiseuille flow to the parabolic profile characteristic c of the pure Couette flow through several intermediate stages inthe Couette– [ Poiseuille flow are described. A critical comparison with the Navier–Stokes solutionoftheproblemiscarriedout. 2 v 1 1. Introduction. Twoparadigmaticstationarynonequilibriumflowsaretheplane 8 CouetteflowandthePoiseuilleflow. IntheplaneCouetteflowthefluid(henceforth 8 assumedtobeadilutegas)isenclosedbetweentwoinfiniteparallelplatesinrelative 2 . motion, as sketched in Fig. 1(a). The walls can be kept at different or equal tem- 9 peraturesbut, evenifbothwalltemperaturesarethesame,viscousheatinginduces 0 atemperaturegradientinthe steady state. Ifthe Knudsennumber associatedwith 0 1 the shear rate is small enough the Navier–Stokes (NS) equations provide a satis- : factory description of the Couette flow. On the other hand, as shearing increases, v non-Newtonian effects (shear thinning and viscometric properties) and deviations i X of Fourier’s law (generalized thermal conductivity and streamwise heat flux com- r ponent) become clearly apparent [20]. These nonlinear effects have been derived a from the Boltzmann equation for Maxwell molecules [14, 28, 34, 40, 57], from the Bhatnagar–Gross–Krook (BGK) kinetic model [6, 19, 43], and also from general- ized hydrodynamic theories [49, 51]. A good agreement with computer simulations [16, 17, 25, 30, 31, 37] has been found. The plane Couette flow has also been ana- lyzedinthe contextofgranulargases[60, 65]. In the caseof platesatrestbut kept 2000 Mathematics Subject Classification. Primary: 76P05,82B40;Secondary: 82C40,82C05. Keywords and phrases. Bhatnagar–Gross–Krookkineticmodel,Couette flow,Poiseuilleflow, non-Newtonianproperties. 1 2 MOHAMED TIJ AND ANDRE´S SANTOS U+ U+ y=L/2 ux(y) Fx=mg Fx=mg ux(y) ux(y) (a) (b) (c) y=-L/2 U_ U_ Figure 1. Sketch of (a) the Couette flow, (b) the force-driven Poiseuille flow, and (c) the Couette–Poiseuille flow. atdifferenttemperatures,theCouetteflowbecomesthefamiliarplaneFourierflow, which also presents interesting properties by itself [4, 16, 17, 24, 29, 33, 40, 41, 42]. ThePoiseuilleflow,whereagasisenclosedinachannelorslabandfluidmotionis induced by a longitudinal pressuregradient,is a classicalproblem in kinetic theory [10, 35]. Essentially the same type of flow field is generated when the pressure gradient is replaced by the action of a uniform longitudinal body force F = mgx (e.g., gravity), as illustrated in Fig. 1(b). This force-driven Poiseuille flow has received a lot of attention both from theoretical [2, 3, 15, 18, 21, 32, 38, 39, 44b, 49, 51, 55, 56, 58, 59, 64, 67] and computational [1, 12, 22, 23, 27, 38, 61, 62, 68] points of view. This interest has been mainly motivated by the fact that the force-driven Poiseuille flow provides a nice example illustrating the limitations of the NS description in the bulk domain (i.e., far away from the boundary layers). In particular, while the NS equations predict a temperature profile with a flat maximum at the center, computer simulations [27] and kinetic theory calculations [55, 56] show that it actually has a local minimum at that point. Obviously, the Couette and Poiseuille flows can be combined to become the Couette–Poiseuille (or Poiseuille–Couette) flow [9, 36, 50, 53]. To the best of our knowledge,allthestudies onthe Couette–Poiseuilleflowassumethatthe Poiseuille part is driven by a pressure gradient, not by an external force. This paper intends to fill this gap by considering the steady state of a dilute gas enclosed between two infinite parallel plates in relative motion, the particles of the gas being subject to the actionof a uniform body force. This Couette–Poiseuilleflow is sketchedin Fig. 1(c). We will study the problem by the tools of kinetic theory by solving the BGK model for Maxwell molecules. The aim of this work is two-fold. First, we want to investigate how the fully developed non-Newtonian Couette flow is distorted by the action of the external force. To that end we will assume a finite value of the Knudsen number related to theshearrateandperformaperturbationexpansiontofirstorderintheforce. Asa secondobjective, we will study how the non-Newtonianforce-drivenPoiseuilleflow is modified by the shearing. This is done by assuming that the shear-rateKnudsen number and the scaled force are of the same order and neglecting terms of third and higher order. In both cases we are interested in the physical properties in the central bulk region of the slab, outside the influence of the boundary layers. COUETTE–POISEUILLE FLOW 3 The organization of the paper is as follows. The Boltzmann equation for the Couette–Poiseuilleflow is presented in Sec. 2. Section 3 deals with the NS descrip- tion of the problem. The main part of the paper is contained in Sec. 4, where the kinetic theoryapproachis workedout. Some technicalcalculationsarerelegatedto Appendix A. The results are graphically presented and discussed in Sec. 5. The paper ends with some concluding remarks in Sec. 6. 2. The Couette–Poiseuille flow. Symmetry properties. Let us consider a dilute monatomic gas enclosed between two infinite parallel plates located at y = ±L/2. The plates are in relative motion with velocities U along the x axis and ± are kept at a common temperature T . The imposed shear rate is therefore ω = w (U −U )/L. Besides, an external body force F=mgx, where m is the mass of a + − particle and g is a constant acceleration, is applied. The geometry of the problem is sketched in Fig. 1(c). In the absence of the externalbforce (g = 0) this problem reduces to the plane Couette flow [see Fig. 1(a)]. On the other hand, if the plates are at rest (ω = 0), one is dealing with the force-driven Poiseuille flow [see Fig. 1(b)]. The general problem with ω 6= 0 and g 6= 0 defines the Couette–Poiseuille flow analyzed in this paper. In the steady state only gradients along the y axis are present and thus the Boltzmann equation becomes ∂ ∂ v +g f(y,v|ω,g)=J[f,f], (1) y ∂y ∂v (cid:18) x(cid:19) where f is the one-particle velocity distribution function and J[f,f] is the Boltz- mann collision operator [7, 8, 11, 13, 26, 48], whose explicit expression will not be written down here. The notation f(y,v|ω,g) emphasizes the fact that, apart from itsspatialandvelocitydependencies,thedistributionfunctiondependsontheinde- pendent external parameters ω and g. As said above, g =0 and ω =0 correspond to the Couette and Poiseuille flows, respectively. In general, Eq. (1) must be solved subjected to specific boundary conditions, which can be expressed in terms of the kernels K (v,v′) defined as follows. When ± aparticlewithvelocityv′hitsthewallaty =L/2,theprobabilityofbeingreemitted withavelocityvwithintherangedvisK (v,v′)dv;thekernelK (v,v′)represents + − the same but at y =−L/2. The boundary conditions are then [13] Θ(±v )|v |f(y =∓L/2,v)=Θ(±v ) dv′|v′|K (v,v′)Θ(∓v′)f(y =∓L/2,v′), y y y y ∓ y Z (2) where Θ(x) is Heaviside’s step function. In the case of boundary conditions of complete accommodation with the walls, so that K (v,v′) = K (v) does not ± ± depend on the incoming velocity v′, the kernels can be written as K (v)=A−1Θ(±v )|v |ϕ (v), A ≡ dvΘ(±v )|v |ϕ (v), (3) ∓ ∓ y y ∓ ∓ y y ∓ Z whereϕ (v)representstheprobabilitydistributionofafictitiousgasincontactwith ∓ the system at y = ∓L/2. Equation (3) can then be interpreted as meaning that whenaparticlehitsawall,itisinstantaneouslyabsorbedandreplacedbyaparticle leaving the fictitious bath. Of course, any choice of ϕ (v) must be consistent with ∓ the imposed wall velocities and temperatures, i.e. U = dvv ϕ (v), (4) ∓ x ∓ Z 4 MOHAMED TIJ AND ANDRE´S SANTOS 1 k T = m dv(v−U )2ϕ (v). (5) B w ∓ ∓ 3 Z Inserting Eq. (3) into Eq. (2), one has Θ(±v )f(y =∓L/2,v)=Θ(±v )n ϕ (v), (6) y y ∓ ∓ where dv′Θ(∓v′)|v′|f(y =∓L/2,v′) y y n ≡ . (7) ∓ dv′Θ(±v′)|v′|ϕ (v′) R y y ∓ Theboundaryconditions(6)areusuallyreferredtoasdiffuse (orstochastic)bound- R ary conditions. The simplest and most common choice is that of a Maxwell– Boltzmann (MB) distribution [52]: m 3/2 m(v−U )2 ϕMB(v)= exp − ∓ . (8) ∓ 2πk T 2k T (cid:18) B w(cid:19) (cid:20) B w (cid:21) The first few moments of f define the densities of conserved quantities (mass, momentum, and temperature) and the associated fluxes. More explicitly, n(y|ω,g)= dvf(y,v|ω,g), (9) Z n(y|ω,g)u(y|ω,g)= dvvf(y,v|ω,g), (10) Z m n(y|ω,g)k T(y|g)=p(y|ω,g)= dvV2(y,v|ω,g)f(y,v|ω,g), (11) B 3 Z P (y|ω,g)=m dvV (y,v|ω,g)V (y,v|ω,g)f(y,v|ω,g), (12) ij i j Z m q(y|ω,g)= dvV2(y,v|ω,g)V(y,v|ω,g)f(y,v|ω,g). (13) 2 Z In these equations n is the number density, u is the flow velocity, V(y,v|a,ω,)≡v−u(y|ω,g) (14) is the peculiar velocity, T is the temperature, k is the Boltzmann constant, p is B the hydrostatic pressure, P is the pressure tensor, and q is the heat flux. Taking ij velocity moments in both sides of Eq. (1) one gets the following exact balance equations ∂ P =0, (15) y yy ∂ P =mng, (16) y xy ∂ q +P ∂ u =0. (17) y y xy y x Henceforth, without loss of generality, we will assume u (0) = 0. In other words, x we will adopt a reference frame solidary with the flow at the midpoint y =0. The symmetry properties of the Couette–Poiseuille flow imply the following in- variance properties of the velocity distribution function: f(y,v ,v ,v |ω,g) = f(−y,−v ,−v ,v |ω,−g) x y z x y z = f(−y,v ,−v ,v |−ω,g) x y z = f(y,v ,v ,−v |ω,g), (18) x y z As a consequence, if χ(y|ω,g) denotes a hydrodynamic variable or a flux, one has χ(y|ω,g) = S χ(−y|ω,−g) g = S χ(−y|−ω,g), (19) ω COUETTE–POISEUILLE FLOW 5 Quantity S S g ω n + + u − + x T + + p + + P + + xx P + + yy P + − xy q − + x q − − y Table 1. Parity factors S and S for the hydrodynamic fields g ω and the fluxes [see Eq. (19)]. where S = ±1 and S = ±1. The parity factors S and S for the non-zero g ω g ω hydrodynamicfieldsandfluxesaredisplayedinTable1. Ingeneral,ifχisamoment of the form χ(y|ω,g)= dvVkx(y|ω,g)vkyv2kzf(y,v|ω,g) (20) x y z Z then Sg =(−1)kx+ky and Sω =(−1)ky. ThegeneralsolutiontothestationaryBoltzmannequation(1)withtheboundary conditions (6) can be split into two parts [45, 46, 47]: f(y,v|ω,g)=f (y,v|ω,g)+f (y,v|ω,g). (21) H B Here, f represents the hydrodynamic, Hilbert-class, or normal contribution to H the distribution function. This means that f depends on space only through a H functional dependence on the hydrodynamic fields, i.e. f (y)=f [n,T,u ]. (22) H H x The contribution f represents the boundary-layer correction to f , so that f = B H f +f verifiesthespecifiedboundaryconditions. Thecorrectionf isappreciably H B B different from zero only in a thin layer (the so-called boundary layer or Knudsen layer),adjacenttotheplates,ofthicknessoftheorderofthemeanfreepath. Conse- quently,iftheseparationLbetweentheplatesismuchlargerthanthecharacteristic mean free path, there exists a well defined bulk region where the boundary correc- tion vanishes and the distribution function is fully givenby its hydrodynamic part. In the boundary layers the hydrodynamic profiles are much less smooth than in the bulk domain. The values of the flow velocity near the walls are different from the velocity of the plates (velocity slip phenomenon), i.e. u (y = ±L/2) 6= U . x ± Besides, the extrapolation of the velocity profile in the bulk to the boundaries, u (y = ±L/2), is also different from both the actual values u (y = ±L/2) and x,H x thewallvelocitiesU . Ofcourse,ananalogoustemperaturejumpeffecttakesplace ± with the temperature profile. The boundary contribution f for small Knudsen B numbers has been analyzed elsewhere [8, 47]. In the remaining of this paper we will focus on the hydrodynamic part f (and H will drop the subscript H), with special emphasis on the corresponding hydrody- namic contributions to the momentum and heat fluxes. In order to nondimension- alize the problem, we choose quantities evaluated at the central plane y = 0 as 6 MOHAMED TIJ AND ANDRE´S SANTOS units: v3(0) v k T(0) f∗(s,v∗|a,g∗)≡ T f(y,v|ω,g), v∗ ≡ , v (0)≡ B , (23) T n(0) v (0) m T r n(y|ω,g) T(y|ω,g) p(y|ω,g) n∗(s|a,g∗)≡ , T∗(s|a,g∗)≡ , p∗(s|a,g∗)≡ , (24) n(0) T(0) p(0) P (y|ω,g) q(y|ω,g) P∗(s|a,g∗)≡ ij , q∗(s|a,g∗)≡ , (25) ij p(0) p(0)v (0) T 1 ∂u g ν a≡ x , g∗ ≡ , ν∗ ≡ . (26) ν(0) ∂y v (0)ν(0) ν(0) (cid:12)y=0 T (cid:12) In the above equations we hav(cid:12)e found it convenient to introduce the dimensionless scaled spatial variable (cid:12) 1 y s(y)≡ dy′ν(y′), (27) v (0) T Z0 whereν(y)isaneffectivecollisionfrequency. Forthesakeofconcreteness,wechoose it as p(y) ν(y)= , (28) η(y) where η is the NS shear viscosity. The change from the boundary-imposed shear rate ω to the reduced local shear rate a is motivated by our goal of focusing on the central bulk region of the system, outside the boundary layers. Note that a represents the Knudsen number associated with the velocity gradient at y = 0. Likewise, g∗ measures the strength of the external field on a particle moving with the thermal velocity along a distance on the order of the mean free path. The relationship (27) can be inverted to yield s ds′ y y∗(s)= , y∗ ≡ . (29) ν∗(s′) v (0)/ν(0) Z0 T The invariance properties (18) translate into f∗(s,v∗,v∗,v∗|a,g∗) = f∗(−s,−v∗,−v∗,v∗|a,−g∗) x y z x y z = f∗(−s,v∗,−v∗,v∗|−a,g∗) x y z = f∗(−s,v∗,v∗,−v∗|a,g∗). (30) x y z Given the symmetry properties (30), we can restrict ourselves to a>0 and g∗ >0 without loss of generality. 3. Navier–Stokes description. To gain some insight into the type of fields one can expect in the Couette–Poiseuille flow, it is instructive to analyze the solution provided by the NS level of description. In the geometry of the problem, the NS constitutive equations are P =P =P =p, (31) xx yy zz P =−η∂ u , (32) xy y x q =0, (33) x q =−κ∂ T, (34) y y where η is the shear viscosity, as said above, and κ is the thermal conductivity. Inserting the NS approximate relations (31)–(34) into the exact conservationequa- tions (15)–(17) one gets p=const, (35) COUETTE–POISEUILLE FLOW 7 (η∂ )2u =−ηmng, (36) y x 5k B (η∂ )2T =−(η∂ u )2, (37) y y x 2mPr where Pr=(5k /2m)η/κ≃ 2 is the Prandtl number. In dimensionless form, Eqs. B 3 (36) and (37) can be rewritten as n∗(s) ∂2u∗(s)=− g∗, (38) s x ν∗(s) 2Pr ∂2T∗(s)=− [∂ u∗(s)]2. (39) s 5 s x For simplicity, let us assume that the particles are Maxwell molecules [7, 11, 63], so ν(y) ∝ n(y) and ν∗(s) = n∗(s). In that case, Eqs. (38) and (39) allow for an explicit solution: 1 u∗(s|a,g∗)=as− g∗s2, (40) x 2 Pr T∗(s|a,g∗)=1− s2 6a2−4ag∗s+g∗2s2 . (41) 30 (cid:16) (cid:17) Here we have applied the Galilean choice u (0) = 0 and the symmetry property x ∂ T| =0. y y=0 Equation(40)showsthat,accordingtotheNSapproximation,thevelocityfieldin theCouette–Poiseuilleflowissimplythesuperpositionofthe(quasi)linearCouette profile and the (quasi) parabolic Poiseuille profile. In the case of the temperature field, however, apart from the (quasi) parabolic Couette profile and the (quasi) quartic Poiseuille profile, a (quasi) cubic coupling term is present. Here we use the term “quasi” because the simple polynomial forms in Eqs. (40) and (41) refer to thescaledvariables. To gobacktothe realspatialcoordinatey oneneedstomake use of the relationship (27), taking into account that for Maxwell molecules ν ∝n. Instead of expressing s as a function of y it is more convenient to proceed in the opposite sense by using Eq. (29). Since 1/ν∗ =T∗ one simply has Pr 1 y∗(s)=s 1− s2 2a2−ag∗s+ g∗2s2 . (42) 30 5 (cid:20) (cid:18) (cid:19)(cid:21) For further use, note that, according to Eq. (41), ∂2T∗ 2 =−Pr a2. (43) ∂y∗2 5 (cid:12)y∗=0 (cid:12) Thus, the NS temperature profile pr(cid:12)esents a maximum at the midpoint y∗ =0. (cid:12) Before closing this section, let us write the pressure tensor and the heat flux profiles provided by the NS description: P∗ (s|a,g∗)=P∗ (s|a,g)=P∗ (s|a,g)=1, (44) xx yy zz P∗ (s|a,g∗)=−a+g∗s, (45) xy q∗(s|a,g∗)=0, (46) x 1 q∗(s|a,g∗)=s a2−ag∗s+ g∗2s2 . (47) y 3 (cid:18) (cid:19) 8 MOHAMED TIJ AND ANDRE´S SANTOS 4. Kinetic theory description. Perturbation solution. Now we want to get the hydrodynamicand flux profiles in the bulk domainofthe system froma purely kinetic approach, i.e., without assuming a priori the applicability of the NS con- stitutive equations. To that end, instead of considering the detailed Boltzmann operator J[f,f] we will make use of the celebrated BGK kinetic model [5, 7, 66]. In the BGK model Eq. (1) is replaced by ∂ ∂ v +g f(y,v|ω,g)=−ν(y|ω,g)[f(y,v|ω,g)−M(y,v|ω,g)], (48) y ∂y ∂v (cid:18) x(cid:19) where ν is the effective collision frequency defined by Eq. (28) and m 3/2 mV2 M(v)=n exp − (49) 2πk T 2k T (cid:18) B (cid:19) (cid:18) B (cid:19) isthelocalequilibriumdistributionfunction. Intermsofthedimensionlessvariables introduced in Eqs. (23)–(27), Eq. (48) can be rewritten as g∗ ∂ 1+v∗∂ f∗(s,v∗|a,g∗)=M∗(s,v∗|a,g∗)− f∗(s,v∗|a,g∗). (50) y s ν∗(s|a,g∗)∂v∗ x Its(cid:0)formal s(cid:1)olution is f∗(v∗) = 1+v∗∂ −1 M∗(v∗)− g∗ ∂ f∗(v∗) y s ν∗∂v∗ (cid:20) x (cid:21) (cid:0)∞ (cid:1) g∗ ∂ = (−v∗∂ )k M∗(v∗)− f∗(v∗) . (51) y s ν∗∂v∗ k=0 (cid:20) x (cid:21) X The formal character of the solution (51) is due to the fact that f∗ appears on the right-hand side explicitly and also implicitly through M∗ and ν∗. The solvability (or consistency) conditions are dv∗ 1,v∗,v∗2 f∗(s,v∗|a,g∗)= dv∗ 1,v∗,v∗2 M∗(s,v∗|a,g∗). (52) Z n o Z n o Let us assume now that g∗ is a small parameter so the solution to Eq. (50) can be expanded as f∗(s,v∗|a,g∗)=f∗(s,v∗|a)+f∗(s,v∗|a)g∗+f∗(s,v∗|a)g∗2+··· . (53) 0 1 2 Likewise, χ∗(s|a,g∗)=χ∗(s|a)+χ∗(s|a)g∗+χ∗(s|a)g∗2+··· , (54) 0 1 2 where χ∗ denotes a generic velocity moment of f∗. The expansions of n∗, u∗, and T∗ induce the corresponding expansion of M∗. The expansion in powers of g∗ allowsthe iterativesolutionofEq.(51)by ascheme similarto thatfollowedinRef. [54] in the case of an external force normal to the plates. 4.1. Zeroth order in g. Pure Couette flow. 4.1.1. Finite shear rates. To zeroth order in g∗, Eqs. (50) and (51) become 1+v∗∂ f∗(s,v∗|a)=M∗(s,v∗|a), (55) y s 0 0 ∞ (cid:0) (cid:1) f∗(s,v∗|a)= (−v∗∂ )kM∗(s,v∗|a), (56) 0 y s 0 k=0 X where p∗ V∗2 M∗(v∗)= 0 exp − 0 , V∗ ≡v∗−u∗. (57) 0 (2π)3/2T0∗5/2 (cid:18) 2T0∗(cid:19) 0 0 COUETTE–POISEUILLE FLOW 9 Thesearejusttheequationscorrespondingtothe pureCouetteflow. The complete solutionhas been obtained elsewhere[6, 20, 25] andso here we only quote the final results. The hydrodynamic profiles are p∗(s|a)=1, (58) 0 u∗ (s|a)=as, (59) x,0 T∗(s|a)=1−γ(a)s2, (60) 0 wherethedimensionlessparameterγ(a)isanonlinear functionofthereducedshear rate a given implicitly through the equation [6, 20] F (γ) a2 =γ 3+2 2 , (61) F (γ) (cid:20) 1 (cid:21) where the mathematical functions F (x) are defined by r F (x)= 2 ∞dtte−t2/2K (2x−1/4t1/2), F (x)= d x rF (x), (62) 0 0 r 0 x dx Z0 (cid:18) (cid:19) K (x)beingthezeroth-ordermodifiedBesselfunction. Equation(62)clearlyshows 0 thatF (x) has an essentialsingularity atx=0 and thus its expansionin powersof r x, ∞ F (x)= (k+1)r(2k+1)!(2k+1)!!(−x)k, (63) r k=0 X is asymptotic and not convergent. However, the series representation (63) is Borel summable[6,25],thecorrespondingintegralrepresentationbeinggivenbyEq.(62). The functions F (x) with r ≥ 3 can be easily expressed in terms of F (x), F (x), r 0 1 and F (x) as 2 1−F (x) 1 0 F (x)= −F (x)− F (x), (64) 3 2 1 8x 4 r−3 1 r−3 1 F (x)= (−1)m+rF (x)−F (x)− F (x), r ≥4. (65) r m r−1 r−2 8x m 4 m=0(cid:18) (cid:19) X It is interesting to compare the hydrodynamic profiles with the results obtained from the Boltzmann equation at NS order (see Sec. 3). We observe that Eq. (58) agrees with Eq. (35) and Eq. (59) agrees with Eq. (40) for g∗ = 0. On the other hand, Eq. (41) with g∗ =0 differs from Eq. (60), except in the limit of small shear rates, in which case γ(a)≈ 1a2 (Note that Pr=1 in the BGK model). 5 The relevant transport coefficients of the steady Couette flow are obtained from the pressure tensor and the heat flux. They are highly nonlinear functions of the reduced shear rate a given by [6, 19, 20, 30] P∗ (s|a)=1+4γ[F (γ)+F (γ)], (66) xx,0 1 2 P∗ (s|a)=1−2γ[F (γ)+2F (γ)], (67) yy,0 1 2 P∗ (s|a)=1−2γF (γ), (68) zz,0 1 P∗ (s|a)=−aF (γ), (69) xy,0 0 a F (γ) q∗ (s|a)= F (γ)−1−10γF (γ)−8γF (γ) 1− 2 s, (70) x,0 2 0 1 2 F (γ) (cid:26) (cid:20) 1 (cid:21)(cid:27) q∗ (s|a)=a2F (γ)s. (71) y,0 0 Notice that, although the temperature gradient is only directed along the y axis (so that there is a response in this direction through q∗), the shear flow induces a y 10 MOHAMED TIJ AND ANDRE´S SANTOS nonzero x component of the heat flux [19, 20, 30, 37]. Furthermore, normal stress differences(absentatNSorder)arepresent. Equations(69)and(71)canbeusedto identify generalizednonlinear shear viscosity andthermal conductivity coefficients. Ingeneral,thevelocitymomentsofdegreekoff∗ arepolynomialfunctionsofthe 0 spatialvariablesofdegreek−2. Anexplicitexpressionforthevelocitydistribution function f∗ has also been derived [20, 25]. 0 4.1.2. Limit of small shear rates. The coefficient γ(a) characterizing the profile of thezeroth-ordertemperatureT∗ isacomplicatednonlinearfunctionofthe reduced 0 shearratea,asclearlyapparentfromEq.(61). Obviously,thezeroth-orderpressure tensor and heat flux given by Eqs. (66)–(71) inherit this nonlinear character. It is illustrative to assume that the reduced shear rate a is small so one can express the quantities of interest as the first few terms in a (Chapman–Enskog) series expansion. From Eqs. (61)–(71) one obtains a2 72 γ(a)= 1+ a2+··· , (72) 5 25 (cid:18) (cid:19) 8a2 198 P∗ (s|a)=1+ 1− a2+··· , (73) xx,0 5 25 (cid:18) (cid:19) 6a2 228 P∗ (s|a)=1− 1− a2+··· , (74) yy,0 5 25 (cid:18) (cid:19) 2a2 108 P∗ (s|a)=1− 1− a2+··· , (75) zz,0 5 25 (cid:18) (cid:19) 18 P∗ (s|a)=−a 1− a2+··· , (76) xy,0 5 (cid:18) (cid:19) 14a3 1836 q∗ (s|a)=− 1− a2+··· s, (77) x,0 5 175 (cid:18) (cid:19) 18 q∗ (s|a)=a2 1− a2+··· s. (78) y,0 5 (cid:18) (cid:19) The terms of order a2, a, and a2 in Eqs. (72), (76), and (78), respectively, agree with the corresponding NS expressions, Eqs. (41), (45), and (47). On the other hand, as noted above, the normal stress differences (P∗ −P∗ and P∗ −P∗ ) and xx yy zz yy the streamwise heat flux component q∗ reveal non-Newtonian effects of orders a2 x and a3, respectively. 4.2. First order in g. Couette–Poiseuille flow. 4.2.1. Finite shear rates. To first order in g∗ Eq. (51) yields f∗(v∗)−M∗(v∗)=Λ(I)(v∗)+Λ(II)(v∗), (79) 1 1 where ∞ ∞ ∂ Λ(I)(v∗)≡ (−v∗∂ )kM∗(v∗), Λ(II)(v∗)≡− (−v∗∂ )kT∗ f∗(v∗), (80) y s 1 y s 0 ∂v∗ 0 k=1 k=0 x X X T∗ V∗2 u∗ M∗(v∗)=M∗(v∗) p∗+ 1 0 −5 + x,1V∗ , (81) 1 0 1 2T∗ T∗ T∗ x,0 (cid:20) 0 (cid:18) 0 (cid:19) 0 (cid:21)