NON-NEWTONIAN COUETTE–POISEUILLE FLOW OF A

7/28/2019 NON-NEWTONIAN COUETTEPOISEUILLE FLOW OF A

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arXiv:1009.2881v1

[cond-mat.stat-m

ech]15Sep2010

Manuscript submitted to Website: http://AIMsciences.orgAIMS JournalsVolume X, Number 0X, XX 200X pp. XXX

NON-NEWTONIAN COUETTEPOISEUILLE FLOW OF A

DILUTE GAS

Mohamed Tij

Departement de Physique, Universite Moulay Ismal,

Meknes, Morocco

Andres Santos

Departamento de Fsica, Universidad de Extremadura,E-06071 Badajoz, Spain

Abstract. The steady state of a dilute gas enclosed between two infinite par-allel plates in relative motion and under the action of a uniform body force

parallel to the plates is considered. The BhatnagarGrossKrook model ki-netic equation is analytically solved for this CouettePoiseuille flow to firstorder in the force and for arbitrary values of the Knudsen number associated

with the shear rate. This allows us to investigate the influence of the exter-nal force on the non-Newtonian properties of the Couette flow. Moreover, the

CouettePoiseuille flow is analyzed when the shear-rate Knudsen number andthe scaled force are of the same order and terms up to second order are retained.In this way, the transition from the bimodal temperature profile characteristic

of the pure force-driven Poiseuille flow to the parabolic profile characteristicof the pure Couette flow through several intermediate stages in the Couette

Poiseuille flow are described. A critical comparison with the NavierStokessolution of the problem is carried out.

1. Introduction. Two paradigmatic stationary nonequilibrium flows are the planeCouette flow and the Poiseuille flow. In the plane Couette flow the fluid (henceforthassumed to be a dilute gas) is enclosed between two infinite parallel plates in relativemotion, as sketched in Fig. 1(a). The walls can be kept at different or equal tem-peratures but, even if both wall temperatures are the same, viscous heating inducesa temperature gradient in the steady state. If the Knudsen number associated withthe shear rate is small enough the NavierStokes (NS) equations provide a satis-factory description of the Couette flow. On the other hand, as shearing increases,non-Newtonian effects (shear thinning and viscometric properties) and deviationsof Fouriers law (generalized thermal conductivity and streamwise heat flux com-ponent) become clearly apparent [16]. These nonlinear effects have been derivedfrom the Boltzmann equation for Maxwell molecules [10, 23, 29, 35, 47], from the

BhatnagarGrossKrook (BGK) kinetic model [5, 15, 38], and also from general-ized hydrodynamic theories [40, 42]. A good agreement with computer simulations[12, 13, 21, 25, 26, 32] has been found. The plane Couette flow has also been ana-lyzed in the context of granular gases [50, 55]. In the case of plates at rest but kept

2000 Mathematics Subject Classification. Primary: 76P05, 82B40; Secondary: 82C40, 82C05.Key words and phrases. BhatnagarGrossKrook kinetic model, Couette flow, Poiseuille flow,

non-Newtonian properties.

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2 MOHAMED TIJ AND ANDRES SANTOS

Figure 1. Sketch of (a) the Couette flow, (b) the force-drivenPoiseuille flow, and (c) the CouettePoiseuille flow.

at different temperatures, the Couette flow becomes the familiar plane Fourier flow,which also presents interesting properties by itself [3, 12, 13, 20, 24, 28, 35, 36, 37].

The Poiseuille flow, where a gas is enclosed in a channel or slab and fluid motion isinduced by a longitudinal pressure gradient, is a classical problem in kinetic theory[8, 30]. Essentially the same type of flow field is generated when the pressuregradient 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 hasreceived a lot of attention both from theoretical [1, 2, 11, 14, 17, 27, 33, 34, 39, 40,42, 45, 46, 48, 49, 54, 57] and computational [18, 19, 22, 33, 51, 52, 58] points of view.This interest has been mainly motivated by the fact that the force-driven Poiseuilleflow 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 NSequations predict a temperature profile with a flat maximum at the center, computersimulations [22] and kinetic theory calculations [45, 46] show that it actually has alocal minimum at that point.

Obviously, the Couette and Poiseuille flows can be combined to become theCouettePoiseuille (or PoiseuilleCouette) flow [7, 31, 41, 43]. To the best of ourknowledge, all the studies on the CouettePoiseuille flow assume that the Poiseuillepart is driven by a pressure gradient, not by an external force. This paper intendsto fill this gap by considering the steady state of a dilute gas enclosed between twoinfinite parallel plates in relative motion, the particles of the gas being subject tothe action of a uniform body force. This CouettePoiseuille flow is sketched in Fig.1(c). We will study the problem by the tools of kinetic theory by solving the BGKmodel 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 bythe action of the external force. To that end we will assume a finite value of theKnudsen number related to the shear rate and perform a perturbation expansion tofirst order in the force. As a second objective, we will study how the non-Newtonianforce-driven Poiseuille flow is modified by the shearing. This is done by assumingthat the shear-rate Knudsen number and the scaled force are of the same order andneglecting terms of third and higher order. In both cases we are interested in thephysical properties in the central bulk region of the slab, outside the influence ofthe boundary layers.


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COUETTEPOISEUILLE FLOW 3

The organization of the paper is as follows. The Boltzmann equation for theCouettePoiseuille flow 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 thekinetic theory approach is worked out. Some technical calculations are relegated toAppendix A. The results are graphically presented and discussed in Sec. 5. Thepaper ends with some concluding remarks in Sec. 6.

2. The CouettePoiseuille flow. Symmetry properties. Let us consider adilute 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 andare kept at a common temperature Tw. The imposed shear rate is therefore =(U+ U)/L. Besides, an external body force F = mgx, where m is the mass of aparticle and g is a constant acceleration, is applied. The geometry of the problemis sketched in Fig. 1(c). In the absence of the external force (g = 0) this problemreduces to the plane Couette flow [see Fig. 1(a)]. On the other hand, if the plates

are at rest ( = 0), one is dealing with the force-driven Poiseuille flow [see Fig.1(b)]. The general problem with = 0 and g = 0 defines the CouettePoiseuilleflow analyzed in this paper.

In the steady state only gradients along the y axis are present and thus theBoltzmann equation becomes

vy

y+ g

vx

f(y, v|, g) = J[f, f], (1)

where f is the one-particle velocity distribution function and J[f, f] is the Boltz-mann collision operator [6, 9], whose explicit expression will not be written downhere. The notation f(y, v|, g) emphasizes the fact that, apart from its spatial andvelocity dependencies, the distribution function depends on the independent exter-nal parameters and g. As said above, g = 0 and = 0 correspond to the Couette

and Poiseuille flows, respectively.The first few moments of f define the densities of conserved densities (mass,

momentum, and temperature) and the associated fluxes. More explicitly,

n(y|, g) =

dv f(y, v|, g), (2)

n(y|, g)u(y|, g) =

dv vf(y, v|, g), (3)

n(y|, g)kBT(y|g) = p(y|, g) =m

3

dv V2(y, v|, g)f(y, v|, g), (4)

Pij(y|, g) = m

dv Vi(y, v|, g)Vj(y, v|, g)f(y, v|, g), (5)

q(y|, g) =

m

2 dv V2(y, v|, g)V(y, v|, g)f(y, v|, g). (6)In these equations n is the number density, u is the flow velocity,

V(y, v|a,, ) v u(y|, g) (7)

is the peculiar velocity, T is the temperature, kB is the Boltzmann constant, p isthe hydrostatic pressure, Pij is the pressure tensor, and q is the heat flux. Takingvelocity moments in both sides of Eq. (1) one gets the following exact balanceequations

yPyy = 0, (8)


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Quantity Sg Sn + +

ux +T + +p + +

Pxx + +Pyy + +Pxy + qx +qy

Table 1. Parity factors Sg and S for the hydrodynamic fieldsand the fluxes [see Eq. (12)].

yPxy = mng, (9)yqy + Pxyyux = 0. (10)

Henceforth, without loss of generality, we will assume ux(0) = 0. In other words,we will adopt a reference frame solidary with the flow at the midpoint y = 0.

The symmetry properties of the CouettePoiseuille flow imply the following in-variance properties of the velocity distribution function:

f(y, vx, vy, vz|, g) = f(y, vx, vy, vz|, g)

= f(y, vx, vy, vz| , g)

= f(y, vx, vy, vz|, g), (11)

As a consequence, if (y|, g) denotes a hydrodynamic variable or a flux, one has

(y|, g) = Sg(y|, g)= S(y| , g), (12)

where Sg = 1 and S = 1. The parity factors Sg and S for the non-zerohydrodynamic fields and fluxes are displayed in Table 1. In general, if is a momentof the form

(y|, g) =

dv Vkxx (y|, g)v

kyy v

2kzz f(y, v|, g) (13)

then Sg = (1)kx+ky and S = (1)ky .In order to nondimensionalize the problem, we will choose quantities evaluated

at the central plane y = 0 as units:

f(s, v|a, g) v3T(0)

n(0)f(y, v|, g), v

v

vT(0), vT(0) kBT(0)m , (14)

n(s|a, g) n(y|, g)

n(0), T(s|a, g)

T(y|, g)

T(0), p(s|a, g)

p(y|, g)

p(0), (15)

Pij(s|a, g)

Pij(y|, g)

p(0), q(s|a, g)

q(y|, g)

p(0)vT(0), (16)

a 1

(0)

uxy

y=0

, g g

vT(0)(0),

(0). (17)


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In the above equations we have found it convenient to introduce the dimensionlessscaled spatial variable

s(y) 1vT(0)

y0

dy (y), (18)

where (y) is an effective collision frequency. For the sake of concreteness, we chooseit as

(y) =p(y)

(y), (19)

where is the NS shear viscosity. The change from the boundary-imposed shearrate to the reduced local shear rate a is motivated by our goal of focusing onthe central bulk region of the system, outside the boundary layers. Note that arepresents the Knudsen number associated with the velocity gradient at y = 0.Likewise, g measures the strength of the external field on a particle moving withthe thermal velocity along a distance on the order of the mean free path.

The relationship (18) can be inverted to yield

y(s) =s

0

ds

(s), y

y

vT(0)/(0). (20)

The invariance properties (11) translate into

f(s, vx, v

y, v

z |a, g) = f(s, vx, v

y, v

z |a, g)

= f(s, vx, v

y , v

z | a, g)

= f(s, vx, v

y , v

z |a, g). (21)

Given the symmetry properties (21), we can restrict ourselves to a > 0 and g > 0without loss of generality.

3. NavierStokes description. To gain some insight into the type of fields onecan expect in the CouettePoiseuille flow, it is instructive to analyze the solutionprovided by the NS level of description. In the geometry of the problem, the NSconstitutive equations are

Pxx = Pyy = Pzz = p, (22)

Pxy = yux, (23)

qx = 0, (24)

qy = yT, (25)

where is the shear viscosity, as said above, and is the thermal conductivity.Inserting the NS approximate relations (22)(25) into the exact conservation equa-tions (8)(10) one gets

p = const, (26)

(y)2 ux = mng, (27)

5kB2m Pr

(y)2 T = (yux)

2 , (28)

where Pr = (5kB/2m)/ 23 is the Prandtl number. In dimensionless form, Eqs.

(27) and (28) can be rewritten as

2su

x(s) = n(s)

(s)g, (29)

2sT(s) =

2 Pr

5[su

x(s)]2 . (30)


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For simplicity, let us assume that the particles are Maxwell molecules [6, 9, 53], so(y) n(y) and (s) = n(s). In that case, Eqs. (29) and (30) allow for an explicit

solution:ux(s|a, g

) = as 1

2gs2, (31)

T(s|a, g) = 1 Pr

30s2

6a2 4ags + g2s2

. (32)

Here we have applied the Galilean choice ux(0) = 0 and the symmetry propertyyT|y=0 = 0.

Equation (31) shows that, according to the NS approximation, the velocity field inthe CouettePoiseuille flow is simply the superposition of the (quasi) linear Couetteprofile and the (quasi) parabolic Poiseuille profile. In the case of the temperaturefield, however, apart from the (quasi) parabolic Couette profile and the (quasi)quartic Poiseuille profile, a (quasi) cubic coupling term is present. Here we use theterm quasi because the simple polynomial forms in Eqs. (31) and (32) refer to

the scaled variable s. To go back to the real spatial coordinate y one needs to makeuse of the relationship (18), taking into account that for Maxwell molecules n.Instead of expressing s as a function of y it is more convenient to proceed in theopposite sense by using Eq. (20). Since 1/ = T one simply has

y(s) = s

1

Pr

30s2

2a2 ags +1

5g2s2

. (33)

For further use, note that, according to Eq. (32),

2T

y2

y=0

= Pr2

5a2. (34)

Thus, the NS temperature profile presents a maximum at the midpoint y = 0.Before closing this section, let us write the pressure tensor and the heat flux

profiles provided by the NS description:

Pxx(s|a, g) = Pyy(s|a, g) = P

zz(s|a, g) = 1, (35)

Pxy(s|a, g) = a + gs, (36)

qx(s|a, g) = 0, (37)

qy(s|a, g) = s

a2 ags +

1

3g2s2

. (38)

4. Kinetic theory description. Perturbation solution. Now we want to getthe hydrodynamic and flux profiles in the bulk domain of the system from a purelykinetic approach, i.e., without assuming a priori the applicability of the NS con-

stitutive equations. To that end, instead of considering the detailed Boltzmannoperator J[f, f] we will make use of the celebrated BGK kinetic model [4, 6, 56].In the BGK model Eq. (1) is replaced by

vy

y+ g

vx

f(y, v|, g) = (y|, g) [f(y, v|, g) M(y, v|, g)] , (39)

where is the effective collision frequency defined by Eq. (19) and

M(v) = n

m

2kBT

3/2exp

mV2

2kBT

(40)


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is the local equilibrium distribution function. In terms of the dimensionless variablesintroduced in Eqs. (14)(18), Eq. (39) can be rewritten as1 + vys f(s, v|a, g) = M(s, v|a, g) g(s|a, g) vx f(s, v|a, g). (41)Its formal solution is

f(v) =

1 + vys1

M(v) g

vxf(v)

=

k=0

(vys)k

M(v)

g

vxf(v)

. (42)

The formal character of the solution (42) is due to the fact that f appears on theright-hand side explicitly and also implicitly through M and . The solvability(or consistency) conditions are

dv 1, v, v2 f(s, v|a, g) = dv 1, v, v2M(s, v|a, g). (43)Let us assume now that g is a small parameter so the solution to Eq. (41) can

be expanded as

f(s, v|a, g) = f0 (s, v|a) + f1 (s, v

|a)g + f2 (s, v|a)g2 + . (44)

Likewise,

(s|a, g) = 0(s|a) + 1(s|a)g

+ 2(s|a)g2 + , (45)

where denotes a generic velocity moment of f. The expansions of n, u, andT induce the corresponding expansion of M. The expansion in powers of g

allows the iterative solution of Eq. (42) by a scheme similar to that followed in Ref.[44] 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. (41) and (42) become

1 + vys

f0 (s, v|a) = M0(s, v

|a), (46)

f0 (s, v|a) =

k=0

(vys)kM0(s, v

|a), (47)

where

M0(v) =

p0

(2)3/2 T05/2

exp

V02

2T0

, V

0 v u0. (48)

These are just the equations corresponding to the pure Couette flow. The completesolution has been obtained elsewhere [5, 16, 21] and so here we only quote the finalresults. The hydrodynamic profiles are

p0(s|a) = 1, (49)

ux,0(s|a) = as, (50)

T0 (s|a) = 1 (a)s2, (51)

where the dimensionless parameter (a) is a nonlinear function of the reduced shearrate a given implicitly through the equation [5, 16]

a2 =

3 + 2

F2()

F1()

, (52)


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where the mathematical functions Fr(x) are defined by

F0(x) = 2x 0 dttet2/2K0(2x1/4t1/2), Fr(x) = ddx xr

F0(x), (53)

K0(x) being the zeroth-order modified Bessel function. Equation (53) clearly showsthat Fr(x) has an essential singularity at x = 0 and thus its expansion in powers ofx,

Fr(x) =k=0

(k + 1)r(2k + 1)!(2k + 1)!!(x)k, (54)

is asymptotic and not convergent. However, the series representation (54) is Borelsummable [5, 21], the corresponding integral representation being given by Eq. (53).The functions Fr(x) with r 3 can be easily expressed in terms of F0(x), F1(x),and F2(x) as

F3(x) = 1 F0(x)

8x F2(x) 1

4F1(x), (55)

Fr(x) =1

8x

r3m=0

r 3

m

(1)m+rFm(x) Fr1(x)

1

4Fr2(x), r 4. (56)

It is interesting to compare the hydrodynamic profiles with the results obtainedfrom the Boltzmann equation at NS order (see Sec. 3). We observe that Eq. (49)agrees with Eq. (26) and Eq. (50) agrees with Eq. (31) for g = 0. On the otherhand, Eq. (32) with g = 0 differs from Eq. (51), except in the limit of small shearrates, in which case (a) 15 a

2 (Note that Pr = 1 in the BGK model).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 [5, 15, 16, 25]

Pxx,0(s|a) = 1 + 4[F1() + F2()], (57)

Pyy,0(s|a) = 1 2[F1() + 2F2()], (58)

Pzz,0(s|a) = 1 2F1(), (59)

Pxy,0(s|a) = aF0(), (60)

qx,0(s|a) =a

2

F0() 1 10F1() 8F2()

1

F2()

F1()

s, (61)

q

y,0(s|a) = a

2

F0()s. (62)Notice that, although the temperature gradient is only directed along the y axis

(so that there is a response in this direction through qy), the shear flow induces anonzero x component of the heat flux [15, 16, 25, 32]. Furthermore, normal stressdifferences (absent at NS order) are present. Equations (60) and (62) can be used toidentify generalized nonlinear shear viscosity and thermal conductivity coefficients.

In general, the velocity moments of degree k off0 are polynomial functions of thespatial variable s of degree k 2. An explicit expression for the velocity distributionfunction f0 has also been derived [16, 21].


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4.1.2. Limit of small shear rates. The coefficient (a) characterizing the profile ofthe zeroth-order temperature T0 is a complicated nonlinear function of the reduced

shear rate a, as clearly apparent from Eq. (52). Obviously, the zeroth-order pressuretensor and heat flux given by Eqs. (57)(62) inherit this nonlinear character.

It is illustrative to assume that the reduced shear rate a is small so one canexpress the quantities of interest as the first few terms in a (ChapmanEnskog)series expansion. From Eqs. (52)(62) one obtains

(a) =a2

5

1 +

72

25a2 +

, (63)

Pxx,0(s|a) = 1 +8a2

5

1

198

25a2 +

, (64)

Pyy,0(s|a) = 1 6a2

5

1

228

25a2 +

, (65)

Pzz,0(s|a) = 1 2a2

51 108

25a2 + , (66)

Pxy,0(s|a) = a

1

18

5a2 +

, (67)

qx,0(s|a) = 14a3

5

1

1836

175a2 +

s, (68)

qy,0(s|a) = a2

1

18

5a2 +

s. (69)

The terms of order a2, a, and a2 in Eqs. (63), (67), and (69), respectively, agreewith the corresponding NS expressions, Eqs. (32), (36), and (38). On the otherhand, as noted above, the normal stress differences (Pxx P

yy and P

zz P

yy) and

the streamwise heat flux component qx

reveal non-Newtonian effects of orders a2

and a3, respectively.

4.2. First order in g. CouettePoiseuille flow.

4.2.1. Finite shear rates. To first order in g Eq. (42) yields

f1 (v) M1(v

) = (I)(v) + (II)(v), (70)

where

(I)(v) k=1

(vys)kM1(v

), (II)(v) k=0

(vys)kT0

vxf0 (v

), (71)

M1(v) = M0(v

)

p1 +

T12T0

V0

2

T0 5

+

ux,1T0

Vx,0

, (72)

and we have already specialized to Maxwell molecules, so that = p/T. In orderto apply the consistency conditions (43) to get the hydrodynamic fields p1, u

x,1,

and T1 it is convenient to define the moments

n1n2n3 = (I)n1n2n3 +

(II)n1n2n3 , (73)

(I)n1n2n3 =

dv Vx,0

n1vyn2vz

n3(I)(v), (74)

(II)n1n2n3 =

dv Vx,0

n1vyn2vz

n3(II)(v). (75)


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Therefore, the consistency conditions are

000 = 0, (76)

100 = 0, (77)

010 = 0, (78)

200 + 020 + 002 = 0. (79)

The evaluation of (I)n1n2n3 and

(II)n1n2n3 is carried out in Appendix A. The first-

order profiles are

p1(s|a) = p(1)1 (a)s, (80)

ux,1(s|a) = u(2)x,1(a)s

2, (81)

T1 (s|a) = T(3)1 (a)s

3, (82)

where

p(1)1 (a) = 1

1 F1F

0T(3)

1 (a), (83)

u(2)x,1(a) =2(F1 + 2F2) 3

6F1

a

F1F0

F2F1

T(3)1 (a), (84)

T(3)1 (a) =

4

3a2F0

4F2 (F1 + 2F2) F1 (1 F0) 6F2D(a)

, (85)

with

D(a) 2F1

F20 2F0 + F1 2(F0 2F1) (F1 + 2F2)

+ a2F0F1 (1 F0)

2a2

F0

F21 + 6F1F2 + 8F2

2

2F21 (F1 + 2F2)

. (86)

In the above equations the functions Fr are understood to be evaluated at x = .As shown in Appendix A, the moment n1n2n3 is a polynomial function of s

of degree n1 + n2 + n3 1. In particular, the non-zero elements of the first-orderpressure tensor are

Pij,1(s|a) = P(1)ij,1(a)s, (87)

withP

(1)xx,1(a) = 3p

(1)1 (a) P

(1)zz,1(a), (88)

P(1)yy,1(a) = 0, (89)

P(1)zz,1(a) =

F0 1

2+ F1 + 2F2

T

(3)1 [2(F1 + 2F2) 1]p

(1)1 , (90)

P(1)xy,1(a) = 1. (91)

As for the heat flux vector, the results are

qi,1(a|s) = q(0)i,1 (a) + q

(2)i,1 (a)s

2, (92)

where

q(0)x,1(a) = a

2F1 F0 182

+ F1 + 26F2

4+ a2 1 F0

83+ 7F

1 16F242

T(3)1+

1 F0

4+ 3F2

1

2F1 + 3a

2 F1 F2

u

(2)x,1 + a

1 F0

4+ 3F2

1

2F1 + a

2 F1 F2

p

(1)1 +

2

3F0 +

1

6F1 +

5

3

10

3(F1 + 2F2)

+a2

1 F04

+3

2F1 + 3F2

, (93)


11/23


q(2)x,1(a) = a

2F0 F2 1

2+ F1 + 2F2 a

2

9F0 2F1 7

82

+3 F1 + 14F2

4T(3)1 32 12 F1 F0 + 2(3F1 + 4F2)

3a2

1 F04

1

2F1 3F2

u

(2)x,1 + a

1 +

1

2F0 +

1

2F1

+2(F1 + 2F2) + a2

1 F0

4+

1

2F1 5F2

p

(1)1

6(4 + 4F0 + 5F1 + 2F2) +

4

32(F1 + 2F2) +

a2

2(F1 F0) , (94)

q(0)y,1(a) =

1 F0

82

7F1 + 2F24

+ a23F1 F2 2F0

22

T

(3)1 + a

F1 F0

u(2)x,1

32 F1 + F2 + a2 F0 F12 p(1)1 2a3 (2F1 + F2), (95)q

(2)y,1(a) =

F1 1

4+

1

2F1 + F2 +

a2

4

1 F0

22

2F0 7F1 + 14F2

T

(3)1

+a (F0 + F1 2F2) u(2)x,1 +

1

4(1 F0) + (F1 + 2F2)

a2

2(F0 3F1 + 2F2)

p

(1)1 +

a

6[1 F0 2(F1 + 2F2)] . (96)

Equations (89) and (91) are consistent with the momentum balance equations(8) and (9), respectively. The energy balance equation (10) requires that

q(2)y,1 = aF0u(2)x,1 1

2 . (97)Taking into account Eqs. (83)(85) it is possible to check that Eqs. (96) and (97)are indeed equivalent.

Let us now get the relationship between the scaled space variable s and the true(dimensionless) coordinate y. From the definition (18) we have

dy

ds=

T(s|a, g)

p(s|a, g)

= T0 (s|a) + [T

1 (s|a) T

0 (s|a)p

1(s|a)] g + O(g2). (98)

Inserting Eqs. (51), (80), and (82) one gets

y(s|a, g) = s (a)

3s3

s2

4 2p

(1)1 (a)

T

(3)1 (a) + (a)p

(1)1 (a)

s2

g + O(g2).

(99)4.2.2. Limit of small shear rates. As done in the case ofg = 0, it is illustrative toobtain the first-order coefficients (83)(85), (88), (90), and (93)(96) in the limit ofsmall shear rates. Taking into account Eqs. (54) and (63) one gets

p(1)1 (a) =

12a

5

1

73

5a2 +

, (100)

u(2)x,1(a) =

1

2

29a2

5

1

23136

725a2 +

, (101)


12/23


0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

- 2 . 0

- 1 . 5

- 1 . 0

- 0 . 5

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

- 2 . 5

- 2 . 0

- 1 . 5

- 1 . 0

- 0 . 5

p

(

1

)

1

,

u

x

,

(

2

)

1

,

T

(

3

)

1

P

x

x

,

(

1

)

1

,

P

z

z

,

(

1

)

1

Figure 2. First-order coefficients p(1)

1 , u(2)

x,1, T(3)

1 (left panel),P

(1)xx,1, and P

(1)zz,1 (right panel) as functions of the reduced shear

rate a. The dashed lines represent the terms shown in Eqs. (100)(104).

T(3)

1 (a) =2a

15

1 +

109

5a2 +

, (102)

P(1)xx,1(a) =

28a

5

1

431

35a2 +

, (103)

P(1)zz,1(a) = 8a51 113

5a2 + , (104)

q(0)x,1(a) = 1 +

216a2

5

1

23329

225a2 +

, (105)

q(2)x,1(a) =

29a2

5

1

1844

145a2 +

, (106)

q(0)y,1(a) =

19a

5

1

8172

95a2 +

, (107)

q(2)y,1(a) = a 1 + 4a

2 + . (108)According to Eqs. (31), (32), and (38), the NS equations yield u(2)x,1 = 12 , T(3)1 =2

15a (taking Pr = 1), and q

(2)y,1 = a, so that they agree with the leading terms

in Eqs. (101), (102), and (108). On the other hand, the NS approximation fails

in accounting for the non-zero values of p(1)1 , P

(1)xx,1, P

(1)zz,1, q

(0)x,1, q

(2)x,1, and q

(0)y,1. In

particular, q(0)x,1 = 0 even in the pure Poiseuille flow (a = 0) [35, 39, 45, 46].

5. Results and discussion.


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0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

- 1 . 5

- 1 . 0

- 0 . 5

q

x

,

(

0

)

1

,

q

x

,

(

2

)

1

q

y

,

(

0

)

1

,

q

y

,

(

2

)

1

Figure 3. First-order coefficients q(0)x,1, q(2)x,1 (left panel), q(0)y,1, andq

(2)y,1 (right panel) as functions of the reduced shear rate a. The

dotted lines represent the terms shown in Eqs. (105)(108).

5.1. Finite shear rates. First order in g. The nonlinear dependence on thereduced shear rate a of the zeroth-order quantities has been analyzed elsewhere[5, 15, 16], so that here we focus on the first-order corrections. Figure 2(a) shows

the coefficients associated with the hydrodynamic profiles, i.e., p(1)1 (a), u

(2)x,1(a),

and T(3)1 (a). The first two quantities are negative, while the third one is positive,

in agreement with what might be expected in view of Eqs. ( 100)(102). On theother hand, the practical range of applicability of the truncated series (100)(102)is restricted to small shear rates (a 0.1). The addition of further terms in the(ChapmanEnskog) expansion in powers ofa would not improve that range becauseof the asymptotic character of the series. Note that the range of applicability of the

NS description (according to which p(1)1 = 0, u

(2)x,1 =

12 , and T

(3)1 =

215 a) is even

much more restrictive, especially in the case of p(1)1 .

The normal-stress coefficients P(1)xx,1(a) and P

(1)zz,1(a) are plotted in Fig. 2(b). Both

coefficients vanish in the NS description. Again, the truncated series (103) and (104)are reliable only for a 0.1. We observe that the xx element has a much largermagnitude than the zz element. The other relevant coefficients of the pressure

tensor are not plotted because they are identically P(1)yy,1 = 0 and P

(1)yy,1 = 1, as a

consequence of the exact momentum balance equations (8) and (9).The coefficients associated with the heat flux vector are plotted in Fig. 3. Ac-

cording to the NS equations, q(0)x,1 = q

(2)x,1 = q

(0)y,1 = 0 and q

(2)y,1 = a, what strongly

contrasts with the nonlinear behavior observed in Fig. 3. This is especially dramatic

in the case of the streamwise coefficient q(2)x,1, which deviates from zero even in the

limit a 0. It is interesting to note that, while q(2)x,10, q

(0)y,1, and q

(2)y,1 have definite

signs (at least in the interval 0 a 1), the coefficient q(0)x,1 changes from negative

to positive around a 0.42.


14/23


- 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8

- 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8

- 1 . 0

- 0 . 5

- 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8

T

*

u

* x

,

q

* i

P

* i

j

Figure 4. Profiles of (a) temperature, (b) elements of the pressuretensor, and (c) flow velocity and components of the heat flux vector.The value of the reduced shear rate is a = 1. Two values of theexternal force are considered: g = 0 (dashed lines) and g = 0.1(solid lines).

In order to illustrate how the Couette-flow profiles are distorted by the action ofthe external force, we will take a = 1 with g = 0 and g = 0.1. While the lattervalue is possibly not small enough as to make the first-order calculations sufficient,our aim here is to highlight the trends to be expected when the fully nonlinearCouette flow coexists with the force-driven Poiseuille flow. The profiles are shown

in Fig. 4. In the pure Couette flow (g = 0) the temperature and pressure profilesare symmetric, while the velocity and heat flux profiles are antisymmetric. Theapplication of the external force breaks these symmetry features since the first-order terms have symmetry properties opposite to those of the zeroth-order terms,in agreement with the signs of Sg in Table 1. As a consequence, the temperaturegradient increases across the channel with respect to that of the Couette flow, asshown by Fig. 4(a). The elements of the pressure tensor are no longer uniformbut exhibit negative gradients, especially in the case of the normal stress Pxx [seeFig. 4(b)]. An exception is Pyy , which is exactly uniform as a consequence of the


15/23


momentum balance equation (8). To first order in g the value of Pyy is the sameas in the Couette flow [see Eq. (89)], but this situation changes when terms of order

g2

are added [35, 45, 46]. We observe from Fig. 4(c) that the flow velocity (in thereference frame moving with the midplane y = 0) is decreased by the action of theexternal force. A similar behavior is presented by qy, what qualitatively correlateswith the increase in the temperature gradient observed in Fig. 4(a). Regarding thecomponent qx, it takes larger values in the CouettePoiseuille flow than in the pureCouette flow. In both cases (g = 0 and g = 0.1) the shearing is so large (a = 1)that the two components of the heat flux have a similar magnitude, i.e., |qx| |q

y |.

An interesting effect induced by the external field is the existence of a non-zero heatflux at y = 0, even though the temperature gradient vanishes at that point. More

specifically, qx(0) = q(0)x,1 > 0 and q

y(0) = q

(0)y,1 < 0. The sign of the former quantity

changes, as noted before, at a 0.42.

5.2. Small shear rates. Second order in g. Thus far, all the results are valid

for arbitrary values of the reduced shear rate a but are restricted to first order inthe reduced external field g. One could continue the perturbation scheme devisedin Sec. 4 to further orders in g but the analysis becomes extremely cumbersomeif one still wants to keep a arbitrary. Furthermore, the perturbation expansion inpowers of g is expected to be only asymptotic [46]. On the other hand, we cancombine the results obtained here [see Eqs. (63)(69) and (100)(108)] with thoseof Ref. [46] to obtain the hydrodynamic and flux profiles to second order in g, byassuming that both the reduced shear rate and the reduced force are of the sameorder, i.e., a g. The results are

p(s|a, g) = 1 12

5ags +

6

5g2s2 + , (109)

ux(s|a, g) = as

1

2

gs2 + , (110)

T(s|a, g) = 1 1

5a2s2 +

2

15ags3 + g2s2

1925

1

30s2

+ , (111)

Pxx(s|a, g) = 1 +

8

5a2

28

5ags + g2

328

25+

14

5s2

+ , (112)

Pyy(s|a, g) = 1

6

5a2

306

25g2 + , (113)

Pzz (s|a, g) = 1

2

5a2

8

5ags g2

22

25

4

5s2

+ , (114)

Pxy(s|a, g) = a + gs + , (115)

qx(s|a, g) = g + , (116)

q

y(s|a, g

) = a2

s ag195 + s2 + 13 g2s3 + . (117)

In these equations the ellipses denote terms that are at least of orders a3, g3, a2g

or ag2. As for the relationship between y and s, from Eqs. (18) or (20) we get

s = y +1

15a2y3

1

30agy2

36 + y2

+

1

150g2y3

22 + y2

+ . (118)

Therefore, we can safely replace s by y in Eqs. (109)(117).Again, it is instructive to compare Eqs. (109)(117) against the NS predictions

worked out in Sec. 3 (with Pr = 1, in consistency with the BGK value of the


16/23


- 8 - 6 - 4 - 2 0 2 4 6 8 - 8 - 6 - 4 - 2 0 2 4 6 8

(

T

*

-

1

)

/

g

*

2

(

T

*

-

1

)

/

g

*

2

Figure 5. Profiles of the scaled temperature difference (T 1)/g2 for |g| 1 and several values of the ratio a/g: a/g = 0

(), a/g = 1 ( ), a/g =

19/5 (- - - ), a/g = 3 ( ),

a/g = 2

19/5 (- - - ), and a/g = 5 (- - - -). The left panelcorresponds to the kinetic-theory predictions, Eq. (120), while theright panel corresponds to the NS predictions, Eq. (32).

Prandtl number). As can be seen from Eqs. (26), (31), (32), and (35)(38), the NSexpressions do not contain terms of order higher than a2, g2, or ag and thereforetheir comparison with the kinetic-theory results (109)(117) is not biased. We see

that only the NS results for u

x and P

xy are supported by kinetic theory. Thisdoes not mean that Newtons law for the shear stress, Eq. (23) is satisfied, since = p/ and the hydrostatic pressure p is not actually uniform. As said before,the NS constitutive equations also fail to account for the existence of normal stressdifferences (typical non-Newtonian effects) as well as of a streamwise componentof the heat flux (failure of Fouriers law). Perhaps the most interesting and subtledifferences refer to the presence in Eqs. (111) and (117) of the extra terms 1925 g

2s2

and 195 ag, respectively, which are absent in their NS counterparts, Eqs. (32) and

(38). The extra term in qy implies that qy(0) = 0, what represents a clear violation

of Fouriers law (25). The term 1925 g2s2 present in the temperature field (111) has

dramatic consequences on the curvature of the temperature profile at the midpointy = 0. From Eq. (111) we get

2T

y2y=0

= 25

a2 + 3825

g2. (119)

Therefore, while the NS temperature profile presents a maximum at y = 0 [see Eq.(34)], Eq. (119) shows that the profile has actually a local minimum if a2 < 195 g

2.To analyze this feature in more detail, let us rewrite Eq. (111) as

T 1

g2

1

5

a

g

2y2 +

2

15

a

g

y3 + y2

19

25

1

30y2

. (120)


17/23


Figure 5(a) shows the scaled temperature difference (T 1)/g2, as given by Eq.

(120), for a/g = 0, 1, 19/5, 3, 219/5, and 5. In the case a/g = 0 (pure

Poiseuille flow) the temperature profile has a minimum at y

= 0 surrounded bytwo symmetric maxima at y = 2

19/5. When a/g = 0 (CouettePoiseuille

flow) several possibilities arise. If 0 < a/g 0 departs from it. This is represented by thecase a/g = 1 in Fig. 5(a). At a/g =

19/5, the left maximum and the central

minimum merge to become an inflection point of zero slope. Next, in the range19/5 < a/g < 2

19/5 the temperature presents a local maximum at y = 0

followed by a minimum and an absolute maximum, both with y > 0. This situationis illustrated by the case a/g = 3 in Fig. 5(a). At a/g = 2

19/5 the minimum

and maximum with y > 0 merge to create an inflection point of zero slope. Finally,ifa/g >

19/5 [see case a/g = 5 in Fig. 5(a)] only the central maximum remains

and the profile becomes more and more symmetric as a/g increases. In the limita/g (or, equivalently, g 0) one recovers the pure Couette flow. This richphenomenology is absent in the case of the NS temperature profile, as shown in Fig.5(b).

Given the physical interest of Eq. (111) or, equivalently, Eq. (120), it is convenientto rewrite it in real units. This yields

T(y) = T(0) (0)

2(0)

uxy

2y=0

y2 +mn(0)

3(0)

uxy

y=0

gy3

m2n2(0)

12(0)(0)g2y4 + CT

m2

k2BT(0)g2y2, (121)

where CT =1925 in the BGK model, while CT 1.0153 in the Boltzmann equation

for Maxwell molecules [35, 39, 45]. Equation (121) still holds in the NS description,except that CT = 0.

6. Concluding remarks. In this paper we have studied the stationary CouettePoiseuille flow of a dilute gas. As illustrated by Fig. 1(c), the gas is enclosedbetween two infinite parallel plates in relative motion (Couette flow) and at thesame time the particles feel the action of a uniform longitudinal force (force-drivenPoiseuille flow) along the same direction as the moving plates. Our main goal hasbeen to assess the limitations of the NS description of the problem and highlightthe importance of non-Newtonian properties.

In order to get explicit results, the complicated Boltzmann collision operator hasbeen replaced by the mathematically much simpler BGK model with a temperature-independent collision frequency (Maxwell molecules). The kinetic model has been

solved to first order in the reduced force parameter g

for arbitrary values of thereduced shear rate a. Moreover, complementing these results with those obtainedin previous works for the pure Poiseuille flow to second order in g, we have beenable to get the solution to second order in both a and g.

Starting from the pure nonlinear Couette flow, we have studied the influence of aweak external force, as measured by the nonlinear shear-rate dependence of the nine

coefficients p(1)1 (a), u

(2)x,1(a), T

(3)1 (a), P

(1)xx,1(a), P

(1)zz,1(a), q

(0)x,1(a), q

(2)x,1(a), q

(0)y,1(a), and

q(2)x,1(a). These functions are plotted in Figs. 2 and 3. A more intuitive picture onthe distortion produced by the force on the Couette profiles for the hydrodynamic


18/23


fields (p, ux, and T), the pressure tensor (Pxx, Pyy , Pzz , and Pxy), and the heatflux (qx and qy) is provided by Fig. 4.

Complementarily, we have obtained the quantities of interest [cf. Eqs. (109)(117)] when the shear rate and the force are treated on the same footing, both tosecond order. This has allowed us to analyze [see Fig. 5(a)] how, by starting fromthe pure Poiseuille flow, the symmetric bimodal temperature profile is stronglydistorted by the shearing until arriving at the symmetric parabola characteristic ofthe pure Couette flow.

Considering the great current interest in the force-driven Poiseuille flow as aplayground to test hydrodynamic theories and theoretical approaches, we expectthat the work presented here may contribute to motivate further studies, boththeoretical and computational, on the CouettePoiseuille flow.

Acknowledgments. We dedicate this paper to the fond memory of Carlo Cer-cignani. The work of A.S. has been supported by the Ministerio de Educacion y

Ciencia (Spain) through Grant No. FIS200760977 (partially financed by FEDERfunds) and by the Junta de Extremadura (Spain) through Grant No. GR10158.

Appendix A. Evaluation of (I)n1n2n3 and

(II)n1n2n3 . From Eqs. (71) and (74) one

gets

(I)n1n2n3 =

n1=0

k=

k +

n1!(a)

(n1 )!(s)

k

dvVx,0

n1vyn2+k+vz

n3M1(v),

(122)where 0 = 1 and = 0 for 1. In Eq.. (122) use has been made of themathematical relations

A(s)(s)k

B(s) =

k

=0

k(s)k sA(s)B(s) , (123)sV

x,0

n1 =n1!(a)

(n1 )!Vx,0

n1. (124)

Next, using the Maxwellian integralsdvVx,0

n1vyn2vz

n3M0(v) = Kn1Kn2Kn3T

0

(n1+n2+n32)/2, (125)

where Kn = (n 1)!! if n = even [with the convention (1)!! = 1], being zero ifn = odd, Eq. (122) becomes

(I)n1n2n3 = Kn3

n1

=0

k=k +

n1!(a)

(n1 )!Kk+n2+(s)

kT0(k+n1+n2+n32)/2

Kn1

p1 +

k + n1 + n2 + n3 2

2

T1T0

+ Kn1+1

ux,1T0

(126)

Before considering the integral (II)n1,n2,n3, it is convenient to make use of the

relation

(s)k [A(s)B(s)] =

km=0

k

m

[(s)

mA(s)]

(s)kmB(s)

(127)


19/23


to rewrite the function (II)(v) as

(II)(v) = (1 s2)(II,0)(v) + 2s(II,1)(v) 2(II,2)(v), (128)

where

(II,m)(v)

vx

k=0

k + m

m

vy

k+m(s)

kf0 (v)

=

vx

k=0

k + m + 1

m + 1

vy

k+m(s)kM0(v

). (129)

In the last step we have made use of Eq. (47) and the mathematical property

k=0

+ m

m

=

k + m + 1

m + 1

. (130)

Insertion of Eq. (128) into Eq. (75) gives

(II)n1n2n3 = (1 s2)(II,0)n1n2n3 + 2s

(II,1)n1n2n3 2

(II,2)n1n2n3 , (131)

where

(II,m)n1n2n3 =

dvVx,0

n1vyn2vz

n3(II,m)(v). (132)

Using again Eqs. (123)(125), one gets

(II,m)n1n2n3 = Kn3

n11=0

k=0

k + + m + 1

m + 1

k +

n1!(a)

(n1 1 )!Kn11

Kk+n2++m(s)kT0

(k+n1+n2+n3+m3)/2. (133)

Once the integrals (I)n1n2n3 and

(II)n1n2n3 are expressed in terms of T

0 (s), p

1(s),

ux,1(s), and T1 (s), we can apply the consistency conditions (76)(79) to get the

hydrodynamic profiles to first order in g. To that end, we first guess the polynomialforms (80)(82), so that only the coefficients p

(1)1 , u

(2)x,1, and T

(3)1 remain to be

determined. It is straightforward to check that (I)000 =

(II)000 = 0, and thus Eq. (76)

is identically satisfied. The remaining relevant quantities in Eqs. (77)(79) turn outto be

(I)100 = 2aT

(3)1

F1 F2

+ 2(u(2)x,1 + ap

(1)1 )F1, (134)

(I)010 = T

(3)1

F1 F0

p(1)1 F0, (135)

s1(I)200 = T

(3)1

10F2 F1

+

F0 1

22

a2 F1 2F2

F0 1

2

+2p

(1)

1 (2F2 F1) a2 (2F2 + F1) + 8u(2)x,1aF2, (136)s1

(I)020 = T

(3)1

F0 F1

p(1)1 (1 F0) , (137)

s1(I)002 = T

(3)1

F0 1

2+ F1 + 2F2

2p

(1)1 (F1 + 2F2) , (138)

(II,0)100 =

1

T0,

(II,1)100 = 0,

(II,2)100 =

1

3(F1 + 2F2) , (139)

(II,0)010 =

(II,1)010 =

(II,2)010 = 0, (140)


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(II,0)200 = 0,

(II,1)200 = 2a (F1 + 2F2) , (141)

(II,2)200 = 2asF1 + 2F2 + F0 1

6 , (142)

(II,0)020 =

(II,1)020 =

(II,2)020 = 0, (143)

(II,0)002 =

(II,1)002 =

(II,2)002 = 0. (144)

In the above equations use has been made of Eqs. (54)(56).Inserting Eqs. (135) and (140) into the consistency condition (78) one simply

gets Eq. (83). Next, insertion of Eqs. (134) and (139) into Eq. (77) allows one toobtain Eq. (84). Finally, use of Eqs. (136)(138) and (141)(144) in Eq. (79) yields

Eq. (85). Note that from Eqs. (83) and (137) one gets s1(I)020 = p

(1)1 .

Taking into account that T

0 , p

1, u

x,1, and T

1 are polynomials in s of degrees 2,1, 2, and 3, respectively, Eqs. (126) and (133) show that

(I)n1n2n3 and

(II,m)n1n2n3 are

polynomials of degrees n1 + n2 + n3 1 and n1 + n2 + n3 + m 3, respectively.Consequently, the moments defined by Eq. (73) are polynomials of degree n1 + n2 +n3 1.

Let us proceed now to the evaluation of the integrals (I)n1n2n3 and

(II)n1n2n3 related

to the pressure tensor and the heat flux. The integrals related to the diagonalelements of the pressure tensor are already given by Eqs. (136)(138) and (141)(144). For instance, Pyy,1 = p

1 + 020, with similar relations for P

xx,1 and P

zz,1.

The results are displayed in Eqs. (87)(90). In the case of the shear stress, one hasPxy,1 = 110. From Eqs. (126) and (133) we obtain

s1

(I)

110 = T

(3)

1 a

F0 3F1 + 2F2

2u

(2)

x,1F1 p

(1)

1 a(2F1 F0), (145)

(II,0)110 = 0,

(II,0)110 = F1,

(II,2)110 =

2

3(F1 F2)s. (146)

This gives

Pxy,1(a|s) =

a

F0 3F1 + 2F2

T(3)

1 2F1u(2)x,1 a(2F1 F0)p

(1)1

+2

3(F1 + 2F2)

s. (147)

Making use of Eqs. (83) and (84), it is easy to check that Eq. (147) reduces to Eq.

(91).In the case of the heat flux, one has

qx,1 =1

2(300 + 120 + 102)

Pxx,0 1

ux,1, (148)

qy,1 =1

2(210 + 030 + 012) P

xy,0u

x,1. (149)

After tedious algebra one obtains Eqs. (92)(96).


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Received xxxx 20xx; revised xxxx 20xx.

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