The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar–Gross–Krook model kinetic equation is analytically solved for this Couette–Poiseuille flow to first order 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 external force on the non-Newtonian properties of the Couette flow. Moreover, the Couette–Poiseuille flow is analyzed when the shear-rate Knudsen number and the 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 characteristic of the pure Couette flow through several intermediate stages in the Couette–Poiseuille flow are described. A critical comparison with the Navier–Stokes solution of the problem is carried out.
Two paradigmatic stationary nonequilibrium flows are the plane Couette flow and the Poiseuille flow.
In the plane Couette flow the fluid (henceforth assumed to be a dilute gas) is enclosed between two infinite parallel plates in relative motion, as sketched in Fig. 1(a).
The walls can be kept at different or equal temperatures but, even if both wall temperatures are the same, viscous heating induces a temperature gradient in the steady state. If the Knudsen number associated with the shear rate is small enough the Navier–Stokes (NS) equations provide a satisfactory description of the Couette flow. On the other hand, as shearing increases, non-Newtonian effects (shear thinning and viscometric properties) and deviations of Fourier’s law (generalized thermal conductivity and streamwise heat flux component) become clearly apparent . These nonlinear effects have been derived 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 generalized 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 analyzed in the context of granular gases [60, 65]. In the case of plates at rest but kept at different temperatures, the Couette flow becomes the familiar plane Fourier flow, which also presents interesting properties by itself [4, 16, 17, 24, 29, 33, 40, 41, 42].
The Poiseuille flow, where a gas is enclosed in a channel or slab and fluid motion is induced by a longitudinal pressure gradient, is a classical problem 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=mgˆx (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, 44, 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  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, all the studies on the Couette–Poiseuille flow assume that the 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 action of a uniform body force. This Couette–Poiseuille flow is sketched in 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 the shear rate and perform a perturbation expansion to first order in the force. As a second objective, we will study how the non-Newtonian force-driven Poiseuille flow is modified by the shearing. This is done by assuming that the shear-rate Knudsen 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.
The organization of the paper is as follows. The Boltzmann equation for the Couette–Poiseuille flow is presented in Sec. 2. Section 3 deals with the NS description of the problem. The main part of the paper is contained in Sec. 4, where the kinetic theory approach is worked out. Some technical calculations are relegated to 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 Tw. The imposed shear rate is therefore ω=(U+−U−)/L. Besides, an external body force F=mgˆx, 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 external force (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≠0 and g≠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
where f is the one-particle velocity distribution function and J[f,f] is the Boltzmann 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 its spatial and velocity dependencies, the distribution function depends on the independent 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 a particle
with velocity v′ hits the wall at y=L/2, the probability of being
reemitted with a velocity v within the range dv is
K+(v,v′)dv; the kernel K−(v,v′)
represents the same but at y=−L/2.
The boundary conditions are then 
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
where φ∓(v) represents the probability distribution of a
fictitious gas in contact with the system at y=∓L/2. Equation
(3) can then be interpreted as meaning that when a particle hits a
wall, it is instantaneously absorbed and replaced by a particle leaving the fictitious
bath. Of course, any choice of φ∓(v) must be consistent with
the imposed wall velocities and temperatures, i.e.
In these equations n is the number density, u is the flow velocity,
is the peculiar velocity, T is the temperature, kB is the Boltzmann constant, p is the hydrostatic pressure, Pij is the pressure tensor, and q is the heat flux.
Taking velocity moments in both sides of Eq. (1) one gets the following exact balance equations
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 Couette–Poiseuille flow imply the following invariance properties of the velocity distribution function:
As a consequence, if χ(y|ω,g) denotes a hydrodynamic variable or a flux, one has
where Sg=±1 and Sω=±1. The parity factors Sg and Sω for the non-zero hydrodynamic fields and fluxes are displayed in Table 1. In general, if χ is a moment of the form
then Sg=(−1)kx+ky and Sω=(−1)ky.
Table 1: Parity factors Sg and Sω for the hydrodynamic fields and the fluxes [see Eq. (19)].
The general solution to the stationary Boltzmann equation (1) with the boundary conditions (6) can be split into two parts [45, 46, 47]:
Here, fH represents the hydrodynamic, Hilbert-class, or normal contribution to the distribution function. This means that fH depends on space only through a functional dependence on the hydrodynamic fields, i.e.
The contribution fB represents the boundary-layer correction to fH, so that f=fH+fB verifies the specified boundary conditions. The correction fB is appreciably different from zero only in a thin layer (the so-called boundary layer or Knudsen layer), adjacent to the plates, of thickness of the order of the mean free path.
Consequently, if the separation L between the plates is much larger than the characteristic mean free path, there exists a well defined bulk region where the boundary correction vanishes and the distribution function is fully given by 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. ux(y=±L/2)≠U±. Besides,
the extrapolation of the velocity profile in the bulk to the boundaries, ux,H(y=±L/2), is also different from both the actual values ux(y=±L/2) and the wall velocities U±. Of course, an analogous temperature jump effect takes place with the temperature profile.
The boundary contribution fB for small Knudsen numbers has been analyzed elsewhere [8, 47].
In the remaining of this paper we will focus on the hydrodynamic part fH (and will drop the subscript H), with special emphasis on the corresponding hydrodynamic contributions to the momentum and heat fluxes.
In order to nondimensionalize the problem, we choose quantities evaluated at the central plane y=0 as units:
In the above equations we have found it convenient to introduce the dimensionless scaled spatial variable
where ν(y) is an effective collision frequency. For the sake of concreteness, we choose it as
where η is the NS shear viscosity.
The change from the boundary-imposed shear rate ω to the reduced local shear ratea 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.
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
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 conservation equations (15)–(17) one gets
where Pr=(5kB/2m)η/κ≃23 is the Prandtl number. In dimensionless form, Eqs. (36) and (37) can be rewritten as
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:
Here we have applied the Galilean choice ux(0)=0 and the symmetry property ∂yT∣∣y=0=0.
Equation (40) shows that, according to the NS approximation, the velocity field in the Couette–Poiseuille flow is simply the superposition of the (quasi) linear Couette 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 the scaled variable s. To go back to the real spatial coordinate y one needs to make 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
For further use, note that, according to Eq. (41),
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:
4 Kinetic theory description. Perturbation solution.
Now we want to get the hydrodynamic and flux profiles in the bulk domain of the system from a purely kinetic approach, i.e., without assuming a priori the applicability of the NS constitutive 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
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∗ allows the iterative solution of Eq. (51) by a scheme similar to that followed in Ref.  in the case of an external force normal to the plates.
These are just the equations corresponding to the pure Couette flow. The complete solution has been obtained elsewhere [6, 20, 25] and so here we only quote the final results. The hydrodynamic profiles are
where the dimensionless parameter γ(a) is a
nonlinear function of the reduced shear rate a given implicitly
through the equation [6, 20]
where the mathematical functions Fr(x) are defined by
K0(x) being the zeroth-order modified Bessel function.
Equation (62) clearly shows that Fr(x) has an essential singularity at x=0 and thus its expansion in powers of x,
is asymptotic and not convergent. However, the series representation (63) is Borel summable [6, 25], the corresponding integral representation being given by Eq. (62).
The functions Fr(x) with r≥3 can be easily
expressed in terms of F0(x), F1(x), and F2(x) as
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)≈15a2 (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 [6, 19, 20, 30]
Notice that, although the temperature gradient is only directed
along the y axis (so that there is a response in this direction
through q∗y), the shear flow induces a nonzero x component of the heat
flux [19, 20, 30, 37]. Furthermore, normal stress differences (absent at NS order) are present. Equations (69) and (71) can be used to identify generalized nonlinear shear viscosity and thermal conductivity coefficients.
In general, the velocity moments of degree k of f∗0 are polynomial functions of the spatial variable s of degree k−2. An explicit expression for the velocity distribution function f∗0 has also been derived [20, 25].
Limit of small shear rates
The coefficient γ(a) characterizing the profile of the zeroth-order temperature T∗0 is a complicated nonlinear function of the reduced shear rate a, as clearly apparent from Eq. (61). Obviously, the zeroth-order pressure 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
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∗xx−P∗yy and P∗zz−P∗yy) and the streamwise heat flux component q∗x reveal non-Newtonian effects of orders a2 and a3, respectively.
and we have already specialized to Maxwell molecules, so that ν∗=p∗/T∗.
In order to apply the consistency conditions (52) in the derivation of the hydrodynamic fields p∗1, u∗x,1, and T∗1, it is convenient to define the moments
Therefore, the consistency conditions are
The evaluation of Φ(I)n1n2n3 and Φ(II)n1n2n3 is carried out in Appendix A. The first-order profiles are
In the above equations the functions Fr are understood to be evaluated at x=γ.
As shown in Appendix A, the moment Φn1n2n3 is a polynomial function of s of degree n1+n2+n3−1. In particular, the non-zero elements of the first-order pressure tensor are