Thermal Segregation Beyond Navier-Stokes

A dilute suspension of impurities in a low density gas is described by the Boltzmann and Boltzman-Lorentz kinetic theory. Scaling forms for the species distribution functions allow an exact determination of the hydrodynamic fields, without restriction to small thermal gradients or Navier-Stokes hydrodynamics. The thermal diffusion factor characterizing sedimentation is identified in terms of collision integrals as functions of the mechanical properties of the particles and the temperature gradient. An evaluation of the collision integrals using Sonine polynomial approximations is discussed. Conditions for segregation both along and opposite the temperature gradient are found, in contrast to the Navier-Stokes description for which no segregation occurs.


I. INTRODUCTION
Consider a granular mixture of two mechanically different species in a steady state with number densities n 0 (r) and n(r), respectively. One component is dilute with respect to the other, n 0 (r)/n(r) << 1, such that this component has negligible effect on the host gas.
Moreover, the latter is at sufficiently low density that the granular Boltzmann kinetic theory applies for its intra-species collisions. The dilute component has negligible intra-species collisions and its collisions with the host gas are described by the granular Boltzmann-Lorentz kinetic theory [1]. The objective here is to provide an exact description of segregation induced by a temperature gradient in this context. The motivation is the description some years ago of an exact solution to the Boltzmann equation for a steady state with constant temperature gradient [2,3]. That analysis is extended here to include the presence of the The particles of the dilute component will be referred to as the "impurities". The hydrodynamic fields obtained for the host gas are zero flow velocity, constant temperature gradient in the x direction, dT (x)/dx = θ , and a constant uniform pressure p = n(x)T (x). The impurities have a temperature profile T 0 (x) proportional to the host temperature T 0 (x) = γT (x), and a non-trivial density n 0 (x) expressed in terms of the host temperature field. In the dilute limit, the concentrations are ρ 0 (x) ≃ n 0 (x)/n(x) and ρ(x) = 1 − ρ 0 (x). They have the relationship dρ 0 /dx = −dρ/dx so any spatial variation of ρ 0 (x) implies the opposite variation of ρ(x) and segregation occurs. Here the segregation is induced by the temperature gradient, and it is common to introduce a thermal diffusion factor Λ defined by This dimensionless factor depends on the properties of the two components, Λ = Λ(α, α 0 , σ/σ 0 , m/m 0 , θ * ), where α, α 0 are the restitution coefficients for the host-host and impurity-host collisions, σ, σ 0 and m, m 0 are the species diameters and masses, and θ * = θ/pσ d−1 is the dimensionless temperature gradient, d being the geometrical dimension of the system. In principle, Λ can be positive or negative within this parameter space. The case Λ = 0 implies no segregation, while Λ positive (negative) implies the impurities increase concentration against (along) the temperature gradient. This is the thermal analogue of the Brazil nut and reverse Brazil nut effects for gravitational segregation [5][6][7][8].
The distribution functions for the two species are of a "normal" form, meaning that their dependence on space and time occurs entirely through the hydrodynamic fields, n(x), T (x), and n 0 (x) [9,10]. Thus, boundary conditions do not occur explicitly but only through the determination of these fields. For example, no external driving source is required in the kinetic equation for a stationary state, since this is implicit in the time independence of the fields. Instead, the stationary form of the fields is determined self-consistently from moments of the kinetic equations. This self-consistency also determines the temperature of the impurities as being proportioal to the host temperature, T 0 (x) = γT (x), with γ = 1 in general. No reference to hydrodynamics is made, although these moment equations are equivalent to the balance equations forming the basis for a hydrodynamical description.
The steady state obtained occurs by establishing a gradient of the heat flux to compensate for local energy loss due to collisional cooling. Thus it is special to granular fluids and links the temperature gradient to the degree of inelasticity rather than to boundary conditions. This is similar to steady uniform shear flow where the steady state is possible due to a balance of viscous heating and collisional cooling, such that the velocity gradient (shear rate) is linked to the degree of inelasticity. In both cases, the control needed to assure Navier-Stokes hydrodynamics is lost. In the present case, smaller gradients entails smaller pressure at constant restitution coefficient, or smaller inelasticity at constant pressure. Such non-Newtonian steady states are a characteristic of granular flows and segregation for such states can be qualitatively different from that from Navier-Stokes hydrodynamics. This has been illustrated recently for thermal segregation under uniform shear flow [11].
The next section defines the system and its kinetic theory description. In section III scaling forms for the distribution functions are introduced and the implications for the hydrodynamic fields are obtained. Three constants must be determined self-consistently.
One of these, the temperature gradient θ has been obtained in [2,3]. Collision integrals for the other two are obtained here. The form of the thermal diffusion factor Λ is given in terms of these constants, and the sign of Λ is discussed based on approximate evaluations of the collision integrals given in the Appendices.
where the collision operator Here g ≡ v −v 1 is the relative velocity of the colliding pair, Θ is the Heaviside step function, d σ is the solid angle element about the direction of the unit vector σ, and α is the restitution Now consider M additional impurity particles in this gas, all the same but mechanically different from the fluid particles. For M ≪ N, the primary collisions for the impurity particles are with the host gas particles, and impurity-impurity collisions and effects of the impurities on the gas distribution function f can be neglected. The distribution function for the impurities, F (r, v 0 , t), is governed by the corresponding Boltzmann-Lorentz equation, where the operator I [v 0 |F, f ] describes changes in F due to binary collisions between the impurity and gas particles, In the above expressions, σ ≡ (σ + σ 0 )/2, and σ 0 , m 0 , and α 0 are the hard sphere diameter, mass, and restitution coefficient for the impurity particles, respectively.
The macroscopic state of this system is described by the fluid number density n(r, t), temperature T (r, t), and flow velocity u(r, t), defined in terms of the distribution function It is convenient to introduce corresponding fields for a macroscopic description of the impurity particles, Instead of an impurity velocity, the more usual number flux notation j 0 = n 0 u 0 has been used.

III. SCALING SOLUTIONS
In reference [2], a solution to the Boltzmann equation was described for the special case of a scaling form in terms of the hydrodynamic variables, Such a solution, where the space and time dependence of the distribution function occurs only through the hydrodynamic fields, is called "normal". The definitions of the fields in (8), and the choice of u = 0 give the self-consistency conditions on φ (c, ) Here, a similar scaling solution for the impurities is sought, The definitions (9) then give the conditions on Φ, In order for (12) to be "normal", it should depend only on the hydrodynamic fields for the gas and impurities, i.e., on n 0 (x), n(x), and T (x). Dimensional analysis then requires that The constant γ must be determined in course of solving the kinetic equation (as discussed below). Further comments on the implications of normal solutions is provided in the last section.
In terms of these scaling solutions and dimensionless velocity variables, the Boltzmann and Boltzmann-Lorentz equations become with the dimensionless collision operators The relative velocities w and w 0 are now The expressions of the dimensionless restituting velocities in Eq. (18) are given in Eq. (A3).
Since the right sides of Eqs. (15) and (16) are independent of x, the left sides must be as well. This will be true if the hydrodynamic fields n(x), n 0 (x), and T (x) satisfy the equations where A, B, and C are constants. The constants A and B are determined by taking moments of the Boltzmann equation (15). Namely, multiplication of the equation by 1, c x , and c 2 , and integration over c yields The zeroes on the right sides of (21) result from conservation of particle number and momentum by the collision operator. The first equation of (21) is satisfied because of conditions The exact hydrodynamic fields for the gas are now given exactly by u = 0 and where p = n(x)T (x) is the uniform pressure, and θ ≡ Bpσ d−1 is the constant temperature gradient.
A similar analysis applies for the impurity constants C and γ. Taking moments of the Boltzmann-Lorentz equation (16) with respect to 1, c 0x , and c 2 0 gives The right hand sides of Eqs. (26) and (27)  In summary, the description of the gas and impurities is completely specified by the kinetic equations for φ (c) and Φ (c 0 ),

IV. SEGREGATION
The segregation of impurity particles relative to the host gas is described by the inhomogeneity of the composition ρ 0 (x) ≃ n 0 (x)/n(x), which follows from (30) and (31) The thermal diffusion factor of (1) is therefore Thus thermal segregation can occur, facilitated by gravity, and depends on the sign of (T 0 /T − m 0 /m) and the direction of ∂T /∂x relative to the gravitational force. This is in sharp contrast to the results obtained in the next section.

V. APPROXIMATE DETERMINATION OF T 0 /T AND Λ
To determine the coefficients B, C, and γ = T 0 /T , the distribution functions φ and Φ are represented as truncated Sonine polynomial expansions The method for determining the coefficients in these expansions is described in [2] and

VI. DISCUSSION
The description of a low density granular gas with a dilute concentration of impurities has been given in terms of solutions to the coupled Boltzmann and Boltzmann-Lorentz kinetic equations. These are normal solutions whose space and time dependence are entirely specified in terms of the hydrodynamic fields n, n 0 ,and T . The special case of a steady state in which the host gas has a constant temperature gradient and constant pressure, described earlier in refs. [2] and [3], has been generalized to include a corresponding steady state of the impurities. In this way the thermal segregation factor is identified in terms of the constants of the hydrodynamic fields, without the limiting approximations of small spatial gradients. The self-consistent kinetic equations (28) and (29) determining these constants was solved using a low order Sonine polynomial approximation for the velocity dependence application typically entails limitations to small spatial gradients, e.g. Navier-Stokes order.
Application of Navier-Stokes hydrodynamics obtained in this way, and specialized to the steady state with constant temperature gradient and constant pressure, leads to the prediction of no segregation. The effects described here therefore are due to contributions from the Chapman-Enskog method beyond the small gradient approximation. In fact, there are no limitations on the temperature gradient in the present analysis.
There are two important clarifications to note. First, the validity of a normal solution both for granular and molecular gases is limited to domains away from the initial preparation time and confining boundaries. For the steady state considered here, this means that there is typically a boundary layer across which the normal solution does not apply. Additional information is then required to connect the physically specified values of the fields or their gradients at the boundary with those values associated with the normal solution. These are the familiar "slip" boundary conditions. The existence of the normal solution described here for a system with finite confinement and associated boundary layer has been demonstrated by molecular dynamics simulation in refs. [2] and [3]. Typically, the size of the bulk interior relative to the boundary layer decreases as the temperature gradient is increased. Investigation of this problem for a molecular gas has demonstrated that the bulk normal solution domain still exists beyond the Navier-Stokes limit [15].
A second clarification is the special nature of the steady state described here as being unique to a granular gas. The analysis of [2] shows that it results from the balance of the heat flux gradient and the cooling rate due to inelastic collisions. In the absence of the latter there is no steady state solution of the type considered here. In contrast to normal fluids, the gradients of such steady states are controlled by internal processes rather than boundary sources. External control of the gradients is therefore lost. In the present case the magnitude of the dimensionless temperature gradient θ/pσ d−1 = B (α) monotonically decreases to zero as α → 1, vanishing in the elastic limit. Consequently, for example, it is not possible for the Navier-Stokes to apply here for strong dissipation.

VII. ACKNOWLEDGMENTS
The research of JJB and NK has been partially supported by the Ministerio de Educación y Ciencia (Spain) through Grant No. FIS2008-01339 (partially financed by FEDER funds).

Appendix A: Reduction of collision integrals
The Boltzmann collision integral appearing on the right side of Eq. (23) is simplified further in [2], with the result The Boltzmann-Lorentz collision integrals can be simplified in a similar way. Consider first the collision integral appearing in Eq. (26), where w 0 is defined in Eq. (19) and the dimensionless restituting velocities following from Eq. (7) are It is easily verified that Also, Eqs. (A3) can be inverted to get the collision rule in dimensionless units, Returning to Eq. (A2), change variables in the first term of the brackets on the right hand side to integrate over the restituting velocities. Using the above relations, the equation Finally (26) becomes The analysis of Eq. (27) is similar with the result the equation is multiplied by c 2 x , c 3 x , and c x c 2 , respectively, and afterwards integrated over c. With these coefficients determined, B is calculated from Eq. (23).
To determine Φ (c 0 ), C, and γ, a similar procedure is followed. First, express C as a collision integral from Eqs. (26) and (27), and use this in the kinetic equation (29). Next, express Φ (c 0 ) as a truncated Sonine polynomial expansion which satisfies the conditions (13) with j 0 = 0. The coefficients, A 01 , B 01 , and B 10 are determined from three equations obtained by taking moments of (29) with respect to c 2 0x , c 3 0x , and c 0x c 2 0 . However, these equations also depend on γ, so they are supplemented by an additional equation relating the above coefficients to γ. It is obtained from a new combination of Eqs. .

(B5)
Since φ and B are known at this point, this gives four independent equations for the coefficients A 01 , B 01 , B 10 , and γ. With these determined, C is calculated from Eq. (B3).
In practice, the above procedure leads to highly nonlinear equations for the coefficients.
In the numerical results to be presented in the following, only terms up to second degree in the coefficients have been kept [3]. As an example, in Figs. 7-9, the parameters obtained for a two-dimensional system (d = 2) with m = m 0 and σ = σ 0 = σ are plotted as a