Impurity in a granular gas under nonlinear Couette flow

Abstract

We study in this work the transport properties of an impurity immersed in a granular gas under stationary nonlinear Couette flow. The starting point is a kinetic model for low-density granular mixtures recently proposed by the authors [Vega Reyes F et al. 2007 Phys. Rev. E 75 061306]. Two routes have been considered. First, a hydrodynamic or normal solution is found by exploiting a formal mapping between the kinetic equations for the gas particles and for the impurity. We show that the transport properties of the impurity are characterized by the ratio between the temperatures of the impurity and gas particles and by five generalized transport coefficients: three related to the momentum flux (a nonlinear shear viscosity and two normal stress differences) and two related to the heat flux (a nonlinear thermal conductivity and a cross coefficient measuring a component of the heat flux orthogonal to the thermal gradient). Second, by means of a Monte Carlo simulation method we numerically solve the kinetic equations and show that our hydrodynamic solution is valid in the bulk of the fluid when realistic boundary conditions are used. Furthermore, the hydrodynamic solution applies to arbitrarily (inside the continuum regime) large values of the shear rate, of the inelasticity, and of the rest of parameters of the system. Preliminary simulation results of the true Boltzmann description show the reliability of the nonlinear hydrodynamic solution of the kinetic model. This shows again the validity of a hydrodynamic description for granular flows, even under extreme conditions, beyond the Navier-Stokes domain.

Rheological properties for inelastic Maxwell mixtures under shear flow

“…Note that the properties of uniform temperature and constant densities and shear rate are enforced in computer simulations by applying the Lees-Edwards boundary conditions [17], regardless of the particular interaction model considered. In the case of boundary conditions representing realistic plates in relative motion, the corresponding non-equilibrium state is the so-called Couette flow, where densities, temperature and shear rate are no longer uniform [18,19]. As said before, the rheological properties of the mixture are obtained from the pressure tensor P = P 1 + P 2 , where the partial pressure tensors P r (r = 1, 2) are defined by Eq.…”

An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux

“…Additionally, for a bounded system like ours (see Fig. 1), the particle velocities are updated during step (a) if they eventually touch the boundaries, which in our case are thermal walls moving with constant and opposite velocities (see more details on the description of these boundary conditions elsewhere [30]). The difference between DSMC for the Maxwell model and DSMC for hard spheres lies in the collision step.…”

“…In the case of Maxwell particles this time turns out to be much longer than for hard spheres since the third-order moments are coupled to the fourth-order moments of the USF, which have very long relaxation times [21]. Also, for better averaging the steady values of the hydrodynamic profiles and transport coefficients, we average twice: in time, by means of repeated measurements in the steady state at different uncorrelated instants; and in space, by averaging the small simulation cells to larger hydrodynamic cells, analogously to how we proceeded in a former work [30]. The rest of technical details are the same as in [30].…”

“…Also, for better averaging the steady values of the hydrodynamic profiles and transport coefficients, we average twice: in time, by means of repeated measurements in the steady state at different uncorrelated instants; and in space, by averaging the small simulation cells to larger hydrodynamic cells, analogously to how we proceeded in a former work [30]. The rest of technical details are the same as in [30]. Figure 3 shows the reduced shear rate a, the reduced shear viscosity η * , and the reduced normal stress coefficients θ x and θ y as functions of the coefficient of restitution α for a threedimensional system.…”

Thermal diffusion segregation of an impurity in a driven granular fluid

“…Collisions between impurity and gas particles are also inelastic and are characterized by another (constant) coefficient of restitution, α 0 ≤ 1. Given that the relative concentration of impurities is much smaller than that of gas particles (and so, impurityimpurity collision events are not statistically relevant compared to gas-impurity collisions), one can assume that the one-particle velocity distribution function of impurities verifies the (linear) Boltzmann-Lorentz kinetic equation [3]. In addition, the collisions between impurity and gas particles can be also neglected in the kinetic equation of granular gas and hence, its one-particle velocity distribution function obeys the nonlinear (closed) Boltzmann equation.…”