Uniform shear flow in dissipative gases: Computer simulations of inelastic hard spheres and frictional elastic hard spheres

Abstract

In the preceding paper (cond-mat/0405252), we have conjectured that the main transport properties of a dilute gas of inelastic hard spheres (IHS) can be satisfactorily captured by an equivalent gas of elastic hard spheres (EHS), provided that the latter are under the action of an effective drag force and their collision rate is reduced by a factor (1 + α)/2 (where α is the constant coefficient of normal restitution). In this paper we test the above expectation in a paradigmatic nonequilibrium state, namely the simple or uniform shear flow, by performing Monte Carlo computer simulations of the Boltzmann equation for both classes of dissipative gases with a dissipation range 0.5 ≤ α ≤ 0.95 and two values of the imposed shear rate a. It is observed that the evolution toward the steady state proceeds in two stages: a short kinetic stage (strongly dependent on the initial preparation of the system) followed by a slower hydrodynamic regime that becomes increasingly less dependent on the initial state. Once conveniently scaled, the intrinsic quantities in the hydrodynamic regime depend on time, at a given value of α, only through the reduced shear rate a * (t) ∝ a/ T (t), until a steady state, independent of the imposed shear rate and of the initial preparation, is reached. The distortion of the steady-state velocity distribution from the local equilibrium state is measured by the shear stress, the normal stress differences, the cooling rate, the fourth and sixth cumulants, and the shape of the distribution itself. In particular, the simulation results seem to be consistent with an exponential overpopulation of the high-velocity tail. These properties are common to both the IHS and EHS systems. In addition, the EHS results are in general hardly distinguishable from the IHS ones if α 0.7, so that the distinct signature of the IHS gas (higher anisotropy and overpopulation) only manifests itself at relatively high dissipations.

A rescaling velocity method for dissipative kinetic equations. Applications to granular media

“…We also notice the stability of the scheme with respect of the space grid. The short time behavior of this quantity is close to the one presented in [3] using DSMC simulations on the more realistic 2d x × 2d v geometry. Indeed, we can see that the thermalized walls heat the system at the boundary and the collisions cool it inside the domain.…”

“…When time evolves, the temperature will eventually become uniform inside the domain with bigger and bigger gradients near the boundary. This large time behavior is different of the one described in [3] where the temperature reaches a space uniform steady state. This might be due to the use of a simplified geometry.…”

“…Test 3: Inelastic Shear Flow Problem. The uniform shear flow problem is perhaps one of the most widely studied nonequilibrum problem for granular gases, both experimentally and theoretically [3]. It can be described as follows: consider a gas enclosed between two infinite planes located at y = −1/2 and y = 1/2, in opposite relative motion with respective velocities −1/2 and 1/2 in the x-direction.…”

“…This problem has been solved numerically using Direct Simulation Monte Carlo methods in [3,43]. We shall consider it using the full deterministic inelastic Boltzmann equation.…”

“…Indeed, A. Astillero and A. Santos derived in [2,3] a very simple approximation of the granular gases equation, containing its main features. It consists in replacing the bilinear collision operator Q I by a nonlinear relaxation operator, supplemented with a drag force in the velocity space, depending on the moment of the particle distribution function itself, in order to match the correct macroscopic dissipation of kinetic energy of the granular gases equation.…”