Stability analysis of the homogeneous hydrodynamics of a model for a confined granular gas

Abstract

The linear hydrodynamic stability of a model for confined quasi-two-dimensional granular gases is analyzed. The system exhibits homogeneous hydrodynamics, i.e., there are macroscopic evolution equations for homogeneous states. The stability analysis is carried out around all these states and not only the homogeneous steady state reached eventually by the system. It is shown that in some cases the linear analysis is not enough to reach a definite conclusion on the stability, and molecular dynamics simulation results are presented to elucidate these cases. The analysis shows the relevance of nonlinear hydrodynamic contributions to describe the behavior of spontaneous fluctuations occurring in the system, that lead even to the transitory formation of clusters of particles. The conclusion is that the system is always stable. The relevance of the results for describing the instabilities of confined granular gases observed experimentally is discussed.

Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres

“…An expression incorporating some effects of the non-Gaussianity of the velocity distribution can be found in [14], and in a more compact form in the Appendix of Reference [16]. Equation (4) predicts the existence of a homogeneous steady state with a temperature…”

“…All these coefficients depend on the temperature, and also on the coefficient of normal restitution α and the characteristic speed ∆. Their expressions can be found in [19], and in a more concise way in the Appendix of [16]. They are given by the normal solutions-in the kinetic theory sense-of first-order partial differential equations.…”

“…This implies that both the eigenvalues and eigenfunctions of the dimensionless problem are time dependent, and the linear stability analysis is more involved than for smooth inelastic hard disks in the homogeneous cooling state [18]. An exception is the equation for the transversal component ω k⊥ of the scaled velocity field (shear mode), ω ≡ u/v 0H (t), that is decoupled from the rest of equations and can be integrated directly, yielding [16] …”

“…The behavior of the other hydrodynamic fields cannot be analyzed analytically, but the linearized equations can be solved numerically. When the initial temperature of the system is smaller than its stationary value such that the temperature of the system increases monotonically in time, it is found [16] that the fields oscillate in time with a decaying amplitude (i.e., the system is linearly stable). On the other hand, when the temperature of the system is larger than the steady value, and the initial value of k ⊥ is larger than the wavenumber of the perturbation, the transversal component of the scaled velocity field initially grows in time, as predicted by Equation (16).…”

“…The spontaneous fluctuations occurring in the system were analyzed. The physical picture emerging from the analysis is [16] that the fluctuations of the traversal component of the velocity field with an amplitude that increases in time actually happen in the system. Their effect can manifest itself in the formation of clusters of particles.…”