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Register a new userUnified hydrodynamic description for driven and undriven inelastic Maxwell mixtures at low density
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A hydrodynamic description for inelastic Maxwell mixtures driven by a stochastic bath with friction is derived. Contrary to previous works where constitutive relations for the fluxes were restricted to states near the homogeneous steady state, here the set of Boltzmann kinetic equations is solved by means of the Chapman--Enskog method by considering a more general time-dependent reference state. Due to this choice, the transport coefficients are given in terms of the solutions of a set of nonlinear differential equations which must be in general numerically solved. The solution to these equations gives the transport coefficients in terms of the parameters of the mixture (masses, diameters, concentration, and coefficients of restitution) and the time-dependent (scaled) parameter $xi^*$ which determines the influence of the thermostat on the system. The Navier--Stokes transport coefficients are exactly obtained in the special cases of undriven mixtures ($xi^*=0$) and driven mixtures under steady conditions ($xi^*=xi_text{st}^*$, where $xi_text{st}^*$ is the value of the reduced noise strength at the steady state). As a complement, the results for inelastic Maxwell models (IMM) in both undriven and driven steady states are compared against approximate results for inelastic hard spheres (IHS) [Khalil and Garzo, Phys. Rev. E textbf{88}, 052201 (2013)]. While the IMM predictions for the diffusion transport coefficients show an excellent agreement with those derived for IHS, significant quantitative differences are specially found in the case of the heat flux transport coefficients.
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The Boltzmann equation for inelastic Maxwell models is considered to determine the rheological properties in a granular binary mixture in the simple shear flow state. The transport coefficients (shear viscosity and viscometric functions) are {em exactly} evaluated in terms of the coefficients of restitution, the (reduced) shear rate and the parameters of the mixture (particle masses, diameters and concentration). The results show that in general, for a given value of the coefficients of restitution, the above transport properties decrease with increasing shear rate.
Many features of granular media can be modelled as a fluid of hard spheres with {em inelastic} collisions. Under rapid flow conditions, the macroscopic behavior of grains can be described through hydrodynamic equations. At low-density, a fundamental basis for the derivation of the hydrodynamic equations and explicit expressions for the transport coefficients appearing in them is provided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this chapter is to give an overview of the recent advances made for binary granular gases by using kinetic theory tools. Some of the results presented here cover aspects such as transport properties, energy nonequipartition, instabilities, segregation or mixing, non-Newtonian behavior, .... In addition, comparison of the analytical results with those obtained from Monte Carlo and molecular dynamics simulations is also carried out, showing the reliability of kinetic theory to describe granular flows even for strong dissipation.
Through an exact analysis, we show the existence of Mpemba effect in an anisotropically driven inelastic Maxwell gas, a simplified model for granular gases, in two dimensions. Mpemba effect refers to the couterintuitive phenomenon of a hotter system relaxing to the steady state faster than a cooler system, when both are quenched to the same lower temperature. The Mpemba effect has been illustrated in earlier studies on isotropically driven granular gases, but its existence requires non-stationary initial states, limiting experimental realisation. In this paper, we demonstrate the existence of the Mpemba effect in anisotropically driven granular gases even when the initial states are non-equilibrium steady states. The precise conditions for the Mpemba effect, its inverse, and the stronger version, where the hotter system cools exponentially faster are derived.
The nature of the velocity distribution of a driven granular gas, though well studied, is unknown as to whether it is universal or not, and if universal what it is. We determine the tails of the steady state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular gas where the rate of collision between particles is independent of the separation as well as the relative velocity. We show that the steady state velocity distribution is non-universal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution are shown to be identical to that of a non-interacting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modelled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.
The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to analyze transport properties in spatially inhomogeneous states close to the simple shear flow. A normal solution is obtained via a Chapman--Enskog--like expansion around a local shear flow distribution f^{(0)} that retains all the hydrodynamic orders in the shear rate. The constitutive equations for the heat and momentum fluxes are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the reference state and the corresponding generalized transport coefficients are {em exactly} determined in terms of the coefficient of restitution alpha and the shear rate a. Since f^{(0)} applies for arbitrary values of the shear rate and is not restricted to weak dissipation, the transport coefficients turn out to be nonlinear functions of both parameters a and alpha. A comparison with previous results obtained for inelastic hard spheres from a kinetic model of the Boltzmann equation is also carried out.
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