Simple shear flow in inelastic Maxwell models

Abstract

The Boltzmann equation for inelastic Maxwell models is considered to determine the velocity moments through fourth degree in the simple shear flow state. First, the rheological properties (which are related to the second-degree velocity moments) are {\em exactly} evaluated in terms of the coefficient of restitution $\alpha$ and the (reduced) shear rate $a^{\ast }$. For a given value of $\alpha$, the above transport properties decrease with increasing shear rate. Moreover, as expected, the third-degree and the asymmetric fourth-degree moments vanish in the long time limit when they are scaled with the thermal speed. On the other hand, as in the case of elastic collisions, our results show that, for a given value of $\alpha$, the scaled symmetric fourth-degree moments diverge in time for shear rates larger than a certain critical value $a_c^{\ast }(\alpha )$ which decreases with increasing dissipation. The explicit shear-rate dependence of the fourth-degree moments below this critical value is also obtained.

18 pages, 7 figures; v2: minor changes

References in corpus (9)

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Cited by in corpus (17)

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• Lattice models for granular-like velocity fields: Hydrodynamic limit
• Hydrodynamics of inelastic Maxwell models
• Rheology of dilute cohesive granular gases
• Non-equilibrium phase transition in a sheared granular mixture
• Impurity in a sheared inelastic Maxwell gas
• An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux
• Granular gas of inelastic and rough Maxwell particles
• Generalized transport coefficients for inelastic Maxwell mixtures under shear flow
• Simple shear flow in granular suspensions: Inelastic Maxwell models and BGK-type kinetic model
• Unsteady non-Newtonian hydrodynamics in granular gases
• Navier-Stokes transport coefficients for driven inelastic Maxwell models
• High-degree collisional moments of inelastic Maxwell mixtures. Application to the homogeneous cooling and uniform shear flow states
• Exact transport coefficients from the inelastic rough Maxwell model of a granular gas
• Tracer diffusion coefficients in a sheared inelastic Maxwell gas
• Inelastic Maxwell models for monodisperse gas-solid flows
• Exact Rheology of Uniform Shear Flow in a Gas of Inelastic and Rough Maxwell Particles