Binary granular gas mixtures: Theory, layering effects and some open questions

Abstract

A theory of dilute binary granular gas mixtures comprising smooth spheres, which is formally valid for all physical values of the pertinent coefficients of restitution, is presented. Constitutive relations are obtained using the Chapman Enskog procedure, in the framework of which relatively high orders in the Sonine polynomial expansion are employed. The latter is made possible by a computer-aided method. The transport coefficients are shown to converge as a function of the number of Sonine polynomials employed, typically requiring 3 polynomials, but in some cases up to 6 polynomials are required for convergence to within 1%. A comparison with results obtained using the Grad method helps reveal the limitations of the Chapman-Enskog expansion. In particular, in some cases both the Chapman Enskog expansion and the Grad method give unphysical results, though they agree with each other. These issues are related to the lack of scale separation in granular gases. Using the constitutive relations we obtain a novel segregation pattern in vertically vibrated granular mixtures, which comprises up to five layers; this is an extension of a previously obtained three-layer configuration. Finally, the question of effective hydrodynamic boundary conditions at the transition to the Knudsen regime is discussed: in particular, it seems that in some cases the boundary condition for the heat flux in the hydrodynamic regime is “unphysical”.