Longitudinal Viscous Flow in Granular Gases

The flow characterized by a linear longitudinal velocity field ux(x,t) = a(t)x, where a(t) = a0/(1+a0t), a uniform density n(t) ∝ a(t), and a uniform temperature T (t) is analyzed for dilute granular gases by means of a BGK-like model kinetic equation in d dimensions. For a given value of the coefficient of normal restitution α , the relevant control parameter of the problem is the reduced deformation rate a∗(t) = a(t)/ν(t) (which plays the role of the Knudsen number), where ν(t) ∝ n(t) √ T (t) is an effective collision frequency. The relevant response parameter is a nonlinear viscosity function η∗(a∗) defined from the difference between the normal stress Pxx(t) and the hydrostatic pressure p(t) = n(t)T (t). The main results of the paper are: (a) an exact first-order ordinary differential equation for η∗(a∗) is derived from the kinetic model; (b) a recursion relation for the coefficients of the Chapman–Enskog expansion of η∗(a∗) in powers of a∗ is obtained; (c) the Chapman–Enskog expansion is shown to diverge for elastic collisions (α = 1) and converge for inelastic collisions (α < 1), in the latter case with a radius of convergence that increases with inelasticity; (d) a simple approximate analytical solution for η∗(a∗), hardly distinguishable from the numerical solution of the differential equation, is constructed.