Asymptotic solution of the Boltzmann equation for the shear flow of smoothinelastic disks

The velocity distribution for a homogeneous shear flow of smooth nearly ela stic disks is determined using a perturbation solution of the linearised Bo ltzmann equation. An expansion in the parameter epsilon(I) =(1 - e)(1/2) is used, where e is the coefficient of restitution. In the leading order appr oximation, inelastic effects are neglected and the distribution function is a Maxwell-Boltzmann distribution. The corrections to the distribution func tion due to inelasticity are determined using an expansion in the eigenfunc tions of the linearised Boltzmann operator, which form a complete and ortho gonal basis set. A normal form reduction is effected to obtain first-order differential equations for the coefficients of the eigenfunctions, and thes e are solved analytically subject to a set of simple model boundary conditi ons. The O(epsilon(I)) and O(epsilon(I)(2)) corrections to the distribution function are calculated for both infinite and bounded shear flows. For a h omogeneous shear flow, the results for the O(epsilon(I)) and O(epsilon(I)(2 )) corrections to the distribution function are different from those obtain ed earlier by the moment expansion method and the Chapman-Enskog procedure, but the numerical value of the corrections are small for the second moment s of the velocity distribution, and the numerical results obtained by the d ifferent procedures are very close to each other. The variation in the dist ribution function due to the presence of a solid boundary is analysed, and it is shown that there is an O(epsilon(I)(2)) correction to the density and an O(epsilon(I)) correction to the mean velocity due to the presence of a wall. (C) 2000 Elsevier Science B.V. Ail rights reserved.