Asymptotic solution of the Boltzmann equation for the shear flow of smooth inelastic disks

The velocity distribution for a homogeneous shear flow of smooth nearly elastic disks is determined using a perturbation solution of the linearised Boltzmann equation. An expansion in the parameter $\varepsilon_I =(1 − e)^{1/2}$ is used, where e is the coecient of restitution. In the leading order approximation, inelastic effects are neglected and the distribution function is a Maxwell-Boltzmann distribution. The corrections to the distribution function due to inelasticity are determined using an expansion in the eigenfunctions of the linearised Boltzmann operator, which form a complete and orthogonal basis set. A normal form reduction is effected to obtain first-order differential equations for the coeffcients of the eigenfunctions, and these are solved analytically subject to a set of simple model boundary conditions. The O$(\varepsilon_I)$ and O$(\varepsilon^2_I)$ corrections to the distribution function are calculated for both infinite and bounded shear flows. For a homogeneous shear flow,the results for the O$(\varepsilon_I)$ and O$(\varepsilon^2_I)$ corrections to the distribution function are different from those obtained earlier by the moment expansion method and the Chapman-Enskog procedure, but the numerical value of the corrections are small for the second moments of the velocity distribution,and the numerical results obtained by the different procedures are very close to each other. The variation in the distribution function due to the presence of a solid boundary is analysed, and it is shown that there is an O$(\varepsilon^2_I)$ correction to the density and an O$(\varepsilon_I)$ correction to the mean velocity due to the presence of a wall