On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow

Abstract

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile $U_x(y)=ay$, where $a$ is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function $f(\mathbf{r},\mathbf{v})=f(\mathbf{V})$, with $\mathbf{V}\equiv \mathbf{v}-\mathbf{U}(\mathbf{r})$, which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with grazing collisions in which the nonlinear Boltzmann collision operator reduces to a Fokker-Planck operator. We have found analytically that for shear rates larger than a certain threshold value the velocity distribution function exhibits an algebraic high-velocity tail of the form $f(\mathbf{V};a)\sim |\mathbf{V}{|}^{-4-\sigma (a)}\Phi (\varphi ;a)$, where $\varphi \equiv \tan V_y/V_x$ and the angular distribution function $\Phi (\varphi ;a)$ is the solution of a modified Mathieu equation. The enforcement of the periodicity condition $\Phi (\varphi ;a)=\Phi (\varphi +\pi ;a)$ allows one to obtain the exponent $\sigma (a)$ as a function of the shear rate. As a consequence of this power-law decay, all the velocity moments of a degree equal to or larger than $2+\sigma (a)$ are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle which rotates counterclock-wise as the shear rate increases.

15 pages, 5 figures; change in title plus other minor changes; to be published in J. Stat. Phys