On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow
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 $U_x(y)=a y$, 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 $f({\bf r},{\bf v})=f({\bf V})$, with ${\bf V}\equiv {\bf v}-{\bf U}({\bf r})$, 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 $f({\bf V};a)\sim |{\bf V}|^{-4-σ(a)}Φ(ϕ; a)$, where $ϕ\equiv \tan V_y/V_x$ and the angular distribution function $Φ(ϕ; a)$ is the solution of a modified Mathieu equation. The enforcement of the periodicity condition $Φ(ϕ; a)=Φ(ϕ+π; a)$ allows one to obtain the exponent $σ(a)$ 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 $2+σ(a)$ 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.