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

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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 Uₓ(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(r, v)=f(V), with V≡ v- U(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(V;a) | V|⁻⁴⁻σ⁽ᵃ⁾Φ(ϕ; a), where ϕ≡ Vy/Vₓ 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.