On the Derivation of a High-Velocity Tail from the Boltzmann–Fokker–Planck Equation for Shear Flow

摘要

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(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 a collision rate K(θ)∝lim ∈→0 ∈ −2 δ(θ−∈), where θ is the scattering angle, in which case 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 a th≃0.3520ν (where ν is an average collision frequency and a th/ν is the real root of the cubic equation 64x 3+16x 2+12x−9=0) 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≡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. It diverges when a→a th and tends to a minimum value σ min≃1.252 in the limit a→∞. As a consequence of this power-law decay for a>a th, 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 ~ϕ(a), which rotates from ~ϕ=−π/4,3π/4 when a→a th to ~ϕ=0,π in the limit a→∞.