## On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow.(English)Zbl 1159.82316

Summary: 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 a collision rate $$K(\theta)\propto\lim_{\varepsilon\to0} \varepsilon^{-2}\delta(\theta-\varepsilon)$$, where $$\theta$$ 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 $$a_{\text{th}}\simeq 0.3520\nu$$ (where $$\nu$$ is an average collision frequency and $$a_{\text{th}}/\nu$$ 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 $$f({\mathbf V};a)\sim|{\mathbf V}|^{-4-\sigma(a)} \Phi(\varphi;a)$$, 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. It diverges when $$a\to a_{\text{th}}$$ and tends to a minimum value $$\sigma_{\min}\simeq 1.252$$ in the limit $$a\to\infty$$. As a consequence of this power-law decay for $$a>a_{\text{th}}$$, 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 $$\widetilde{\varphi}(a)$$, which rotates from $$\widetilde{\varphi}= -\pi/4,3\pi/4$$ when $$a\to a_{\text{th}}$$ to $$\widetilde{\varphi}=0$$, $$\pi$$ in the limit $$a\to\infty$$.

### MSC:

 82C40 Kinetic theory of gases in time-dependent statistical mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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