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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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