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29 March 2024
 
  » arxiv » cond-mat/0109490

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On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow
L. Acedo ; A. Santos ; A. V. Bobylev ;
Date 26 Sep 2001
Journal J. Stat. Phys. 109 (5/6), 1027-1050 (2002) DOI: 10.1023/A:1020424610273
Subject Statistical Mechanics | cond-mat.stat-mech
AbstractUniform 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 $f({f r},{f v})=f({f V})$, with ${f V}equiv {f v}-{f U}({f r})$, 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 $f({f V};a)sim |{f V}|^{-4-sigma(a)}Phi(phi; a)$, where $phiequiv an V_y/V_x$ and the angular distribution function $Phi(phi; a)$ is the solution of a modified Mathieu equation. The enforcement of the periodicity condition $Phi(phi; a)=Phi(phi+pi; a)$ allows one to obtain the exponent $sigma(a)$ 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 $2+sigma(a)$ are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle which rotates counterclock-wise as the shear rate increases.
Source arXiv, cond-mat/0109490
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