On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow
| dc.creator | Acedo, L. | |
| dc.creator | Santos, A. | |
| dc.creator | Bobylev, A. V. | |
| dc.date | 2001-09-26 | |
| dc.date | 2002-07-08 | |
| dc.date.accessioned | 2026-07-07T02:42:50Z | |
| dc.date.available | 2026-07-07T02:42:50Z | |
| dc.description | 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 $U_x(y)=a y$, 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 $f({\bf r},{\bf v})=f({\bf V})$, with ${\bf V}\equiv {\bf v}-{\bf U}({\bf r})$, 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 $f({\bf V};a)\sim |{\bf V}|^{-4-σ(a)}Φ(ϕ; a)$, where $ϕ\equiv \tan V_y/V_x$ 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. 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. | |
| dc.description | 15 pages, 5 figures; change in title plus other minor changes; to be published in J. Stat. Phys | |
| dc.identifier | https://arxiv.org/abs/cond-mat/0109490 | |
| dc.identifier | http://arxiv.org/abs/cond-mat/0109490 | |
| dc.identifier | J. Stat. Phys. 109 (5/6), 1027-1050 (2002) | |
| dc.identifier | doi:10.1023/A:1020424610273 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/18301 | |
| dc.subject | Statistical Mechanics | |
| dc.title | On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow | |
| dc.type | text |