{
  "id": "cond-mat/0109490",
  "version": "v2",
  "published": "2001-09-26T17:33:57.000Z",
  "updated": "2002-07-08T07:42:49.000Z",
  "title": "On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow",
  "authors": [
    "L. Acedo",
    "A. Santos",
    "A. V. Bobylev"
  ],
  "comment": "15 pages, 5 figures; change in title plus other minor changes; to be published in J. Stat. Phys",
  "journal": "J. Stat. Phys. 109 (5/6), 1027-1050 (2002)",
  "doi": "10.1023/A:1020424610273",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "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 $f({\\bf V};a)\\sim |{\\bf V}|^{-4-\\sigma(a)}\\Phi(\\phi; a)$, where $\\phi\\equiv \\tan V_y/V_x$ 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-07-08T07:42:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shear flow",
      "high-velocity tail",
      "boltzmann-fokker-planck equation",
      "shear rate",
      "state exhibiting non-newtonian behavior"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}