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Nonlinear viscosity and Grad's method.

F J Uribe1, L S García-Colín

  • 1Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, 09340 México, Distrito Federal, Mexico.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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This study analyzes the Grad ten-moment approximation for viscosity under cylindrical symmetry. Linear and nonlinear collision terms yield similar viscosity for positive velocity gradients, but differ significantly for negative gradients.

Area of Science:

  • Plasma physics
  • Transport phenomena

Background:

  • The Grad ten-moment approximation is a theoretical framework used to describe plasma behavior.
  • Understanding transport properties like viscosity is crucial for modeling plasma dynamics.

Purpose of the Study:

  • To analyze the Grad ten-moment approximation (no heat flux) under cylindrical symmetry in a stationary state.
  • To derive and compare linear and nonlinear expressions for viscosity based on velocity gradients.

Main Methods:

  • Linearization of the collision term in the transport equation for small flux gradients.
  • Inclusion of nonlinear flux contributions to the collision term.
  • Derivation of two viscosity expressions: eta(l) (linear) and eta(nl) (nonlinear).
  • Comparison of eta(l) and eta(nl) for varying velocity gradients.

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Main Results:

  • A linear viscosity (eta(l)) was derived, showing correct limiting behavior for small velocity gradients.
  • A nonlinear viscosity (eta(nl)) was derived by considering nonlinear flux contributions.
  • Maximum deviation between eta(l) and eta(nl) was 0.52% for positive velocity gradients.
  • Significant differences were observed between eta(l) and eta(nl) for negative velocity gradients.

Conclusions:

  • The Grad ten-moment approximation provides a framework for understanding viscosity in plasmas.
  • Linearization of the collision term is a reasonable approximation for positive velocity gradients.
  • Negative velocity gradients introduce significant nonlinear effects impacting viscosity calculations.