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Nonlinear viscosity and Grad's method
1Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, 09340 México, Distrito Federal, Mexico.
Summary
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.
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.