Nonlinear viscosity and Grad's method.

Abstract

The Grad ten-moment approximation (no heat flux) is analyzed for cylindrical symmetry in a stationary situation in which the gradients of the fluxes are assumed to be small. We show that if the collision term in the transport equation, resulting from the ten-moment approximation, is linearized in the fluxes, we can obtain a viscosity (etal) that depends on the gradient of the velocity with the correct limiting behavior for small gradients. The nonlinear contribution of the fluxes to the collision term are then taken into account to derive an expression for the viscosity (eta(nl)) as a function of the gradient of the velocity. A comparison between etal and eta(nl) is performed finding that the maximum percentage deviation between them is 0.52% when the gradient of the hydrodynamic velocity is positive, but when the gradient is negative the situation changes dramatically.