8 - Viscid Hydrodynamics

Summary

This chapter returns to the zero-field limit of MHD replacing the isotropic pressure force density in ideal HD with force densities arising from the viscous stress tensor for viscid HD. As tensor analysis is not a prerequisite for this course, the stress tensor is developed purely from a vector analysis of all stresses applied at a single point in a viscid fluid. This leads to the introduction of bulk and kinetic viscosity in a Newtonian fluid and the identification of ordinary thermal pressure with the trace of the stress tensor. Various flavours of the Navier–Stokes equation are developed including compressible and incompressible forms. The Reynold’s number is introduced as a result of scaling the Navier–Stokes equation which leads to a qualitative discussion on turbulent and laminar flow. Numerous examples are given in which a simplified form of the Navier–Stokes equation can be solved analytically, including plane-parallel flow, open channel flow, Hagen–Poiseuille flow, and Couette flow.