Stability of Normal Shock Waves with Viscosity and Heat Conduction

The stability problem, for small arbitrary one-dimensional disturbances, of a normal shock wave with viscosity and heat conduction in a thermodynamically perfect gas with a Prandtl number of 0.75 is treated, and is formulated explicitly as an eigenvalue problem involving ordinary linear differential equations with polynomial coefficients in a fixed finite domain whose end points are singular points of the differential equations. It is shown by a simple general type of mathematical argument that one possible mode shape for the perturbations is a translation of the shock structure, and that such a disturbance is neutrally stable. For the limiting case of a weak-shock structure, the equations developed here are shown to reduce systematically to a perturbed form of Burgers' equation. The weak shock structure is shown to be stable for any Prandtl number and general equation of state, and a complete solution for the disturbance eigenvalues and eigenfunctions in this case is derived and discussed.