Kinematics of a multi-dimensional shock of arbitrary strength in an ideal gas

Abstract

It has been shown that the kinematics of a shock front in an ideal gas with constant specific heat can be completely described by a first order nonlinear partial differential equation (called here - shock manifold equation or SME) which reduces to the characteristic partial differential equation as the shock strength tends to zero. The condition for the existence of a nontrivial solution of the jump relations across the shock turns out to be the Prandtl relation. Continuing the functions representing the state on the either side of the shock to the other side as infinitely differentiable functions and embedding the shock in a one parameter family of surfaces, it has been shown that the Prandtl relation can be treated as a required shock manifold equation for a function Φ , where Φ=0 is the shock surface. We also show that there are other forms of the SME and prove an important result that they are equivalent. Shock rays are defined to be the characteristic curves of a SME and it has been shown that when the flow on either side of the shock is at rest, the shock rays are orthogonal to the successive positions of the shock surface. Certain results have been derived for a weak shock, in which case the complete history of the curved shock can be determined for a class of problems.