Shock Interaction with a Contact Surface*

In this work the classical problem of the interaction of a shock wave and a contact surface (or contact discontinuity) is analyzed using modern methods of infinitesimal analysis. A shock wave and a contact surface, defined on planar characteristics in space and time, are modeled as traveling waves in a one-dimensional, inviscid, ideal gas. The shock wave and the contact surface are represented by nonstandard Heaviside jump functions and are defined using infinitesimal analysis. For the case in which the shock wave is incident on a denser gas, it is shown that the interaction of the shock with the contact surface admits a solution that may be expressed using a third nonstandard Heaviside function that represents a shock wave reflected off the contact surface. Nonstandard analysis is applied to describe the jump functions and their derivatives. Nonstandard analysis is an area of modern mathematics that studies extensions of the real number system to number systems that contain both infinitesimal numbers and infinitely large numbers and provides a rigorous framework for infinitesimal analysis. It is assumed that the shock wave and contact surface thicknesses occur on idealized infinitesimal intervals and that the nonstandard jump functions in the thermodynamic and kinematic parameters vary smoothly across these idealized discontinuities. The equations of motion are cast in nonconservative form and applied to derive unambiguous relationships between the nonstandard jump functions and their products for the flow parameters in the regions surrounding the shock wave and the contact surface.