Solutions to Forced and Unforced Lin–Reissner–Tsien Equations for Transonic Gas Flows on Various Length Scales

Abstract

The Lin–Reissner–Tsien equation is useful for studying transonic gas flows, and has appeared in both forced and unforced forms in the literature. Defining arbitrary spatial scalings, we are able to obtain a family of exact similarity solutions depending on one free parameter in addition to the model parameter holding the scalings. Numerical solutions compare favorably with the exact solutions in regions where the exact solutions are valid. Mixed wave-similarity solutions, which describe wave propagation in one variable and self-similar scaling of the entire solution, are also given, and we show that such solutions can only exist when the wave propagation is sufficiently slow. We also extend the Lin–Reissner–Tsien equation to have a forcing term, as such equations have entered the physics literature recently. We obtain both wave and self-similar solutions for the forced equations, and we are able to give conditions under which the force function allows for exact solutions. We then demonstrate how to obtain these exact solutions in both the traveling wave and self-similar cases. There results constitute new and potentially physically interesting exact solutions of the Lin–Reissner–Tsien equation and in particular suggest that the forced Lin–Reissner–Tsien equation warrants further study.