Asymptotic equations of gas dynamics: qualitative analysis, construction of solutions, and applications

In this paper, we propose asymptotic expansions for the velocity potential and obtain asymptotic equations of gas dynamics for irrotational isentropic flows of an ideal gas: an equation of linear theory, a nonlinear equation for supersonic flows, and a nonlinear transonic equation. We construct some exact particular solutions for the asymptotic nonlinear transonic equation, which takes into account transverse perturbations. Based on a linear asymptotic equation, we examine the dynamic stability of an elastic deformable element of a channel at a subsonic flow rate of a gas or liquid. The study of stability is carried out in a statement corresponding to small perturbations of a homogeneous flow and small deformations of an elastic element, and is based on the construction of a positive definite functional. Sufficient stability conditions are obtained.