On Admissible Positions of Transonic Shocks for Steady Isothermal Euler Flows in a Horizontal Flat Nozzle under Vertical Gravity

TL;DR

This paper investigates the existence and positioning of transonic shocks in steady isothermal Euler flows within a horizontal flat nozzle under vertical gravity, using a free boundary problem approach and perturbation analysis.

Contribution

It introduces a new free boundary problem for the linearized Euler system with gravity, addressing the shock position determination under small perturbations.

Findings

Existence of solutions to the free boundary problem under small perturbations.

Shock front position can be approximated and refined through a nonlinear iteration scheme.

Vertical gravity significantly influences the shock front location and the solution structure.

Abstract

In this paper we are concerned with the existence of transonic shocks for 2-D steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. In particular, we focus on the contribution of the vertical gravity in determining the position of the shock front. For steady horizontal flows, the existence of normal shocks with the position of the shock front being arbitrary in the nozzle can be easily established. This paper will try to determine the position of the shock front as the state of the flow at the entrance of the nozzle and the pressure at the exit are slightly perturbed. To this end, this paper proposes a free boundary problem of the linearized Euler system with vertical gravity, whose solution could be an initial approximation for the shock solution with the free boundary being the approximation for the shock front. Due to the existence of the vertical gravity, difficulties arise in solving the boundary value problem in the approximate subsonic domain behind the shock front. The linearized Euler system is elliptic-hyperbolic composite for subsonic flows, and the elliptic part and the hyperbolic part are coupled in the $0$-order terms depending on the acceleration of gravity $g$. Moreover, the coefficients are not constants since the unperturbed shock solution depends on the vertical variable. New ideas and techniques are developed to deal with these difficulties and, under certain sufficient conditions on the perturbation, the existence of the solution to the proposed free boundary problem is established as the acceleration of gravity $g>0$ and the perturbation are sufficiently small. Then, with the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed which leads to a transonic shock solution with the position of the shock front being close to the initial approximating position.