Contact discontinuities for 3-D axisymmetric inviscid compressible flows in infinitely long cylinders

Abstract

We prove the existence of a subsonic axisymmetric weak solution (u, rho, p) with u = u(x)e(x)+ u(r)e(r) + u(theta)e(theta) ( )to steady Euler system in a three-dimensional infinitely long cylinder N when prescribing the values of the entropy (= P/rho(gamma) ) and angular momentum density (= ru(theta)) at the entrance by piecewise C-2 functions with a discontinuity on a curve on the entrance of N. Due to the variable entropy and angular momentum density (=swirl) conditions with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity, nonzero swirl, and contains a contact discontinuity r = g(D)(x). We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a solution and analyze the asymptotic behavior of the solution at far field. (C) 2019 Elsevier Inc. All rights reserved.