Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit

Abstract

In this paper we show that, as the viscosity is properly small, there exists a viscous transonic shock solution for the steady 1-D Navier-Stokes system with prescribed pressure at the exit, and it converges to a transonic shock solution to the 1-D steady Euler system as the viscosity goes to zero. Moreover, the position of the shock front is also derived. The key step is to reduce the pressure condition at the exit into a nonlinear boundary condition on the velocity, such that the boundary value problem for the Navier-Stokes system can be reformulated as a boundary value problem for an ODE with an unknown parameter.