Asymptotic analysis of a two-dimensional coupled problem for compressible viscous flows

We consider a two-dimensional coupled transmission problem with the conservation laws for compressible viscous flows, where in a subdomain Omega(1) of the flow-field domain Omega the coefficients modelling the viscosity and heat conductivity are set equal to a small parameter epsilon > 0. The viscous/viscous coupled problem, say P-epsilon, is equipped with specific boundary conditions and natural transmission conditions at the artificial interface Gamma separating Omega(1) and Omega\Omega(1). Here we choose Gamma to be a line segment. The solution of P-epsilon can be viewed as a candidate for the approximation of the solution of the real physical problem for which the dissipative terms are strongly dominated by the convective part in Omega(1). With respect to the norm of uniform convergence, P-epsilon, is in general a singular perturbation problem. Following the Vishik-Ljusternik method, we investigate here the boundary layer phenomenon at Gamma. We represent the solution of P-epsilon, as an asymptotic expansion of order zero, including a boundary layer correction. We can show that the first term of the regular series satisfies a reduced problem, say P-0, which includes the inviscid/viscous conservation laws, the same initial conditions as P-epsilon, specific inviscid/viscous boundary conditions, and transmission conditions expressing the continuity of the normal flux at Gamma. A detailed analysis of the problem for the vector-valued boundary layer correction indicates whether additional local continuity conditions at Gamma are necessary for P-0, defining herewith the reduced coupled problem completely. In addition, the solution of P-0 (which can be computed numerically) plus the boundary layer correction at Gamma (if any) provides an approximation of the solution of P-epsilon and, hence, of the physical solution as well. In our asymptotic analysis we mainly use formal arguments, but we are able to develop a rigorous analysis for the separate problem defining the correctors. Numerical results are in agreement with our asymptotic analysis.