
Abstract
We consider a twodimensional coupled
transmission problem with the conservation laws for compressible
viscous flows, where in a subdomain $\Omega_1$ of the flowfield
domain $\Omega$ the coefficients modelling the viscosity and
heat conductivity are set equal to a small parameter
$\varepsilon>0$. The viscous/viscous coupled problem, say
$P_\varepsilon$, is equipped with specific boundary conditions
and natural transmission conditions at the artificial
interface $\Gamma$ separating $\Omega_1$ and $\Omega \setminus
\Omega_1$. Here we choose $\Gamma $ to be a line segment. The
solution of $P_\varepsilon$ 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, $P_\varepsilon$ is in general a singular
perturbation problem. Following the VishikLjusternik method, we
investigate here the boundary layer phenomenon at $\Gamma$. We
represent the solution of $P_\varepsilon$ 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
$P_\varepsilon$, 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 vectorvalued 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
$P_\varepsilon$ 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.
Mathematics Subject Classification: 34E05, 34E10, 34E15, 35Q30, 76M45, 76N17, 76N20.
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