# American Institute of Mathematical Sciences

February  2004, 10(1&2): 137-163. doi: 10.3934/dcds.2004.10.137

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

 1 Robert Bosch GMBH, FV/PTS, Postfach 30 02 40, D-70442 Stuttgart, Germany 2 IRS, Universtät Stuttgart, D-70550 Stuttgart, Germany 3 Mathematics Department, Central European University, H-1051 Budapest, Hungary 4 Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, 70550 Stuttgart, Germany

Received  March 2002 Revised  February 2003 Published  October 2003

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 $\varepsilon>0$. 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 Vishik--Ljusternik 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 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 $P_\varepsilon$ 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.
Citation: Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137
 [1] O. Guès, G. Métivier, M. Williams, K. Zumbrun. Boundary layer and long time stability for multi-D viscous shocks. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 131-160. doi: 10.3934/dcds.2004.11.131 [2] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161 [3] John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501 [4] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [5] Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014 [6] Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741 [7] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [8] Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3615-3627. doi: 10.3934/dcds.2021009 [9] Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037 [10] Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581 [11] Lili Fan, Lizhi Ruan, Wei Xiang. Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1971-1999. doi: 10.3934/dcds.2020349 [12] Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045 [13] Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065 [14] Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic & Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032 [15] Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419 [16] Antoine Sellier. Boundary element approach for the slow viscous migration of spherical bubbles. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1045-1064. doi: 10.3934/dcdsb.2011.15.1045 [17] Hassen Arfaoui, Faker Ben Belgacem, Henda El Fekih, Jean-Pierre Raymond. Boundary stabilizability of the linearized viscous Saint-Venant system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 491-511. doi: 10.3934/dcdsb.2011.15.491 [18] Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 [19] Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations & Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070 [20] Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020

2020 Impact Factor: 1.392