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

Abstract

We consider a two-dimensional coupled transmission problem with the conservation laws for compressible viscous flows, where in a subdomain Ω 1 of the flow-field domain Ω the coefficients modelling the viscosity and heat conductivity are set equal to a small parameter ε > 0. The viscous/viscous coupled problem, say P ε, is equipped with specific boundary conditions and natural transmission conditions at the artificial interface Γ separating Ω 1 and Ω \ Ω 1 . Here we choose Γ to be a line segment. The solution of Pε 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 Ω 1 . With respect to the norm of uniform convergence, Pε is in general a singular perturbation problem. Following the Vishik-Ljusternik method, we investigate here the boundary layer phenomenon at Γ. We represent the solution of Pε 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 Pε, specific inviscid/viscous boundary conditions, and transmission conditions expressing the continuity of the normal flux at Γ. A detailed analysis of the problem for the vector-valued boundary layer correction indicates whether additional local continuity conditions at Γ 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 Γ (if any) provides an approximation of the solution of Pε 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.

On some mathematical modelling of self‐field MPD thrusters

“…For the subsonic case M 1 < 1, we could not yet prove corresponding results, but taking into account that the matrix A has a positive eigenvalue as well as the special form of system (4.28), we may conjecture that probably all c i are necessarily zero. This conjecture has been checked by numerical tests, see [11]. In other words, in the case of a local subsonic inviscid/viscous flow across the artificial interface Γ, i.e.…”

“…The method of characteristics is very appropriate for establishing boundary conditions associated with the Euler system, corresponding to open (free-stream) parts of the outer boundary. In order to establish transmission conditions at the artificial interface Γ we shall use a different approach (see [10,11]), which is based on singular perturbation theory. In fact, this approach is again very natural and consistent with physical principles.…”

“…Then we equate the coefficients of like powers of ε and separate those depending on τ andτ from the other terms which are independent of τ andτ . After some tedious computations (see [11]) we can show that z = (z − , z + ) (or, equivalently, w = (w − , w + )) satisfies the following problem:…”

“…in the case of a locally supersonic compressible flow entering the hyperbolic Euler subdomain Ω − , one needs to add the continuity conditions (4.30) to problem (P 0 ) (Theorem 4.2). The corresponding numerical tests presented in [11] show that the approximate solution (c 2 , c 4 ) tends to infinity if (c 2 , c 4 )(0, x 2 , t) = (0, 0), hence this solution does not satisfy the condition c 2,4 (∞, x 2 , t) = 0. Therefore, in this case, the components c 2 and c 4 must vanish identically.…”

“…At the same time, it is a survey on the topic, including also known literature. In many respects this work depends on further contributions of the authors [3,[6][7][8][9][10][11][12][13][14][15][16][17][18]26].…”