Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions

Abstract

The global well-posedness in time is proved, with no restriction on the size of the data, for the Stokes/Brinkman and Stokes/Darcy coupled flow problems with new jump interface conditions recently derived by Angot et al. [Phys. Rev. E 95 (2017) 063302-1–063302-16] using asymptotic modelling and shown to be physically relevant. These original conditions include jumps of both stress and tangential velocity vectors at the fluid–porous interface. They can be viewed as generalizations for the multi-dimensional flow of Beavers and Joseph’s jump condition of tangential velocity and Ochoa-Tapia and Whitaker’s jump condition of shear stress. Therefore, they are different from those most commonly used in the literature. The case of Saffman’s approximation is also studied, but with a force balance for the cross-flow including the Darcy drag and inducing a law of pressure jump different from the usual one. The proof of these results follows the general framework briefly introduced by Angot [C. R. Math. Acad. Sci. Paris, Ser. I 348 (2010) 697–702; Appl. Math. Lett. 24 (2011) 803–810.] for the steady flow.

A nonlinear asymptotic model for the inertial flow at a fluid-porous interface

“…(18) to prove that the whole fluid-porous coupled problem is well-posed. Indeed, the mathematical analysis of solvability is recently detailed in [74].…”

“…For the linear terms, we follow the mathematical analysis made in [74], see also [68,73], for the well-posedness study of the fluid-porous Stokes/Darcy problem with the asymptotic interface conditions without inertia (16) and we include the contributions of all the nonlinear inertial terms. To deal with the nonlinear terms, it is more suitable to write the inertial term in the Navier-Stokes equation with constant density ρ in Ω f as:…”

“…Then, we follow the previous works [68,73,74] for the mathematical analysis of the fluid-porous flow without inertia. Using now the inertial interface conditions (32) instead of (18), we get after standard integrations by parts the following energy balance:…”

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

“…Next we apply CT (27). It can be directly checked that bijective Properties (34) and (35), and hence Assumptions (D1) and (D2), hold true.…”

“…The constraint problems are related to parameter identification problems (see the theory in References [13][14][15] and application to biological systems in Reference [16]), to inverse problems by the mean of observation data used in mathematical physics [17,18] and in acoustics [19][20][21], to overdetermined and free-boundary problems [22,23]. As an application, in the current paper we focus on the incompressible Brinkman flow problem under a divergence-free condition (see the related modeling of porous medium in References [24,25], well-posedness analysis in Reference [26], and fluid-porous coupling with numerics in References [27,28]).…”