Boundary integral equations for a three‐dimensional Brinkman flow problem

Abstract

The purpose of this paper is to prove existence and uniqueness in Sobolev or Hölder spaces for a transmission problem which describes the flow of a viscous incompressible fluid past a porous particle embedded in a second porous medium, by using the Brinkman model and potential theory. Some particular cases, which refer to Stokes flow past a porous particle, or to Brinkman's flow past a void, are also presented together with corresponding asymptotic results for the flow velocity field and the hydrodynamic force exerted on the particle.

The Cauchy problem associated to the regularized Brinkman flow equation in L2(R)

“…In this case, if the compatibility condition ∂ t ρ + ∇ · ρv = 0, (1.5) we obtain the equation ∂ t ρ = ∇ · (ρ(1 − ∆) −1 ∇P(ρ)), (1.6) which is called the Brinkman flow equation. See for example [2][3][4][5][6][7] and references therein. The Cauchy Problem associated to the Brinkman flow equation…”

Adaptive FE–BE coupling for an electromagnetic problem in ℝ3—A residual error estimator

“…Then the general analysis of pseudo-differential operators [8,31,32] guarantees that the mapping properties for the integral operators also hold on . Application of the approach in [33] might show that this also holds for Lipschitz boundaries, but this is still to be investigated and therefore assumed here.…”

“…We only consider smooth surfaces and use the Fourier transformation and some results of Costabel and Stephan [45] together with properties of pseudo-differential operators as developed by Seeley [31] (see also Hsiao and Wendland [32]) and applied by MacCamy and Stephan [8]. These results can be extended for − 1 2 <s< 1 2 to polyhedral domains with the methods in [33] as will be shown in a forthcoming paper.…”

Two-dimensional Stokes–Brinkman cell model – a boundary integral formulation

“…Dindosˇand Mitrea [16] used a layer potential approach to treat the Poisson problem for the Stokes system, as well as the nonlinear Navier-Stokes equations on C 1 or even Lipschitz domains in a smooth compact Riemannian manifold, with data in Sobolev or Besov spaces. Kohr and Wendland [17] have used the Brinkman model and the layer potential theory for the Brinkman operator in order to obtain existence and uniqueness in Ho¨lder or Sobolev spaces for a transmission problem that describes the flow of a viscous incompressible fluid past a porous particle embedded in a second porous medium, by assuming that the boundary of the porous particle is a closed Lyapunov surface or, more generally, a Lipschitz surface. Extensions of these results to transmission problems for Stokes and Brinkman operators on Lipschitz domains in compact Riemannian manifolds have been recently considered by Kohr et al [18,19].…”