Boundary integral equations for two‐dimensional low Reynolds number flow past a porous body

Abstract

SUMMARYIn this paper we use the method of matched asymptotic expansions in order to study the two-dimensional steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape. One assumes that the flow inside the porous body is described by the Brinkman model, i.e. by the continuity and Brinkman equations, and that the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. By considering some indirect boundary integral representations, the inner problems are reduced to uniquely solvable systems of Fredholm integral equations of the second kind in some Sobolev or Hölder spaces, while the outer problems are solved by using the singularity method. It is shown that the force exerted by the exterior flow on the porous body admits an asymptotic expansion with respect to low Reynolds number, whose terms depend on the solutions of the abovementioned system of boundary integral equations. In addition, the case of small permeability of the porous body is also treated.

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

“…Thus, the main properties of the layer potential operators for the Brinkman system are provided by those of the layer potentials for the Stokes system. Then [3,4,7,11,13,29] we have the following theorem. LEMMA 5.1 For any !…”

“…[7,29,30]). Also, for all ðu, Þ 2 H 1 loc ðD þ , L St Þ and w 2 H 1 comp ðD þ Þ, one has the Green formula 3 We are referring to a two-dimensional cell model consisting of a porous particle (D 1 ) with an impermeable core (D 2 ) located in a bounded fluid region (D 0 ).…”

Stokes–Brinkman formulation for prediction of void formation in dual-scale fibrous reinforcements: a BEM/DR-BEM simulation

“…On the other hand, for the anisotropic Brinkman equation, Khor et al [26] deduced fundamental solutions in the transformed Fourier space, which reduce to the isotropic fundamental solutions in the real space when χ 1 = χ 2 [27,62]. However, the transformation of these functions to the real space in the anisotropic case is not a trivial problem and to avoid these difficulties, a boundary-domain integral formulation in terms of the Stokes fundamental solutions is considered for the Brinkman equation and the resulting domain integral is transformed into a boundary integral using DR-BEM [42].…”

Boundary integral formulation for flows containing an interface between two porous media

“…When describing a porous medium flow by the Brinkman equations instead of Darcy's law, more boundary conditions are required at the interface due to the spatial derivatives of the velocity. Continuity of both the velocity and the stress have been Boundary integral formulation for porous-porous system 73 used at the interface (Neale & Nader 1974;Kohr, Wendland & Sekhar 2009;Chen, Gunzburger & Wang 2010). The approach taken by Ochoa-Tapia & Whitaker (1995) is to impose a jump condition on the tangential components of the stress using the volume-averaged method on the Stokes equation at the interface:…”