Existence and uniqueness result for the problem of viscous flow in a granular material with a void

Abstract

Abstract. The purpose of this paper is to obtain an indirect boundary integral formulation for the three-dimensional viscous flow problem in a granular material with a void. The corresponding existence and uniqueness result of the classical solution to this problem is proved by using the theory of hydrodynamic potentials.

Boundary integral equations for a three‐dimensional Brinkman flow problem

“…Certain particular cases, which describe Stokes flow past a porous particle, or Brinkman flow past a void, are also presented, together with the corresponding asymptotic results for the hydrodynamic force on the particle. In the particular case of a porous sphere embedded in a second porous medium, we recover some known asymptotic results that have been previously obtained in [21,22].…”

“…Let us now consider the case of Brinkman flow in a porous medium with large permeability and in the presence of a void, i.e., χ 1 1 and χ 0 = 0 (see also [21]). Using similar arguments as in the previous case (χ 0 1, χ 1 = 0), one obtains the following asymptotic formula for the exterior velocity field:…”

“…Since the Brinkman equation is mathematically equivalent to the Stokes resolvent equation, the potential theory for the Stokes resolvent system may be used to treat several porous media flow problems which involve the Brinkman model. In view of this property, Kohr and Sekhar [20,21] have obtained existence and uniqueness for the problem of Stokes flow in a granular material with a void, or for the problem of two-dimensional porous media flows with porous inclusions based on Brinkman's equation, by assuming that the boundary of the flow domain is of class C 2,α , α ∈ (0, 1]. Also, Kohr, Sekhar, and Wendland [22] have used an indirect boundary integral method for the Stokes flow past a porous body, and have obtained some asymptotic results in both cases of large and, respectively, of low permeability of the porous body.…”

Viscous flow around three-dimensional macroscopic cavities in a granular material

“…(2.12) Equation (2.12) is the same as that given by Kohr & Raja Sekhar (2007). The asymptotic forms are…”

Boundary integral method for Stokes flow past a porous body

“…According to the fact that the Stokes resolvent equation and the Brinkman equation have the same form, the potential theory for the Stokes resolvent equation may be used to study various porous media flow problems which involve the Brinkman model (see, e.g. [31,32]).…”