An indirect boundary integral method for an oscillatory Stokesflow problem

Abstract

The purpose of this paper is to present an indirect boundary integral method for the oscillatory Stokes flow provided by the translational oscillations of two rigid spheres in an incompressible Newtonian fluid of infinite expanse

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

“…Briceno and Power [3] obtained a completed boundary integral equation approach for the numerical solution of the boundary value problem corresponding to the motion of N solid particles in the interior of a deformable viscous drop. Kohr [12] obtained a boundary integral method in order to study the oscillatory Stokes flow due to the oscillations of two solid spheres in an incompressible Newtonian fluid. On the other hand, Varnhorn [39], [40], [41] developed a complete potential theory for the Stokes resolvent system whose unknowns are defined only on domains in R n with connected boundaries.…”

Asymptotic formula for the solution of the Stokes problem with a small perturbation of the domain in two and three dimensions

“…We consider here λ ∈ C \ {z ∈ C : Rez ≤ 0, Imz = 0}. The fundamental tensors (Γ, F ) of the Stokes resolvent system (1.1) can be obtained by the Fourier transform method in the following forms (see [3] for d = 2 and [4], [8] …”

“…By using the notationsx = x − y = (x 1 , · · · ,x d ) and r = |x|, we introduce the stress tensor S associated to the fundamental tensors (Γ, F ) and having the following components (see in [8]):…”

“…We will continue using this convention throughout this paper. Taking into account the relation (2.8) and (2.9) we obtain the following explicit forms (see in [8]) as follows. In two dimensions, for ∀x, y ∈ R 2 , x = y we have:…”

“…The asymptotic expansion is derived due to the theory of layer potentials and Fredholm's alternative, and the properties of small perturbation of an interface. In connection with this, we refer to recent works in the context of interface problems [1], [2], [7], the continuity of the solution due to a small perturbation of an interface [7] and layer-potential theory for Stokes problem [3], [4], [5], [6], [8], [9]. In the work of H. Ammari, H. Kang, M. Lim, and H. Zribi [1] , they derived the asymptotic expansion of the boundary perturbations of steady-state voltage potentials, and now in our work, we derive the asymptotic expansion for the Stokes problem with Dirichlet boundary condition.…”