Justification of lubrication approximation: An application to fluid/solid interactions

Abstract

We consider the stationary Stokes problem in a three-dimensional fluid domain F with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle B in a bounded or an exterior smooth container Ω. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small.

Numerical simulation of rigid particles in Stokes flow: lubrication correction for general shapes of particles

“…In the present article, we propose a new explicit expression for the singular field which is based on the same asymptotic flow (u in , q in ). Our asymptotic field is similar to the one proposed in [6], which is modified to be more tractable from a numerical point of view: the singular field we propose presents less oscillations and vanishes outside the gap between the particles. Here, we use estimations proved in [6] as an intermediate result to show that our remaining regular field is bounded independently of the distance.…”

“…Our asymptotic field is similar to the one proposed in [6], which is modified to be more tractable from a numerical point of view: the singular field we propose presents less oscillations and vanishes outside the gap between the particles. Here, we use estimations proved in [6] as an intermediate result to show that our remaining regular field is bounded independently of the distance. Doing so, we obtain numerical results for which the error is independent of the distance which allows to use coarse meshes, independently of the size of the gap.…”

“…In this article, the singular part is computed using an explicit asymptotic expansion of the solution when the distance goes to zero. This expansion is similar to the asymptotic expansion proposed in [6] but is more appropriate for numerical simulations of suspensions. It can be computed for any locally convex (that is the particles have to be convex close to the contact point) and regular shape of particles.…”

“…The outline of the article is the following: after this introduction, the second section is dedicated to the case of two single particles. We explain how we construct the singular field (u sing , p sing ) and how the results of [6] are used to bound the remaining regular field. Numerical simulations are presented in Section 3 to validate the method.…”

On the Derivation of a Stokes–Brinkman Problem from Stokes Equations Around a Random Array of Moving Spheres

“…Combining then that γ t (r) − γ b (r) ≥ h j + cr 2 on (0, δ 0 ) for some c > 0 (since δ 0 < 1/2) and a change of variable r = √ h j s in the integral, we obtain (A.8). More details on these computations can be found in [14].…”