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

Abstract

Abstract. In this paper we consider the resolvent Stokes problem in the case there is a small perturbation of the domain caused by a perturbed boundary. Firstly, we prove that the solution of Stokes problem is continuous due to this small perturbation. Secondly, we derive the first-order term in the displacement field perturbation that due to the deformation of the domain. It is worth emphasizing that even though only the first-order term is given, our method enables us to derive higher-order terms as well. The derivation is rigorous and based on layer potential techniques.

Effects of small boundary perturbation on flow of viscous fluid

“…The case of periodically corrugated boundary was studied by many authors (see e.g. [2][3][4][5][6][7][8][9][10][11]). Recently, non-Newtonian flow near rough surface was studied in [5] and [9], starting from a version of the Prandtl system and using von Mises variables.…”

“…The behavior of the sequence of solution in sequence of domains is studied as the domains approach an open set. The domain perturbation for the Stokes system was treated in using the method of layer potentials. The concept of shape derivative, that we use here, is frequently used in shape optimization, see for instance Mohammadi and Pironneau .…”