Stokes flows in 3D containers

Abstract

This study consists of two parts. First we consider an analytical approach for solving the problem of steady Stokes flow in some 3D containers with arbitrary velocities prescribed over the surfaces. The approach is based on the superposition method. First we discuss the Stokes problem solution in a finite cylinder. This is the simplest problem because the flow domain is restricted with only two families of coordinate surfaces and the edge (rim) is a smooth line. Then we discuss the analytical solution of the Stokes problem in more complicated domains, such as a circular cone, a rectangular trihedral corner and a 3D rectangular cavity. The Moffatt eddies in such domains are described. In the second part of the study we consider the laminar mixing process in the Stokes flow in a 3D container. We show that in 3D flows a much richer variety of mixing regimes is observed than in 2D flow configurations. The mixing processes in a 3D flow, containing periodic lines, possess essentially two-dimensional characteristics. In the flows, where only isolated periodic points exist, the liquid elements stretch or compress in all three directions.