Creeping motion of a sphere along the axis of a closed axisymmetric container

Abstract

The creeping flow around a sphere settling along the axis of a closed axisymmetric container is obtained both theoretically and experimentally. The numerical technique for solving the Stokes equations uses the classical Sampson expansion; the boundary conditions on the sphere are satisfied exactly and those on the container walls are applied in the sense of least squares. This is an extension to the axisymmetric case of the technique for solving various two-dimensional flow problems. Two types of axisymmetric container are considered here as examples: circular cylinders closed by planes at both ends, and cones closed by a base plane. Calculated streamlines patterns show various sets of eddies, depending upon the geometry and the sphere position. Results are in agreement with earlier Stokes flow calculations of eddies in corners and in closed containers. Experiments use laser interferometry to measure the vertical displacement of a steel bead a few millimetres in diameter settling in a container filled with a very viscous silicone oil. The Reynolds number based on the sphere radius is typically of the order of 10−5. The accuracy on the vertical displacement is 50nm. Experiments show that the motion towards the apex of a cone is much slower than that towards a plane; the bead takes hours to reach the micrometre size roughness asperities on a conical wall, as compared with minutes to reach those on a plane wall. The numerical results for the drag force are in excellent agreement with experiments both for the cylindrical and the conical containers. With standard computer accuracy, the present numerical technique applies when the gap between the sphere and the nearby wall is larger than about one radius. For a sphere in the vicinity of any plane horizontal wall, these results also match with a previous analytical solution. That solution is in excellent agreement with our experimental results at small distances from the wall (typically less than a diameter, depending on the sphere size).