Axisymmetric Stokes flow due to the motion of a thin disk along the axis of a circular tube

The axisymmetric Stokes flow around a disk in a circular tube is investigated based on a Stokes approximation, where the radius of the disk, located coaxially with the circular tube, is arbitrary. The disk translates perpendicularly to its own plane along the centerline of the tube, and a Hagen-Poiseuille flow exists far upstream and downstream from the disk. The velocities of the translating disk and the Hagen-Poiseuille flow are given arbitrarily. The problem is investigated through an analysis of the Stokes equation using a complex eigenfunction expansion and the least squares method. The streamline patterns from the stream function, and the pressure distributions of the flow fields, are shown for some typical radii of the disk. The force exerted on the disk, and the pressure drop created by the disk itself, are both calculated as functions of the disk radius. For a small disk radius, the results of the force are coincident with previous asymptotic expressions. For a given velocity of the Hagen-Poiseuille flow in the tube, the translational drift velocity of the disk is determined as a function of the disk radius. The results show that this drift velocity is slightly lower than the mean velocity of the Hagen-Poiseuille flow component projected by the disk. The induced pressure drop from the disk drifting in the Hagen-Poiseuille flow is quite small. When the disk translates in a stagnant circular tube, a series of viscous toroidal eddies appears in the tube apart from the disk, as expected.