On rotating laminar boundary layers

Abstract

The theory of the laminar boundary layer on a sphere of radius a rotating, without translation, about a diameter with angular velocity Ω was first considered by Howarth [6]. He showed that a consistent solution can be obtained on the assumption that there are two boundary layers, originating at opposite ends of the diameter of rotation and converging towards each other at the equator. Near the poles each boundary layer is the same as that on a disc rotating about its axis in a fluid otherwise at rest. In this boundary layer, first discussed by Von Kármán [8], fluid is drawn towards the disc and also moves radially outwards from the axis. As a result of the curvature of the sphere the boundary layers on the sphere must be modified as they spread outwards from the poles and by applying the momentum integral Howarth was able to extend them as far as the equatorial plane. According to his solution the boundary layers at the equator were infinitely thick with zero longitudinal components of velocity but his view was that this would be unlikely in the exact solution. In any case the boundary layers from the two poles must meet at the equator and the parabolic nature of their governing equations made them inadequate to describe the flow in its vicinity.