The Structure of an Asymmetric Stokes Flow

Some Stokes flows exterior to a spherical boundary

“…There is now an extensive literature on slow viscous fluid motions which exhibit flow separation (see, e.g., O'Neill [1] for a review of recent work), but most of the situations studied have been of a twodimensional or of an axisymmetric nature. Thus there is a relative scarcity of truly three-dimensional asymmetric solutions, and in an attempt to remedy this deficiency Hackborn, O'Neill and Ranger [2] considered the structure of the Stokes flow of fluid confined to a fixed spherical container when the motion is induced by a rotlet whose axis is perpendicular to the sphere radius drawn through it. It was shown in [2] that separation with flow reversal occurs when the distance of the rotlet from the centre of the sphere exceeds 0-3788 radii.…”

“…Thus there is a relative scarcity of truly three-dimensional asymmetric solutions, and in an attempt to remedy this deficiency Hackborn, O'Neill and Ranger [2] considered the structure of the Stokes flow of fluid confined to a fixed spherical container when the motion is induced by a rotlet whose axis is perpendicular to the sphere radius drawn through it. It was shown in [2] that separation with flow reversal occurs when the distance of the rotlet from the centre of the sphere exceeds 0-3788 radii. The results of Hackborn et al were subsequently rederived by Shail [3] using appropriate sphere theorems for the calculation of the two scalar functions </ » and x appearing in the representation of the velocity and pressure field (see equations (2)-(6) below) rather than the infinite series of separated solutions employed in [2].…”

“…It was shown in [2] that separation with flow reversal occurs when the distance of the rotlet from the centre of the sphere exceeds 0-3788 radii. The results of Hackborn et al were subsequently rederived by Shail [3] using appropriate sphere theorems for the calculation of the two scalar functions </ » and x appearing in the representation of the velocity and pressure field (see equations (2)-(6) below) rather than the infinite series of separated solutions employed in [2]. Shail also added the solution of the problem in which the rotlet is replaced by a Stokeslet, and gives a description of the motion (which does not exhibit flow reversal), together with a formula for the force experienced by a slowly moving arbitrary particle which sediments through the fluid within the sphere.…”

“…Basic formulation and the sphere theorems. We follow the notations used in [1,2] and consider a region of incompressible viscous fluid which extends to infinity in all directions and is bounded internally by a rigid sphere of unit radius. When the flow is sufficiently slow to permit the Stokes linearisation of the equations of steady motion, the velocity and pressure fields v and p in a singularity-free fluid region satisfy the equations…”

“…where 17 is the coefficient of viscosity of the fluid. In terms of spherical polar coordinates (r, 0, <p), with origin at the centre of the sphere, a suitable solution of(l)is [2] v = curl 2 cos cp +curl --sin <p \,…”