A. M. J. Davis

Separation from the surface of two equal spheres in Stokes flow

In this paper, it is shown that if two spheres of equal radii are placed axisymmetrically in a steady Stokes stream, separation of the flow from the spheres occurs if the distance between their centres is less than approximately 3-67 times the sphere radius. For spheres whose spacing is less than this value, wakes form on both spheres and the fluid within the wakes moves in closed eddy type motion. When the distance between the centres of the spheres is less than approximately 3.22 times the sphere radius, a cylinder of fluid links both spheres, and within this cylinder the fluid rotates in one or more ring vortices, the number of vortices increasing as the distance between the spheres is decreased. When the spheres are in contact, the fluid rotates in an infinite set of nested ring vortices.

Axisymmetric Stokes flows due to a rotlet or stokeslet near a hole in a plane wall: filtration flows

The axially symmetric Stokes-flow problems occurring when a point source, rotlet or stokeslet is situated along the axis through the centre of a circular hole in a solid plane wall are examined. Exact solutions of the governing equations are obtained in terms of toroidal coordinates and their use in modelling the flows caused by a small particle translating and rotating near to a filter pore is considered. First-order expressions are derived for the effects of the wall and hole upon the hydrodynamic force and torque on the particle for situations in which the particle dimensions are small in comparison with its distance from the solid portion of the plane wall. The resulting expressions apply to any centrally symmetric particle, not necessarily axisymmetric. Finally, expressions are derived for the motion of a neutrally buoyant sphere suspended in a flow through a hole. It is demonstrated that such a particle will generally migrate across the streamlines of the undisturbed flow - away from or towards the symmetry axis of the flow, according as the particle is approaching or receding from the hole. Such migratory motion may be of importance in the flow of suspensions through orifices and stenoses.

The slow rotation of a sphere submerged in a fluid with a surfactant surface layer

Abstract The effect of a layer of an adsorbed surfactant monomolecular film of fluid which covers the surface of a large volume of a different substrate fluid is considered with respect to the fluid motion caused by the slow rotation of a submerged sphere. For a semi-infinite substrate, the boundary value problem posed with the surfactant boundary condition of Scriven and Goodrich is solved exactly for any depth of the submerged sphere. Comprehensive numerical calculations are given for the torque and surface velocity for various values of the parameters defining the depth of the sphere and the surface shear viscosity. Asymptotic expressions for the solution are given for the cases of a deeply submerged sphere or when the substrate has a finite depth. The relevance of the work to providing an experimental technique for measuring surface shear viscosity is also considered.

Force and torque formulae for a sphere moving in an axisymmetric Stokes flow with finite boundaries: asymmetric stokeslets near a hole in a plane wall

Abstract A sphere is allowed to move with three degrees of freedom in an axisymmetric flow field and general formulae, correct to the third power of the spheres radius, are developed for the Stokes resistance experienced by the sphere. These are shown to depend on the behaviour within the sphere of the reflected velocity fields which arise from the presence of fixed boundaries at finite distances from stokeslets placed at the spheres center. Application is made to the stagnation flow at a plane. Poiseuille flow and flow past a sphere and some comparisons made with exact formulae. Solutions are given for asymmetrically placed stokeslets near a hole in a plane wall or a disk.

Steady rotation of a tethered sphere at small, non-zero Reynolds and Taylor numbers: wake interference effects on drag

Matched asymptotic expansion methods are used to solve the title problem. First-order Taylor number corrections to both the Stokes-law drag and Kirchhoffs-law couple on the sphere are obtained for Rossby numbers of order unity. This calculation fills a gap between the Proudman-Pearson (1957) rectilinear trajectory analysis, which includes Reynolds-number effects but does not address Taylor-number effects arising from the curvilinear trajectory, and the Herron, Davis & Bretherton (1975) curvilinear-trajectory analysis, which incorporates Taylor-number effects but ignores those arising from the Reynolds number. At the same Reynolds number, the drag on the sphere is found to be greater or less than the classical Oseen (1927)-Proudman & Pearson (1957) value, depending upon the magnitude of a certain dimensionless length parameter B measuring the tether radius to the sphere radius. This drag difference is attributed, in part, to the fact that the sphere runs into the disturbance created by its own wake.

A relation between the radiation and scattering of surface waves by axisymmetric bodies

Results corresponding to those of Newman (1975) for wholly or partially immersed cylinders are obtained, namely relations between the phase angles of the cylindrical outgoing surface waves generated either by the forced oscillations of an immersed axisymmetric body or by the scattering effect of the body on a plane wave.

The influence of relative velocity on the eddy structure between two spheres in Stokes flow

Abstract Davis et al. (1976) have shown that if two solid spheres move together in an axisymmetric Stokes flow, then provided they are sufficiently close, a body of fluid becomes trapped between the spheres. Here it is shown how the small eddy motions induced in this trapped fluid are significantly disrupted when one sphere moves relative to the other.