M. E. O'Neill

A sphere in contact with a plane wall in a slow linear shear flow

Abstract An exact solution of the linearised Stokes flow equations is derived for a viscous flow about a fixed sphere in contact with a fixed plane wall when the fluid motion in the absence of the sphere is assumed to be a uniform linear shear flow. The values of the forces and couples which the fluid exerts on both the sphere and the wall are determined explicitly.

On the slow motion of a sphere parallel to a nearby plane wall

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a . An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that e = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α 0 +α 1 e) log e + β 0 + O (e) where α 0 , α 1 and β 0 are constants which have been determined explicitly.

On the hydrodynamic resistance to a particle of a dilute suspension when in the neighbourhood of a large obstacle

A method is presented for estimating the hydrodynamic force acting on a small particle of a dilute suspension when in a slow streaming motion past a large spherical or cylindrical obstacle for the situation when the particle is moving close to the obstacle. If ap and af denote respectively the radius of the particle and the obstacle, it is shown that the force tangential to the obstacle correct to O(ap/af) and the force normal to the obstacle correct to O(aP2/af2) can be evaluated by using a model in which the particle is in flow of semi-infinite fluid bounded by a rigid plane with the undisturbed flow at infinity determined from the first two non-zero terms in the expansion of the slow motion solution for the flow past the obstacle in a power series about the nearest point of the obstacle to the particle. The deviation of particles from undisturbed flow stream line is shown to be insignificant for particle separations from the obstacle exceeding 2 or 3 particle radii. The mechanical action on the sphere in a semi-infinite fluid, bounded by a plane, whose motion is a general second degree slow flow is also determined.

A slow motion of viscous liquid caused by the rotation of a solid sphere

A steady motion of incompressible viscous liquid caused by the slow rotation of a rigid sphere of radius a is considered. The medium is bounded by an infinite rigid plane and the axis of rotation is parallel to, and at a distance d from, this plane. To complete the analysis the solution by successive approximation of an infinite set of linear equations is required. Satisfactory solutions have been found numerically for four values of d / a , of which 1·13 is the smallest; we gratefully acknowledge valuable help from Miss S. M. Burrough in this part of the work.

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.

Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part I: The determination of exact solutions for any values of the ratio of radii and separation parameters

ZusammenfassungFür die Probleme der langsamen zähen Flüssigkeitsbewegung, die durch die Bewegung von zwei sphärischen Begrenzungen verursacht warden, sind exakte Lösungen angegeben worden. Die Bewegung der Kugeln kann entweder eine Translation entlang oder eine Rotation um die Achse sein, welche senkrecht zur Verbindungslinie der Mittelpunkte geht. Die Kugeln können sich ausserhalb von einander in einer unbegrenzten Flüssigkeit befinden, oder die Flüssigkeit kann zwischen zwei exzentrischen Kugeln eingeschlossen sein. Die Theorie ist anwendbar für alle Werte der Radienverhältnisse (k) und der minimalen Distanz (ε) zwischen den Kugeln.

Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II: Asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero

ZusammenfassungIn dieser Arbeit handelt es sich um eine Analyse der asymptotischen Formen für die Lösung von Problemen der langsamen zähen Strömung, die durch die Bewegung von zwei starren sphärischen Grenzen verursacht wird, wenn diese sich entweder entlang von Axen bewegen, die senkrecht zu der Verbindungslinie ihrer Zentren liegen, oder sich um diese drehen, Es wird der Grenzfall untersucht, bei dem der den Mindestspielraum zwischen den Sphären definierende Parameter ε sich Null nähert. Es wird gezeigt, dass für Kugeln, die ausserhalb einander liegen, die Kräfte und die Momente, die auf jede der beiden Kugeln einwirken, singulär werden, und zwar wie log ε für ε→0, für alle Werte der Radien. Wenn eine Kugel die andere einschliesst, findet man im allgemeinen dasselbe Verhalten, aber wenn eine stationäre Kugel eine drehende einschliesst, so verschwinden die Kräfte für ε»0, wenn der Radius der äusseren Kugel viermal so gross als der der inneren ist. Ein entsprechendes Verhalten zeigen auch die Momente, wenn sich die innere Kugel fortbewegt.

Electrohydrodynamic instability in plane layers of fluid

The instability of a plane layer of non-conducting fluid which is in hydrostatic equilibrium between two semi-infinite conducting fluids with surface charges is discussed for both inviscid and viscous fluid models. It is shown that for both the inviscid and viscous fluid cases, the criteria for instability are the same. Consideration is given to the relevance of the results in explaining the mechanism by which the presence of an electric field promotes more readily the coalescence of water droplets on a water surface by viewing the onset of disruption of the air film as the instability of the air film under the action of the electrostatic field produced by the surface charges on the water surfaces.

Asymmetric creeping motion of an open torus

This paper presents exact solutions using toroidal co-ordinates to the equations of creeping fluid motion with the no-slip boundary conditions for a toroidal particle translating in a direction normal to the axis of symmetry or rotating about an axis normal to the axis of symmetry through an otherwise infinite expanse of quiescent fluid. The associated resisting force and resisting torque are computed for toroids of various geometrical ratios b / a, b being the smallest radius of the open hole and ( b + 2 a ) being the radius to the outermost rim of the torus. These results are compared with approximate calculations based on slender-body theory and on the theory for interacting beads. The exact and approximate calculations become asymptotically equal as b / a becomes very large, but departures from the exact calculations are apparent for b / a less than 10−100 depending on the mode of motion and the method of approximation and the approximations are unreliable for b / a less than 2·0.

Two-dimensional Stokes flows with cylinders and line singularities

A study is made of Stokes flows in which a line rotlet or stokeslet is in the presence of a circular cylinder in a viscous fluid. In contrast to the Stokes Paradox for flow past an isolated cylinder, it is shown that if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity. A similar phenomenon can occur when two equal cylinders rotate with equal and opposite angular velocities, and the flow pattern is then such that there is a closed streamline enclosing both cylinders. §