H. P. Greenspan

On the motion of a small viscous droplet that wets a surface

A model for the movement of a small viscous droplet on a surface is constructed that is based on the lubrication equations and uses the dynamic contact angle to describe the forces acting on the fluid at the contact line. The problems analysed are: the spreading or retraction of a circular droplet; the advance of a thin two-dimensional layer; the creeping of a droplet or cell on a coated surface to a region of greater adhesion; the distortion of droplet shape owing to surface contamination. Relevant biological problems concerning cell movement and adhesion are described.

A simple laboratory model for the oceanic circulation

A linear theory is developed for the motion of a viscous, incompressible fluid in a rotating cylinder with a sloping bottom. An analysis of the normal modes of oscillation reveals that the presence of the bottom slope introduces a new set of low frequency inertial oscillations to replace the purely geostrophic modes which are not allowed in this geometry. The new waves possess mean circulation and are the mechanism by which the fluid adjusts to changes in the rotation rate of the container, a process discussed in detail. The steady motion produced in the cylinder when the cylinders upper surface rotates at a different rate than the bottom surface is studied. It is shown that the presence of the bottom slope inhibits the steady fluid motion in the body of the cylinder and introduces a non-symmetric, high velocity side wall boundary layer. Experimental evidence, presented to validate the theory, reproduces certain important features of the oceanic circulation.

On the general theory of contained rotating fluid motions

A general linear theory is developed to describe the manner in which rigid fluid rotation is established from a prescribed initial state of motion in a container of arbitrary shape. The container rotates with uniform angular velocity and is filled with a viscous incompressible fluid. A new mean circulation theorem is proved and used to separate the flow into geostrophic motion and inertial oscillations. The basic eigenvalue problem is studied and important properties concerning spectrum, orthogonality and completeness are deduced. The effect of viscosity on the inviscid modes is calculated in a manner that maintains the solution uniformly valid through the spin-up time. All modes decay in this time scale which characterizes the entire transition to rigid rotation in all configurations.

On hindered settling of particles of different sizes

Abstract The flow of a non-dilute fluid suspension is considered in which the dispersed phase consists of particles or droplets of different sizes. A phenomenological two-phase flow theory is formulated for both continuous and discrete distributions of particle sizes and illustrated by considering the batch settling of such a mixture. The volume fractions and particle distribution functions are determined, as well as the composition of the sedimentary layer.

On the expansion of a gas into vacuum

Abstract : A study is made of the flow into vacuum of a gas initially at rest in a state of uniform pressure and density; the analysis is based on a continuum model. Among the topics discussed are the motion of the gasvacuum interface, the reflexion of a plane front off a rigid wall, the propagation of compressive waves within such expansions, the escape from a sphere and the collapse of a spherical cavity.

Flow over a containment dyke

The wall of a large tank or reservoir breaks, sending fluid against a secondary containment dyke. The impact of the surging fluid against the safety barrier is studied. The results of theoretical analysis and numerical simulation (for vertical dykes) are in good agreement with experimental data concerning overflow and total spillage as well as the fluid motion after collision, including the development and formation of a strong shock. The dependence of spillage on the inclination of the dyke is also determined by experiment.

On the centrifugal separation of a bulk mixture

Abstract The centrifugal separation of a mixture of particles and fluid in an axisymmetric container is examined. The flow consists of three distinct regions—mixture, sediment and purified fluid—with Ekman boundary layers at the interfaces and walls. In the settling process, the mixture and pure fluid acquire retrograde and prograde rotations relative to the tank. This flow pattern, and the shape and locus of the interface which are easily determined, provide another simple means to compare mixture theory and experiment. It is shown that when the Coriolis force is important, the pure fluid layer on the “outwardly” inclined wall is not thin. Moreover the interface between the mixture and the pure fluid is not perpendicular to the centrifugal force. Both features contrast those of the gravitational Boycott effect. As a consequence, there is no obvious enhancement of settling due to geometrical configuration.

On the enhancement of centrifugal separation

We consider the two-phase flow of a suspension in a rotating cylinder with inclined endplates for which inertial and viscous effects are small. It is shown that, when the Coriolis force is dominant, flow in the core is essentially unaffected by geometry. If a fluid particle can make a complete circuit about the rotation axis, the sedimentation velocity cannot be augmented by geometrical effects as it can in gravitational settling. However, with the insertion of a complete meridional barrier to block movement around the centre, separation becomes more sensitive to the shape of the container walls. In this case, behaviour similar to that in a gravitational field is possible once again.

On a rotational flow disturbed by gravity

We examine a rapidly rotating flow that exhibits periodic vortex detachment. Specifically, the rotation/symmetry axis of a fluid-filled cylinder is set perpendicular to gravity. A free buoyant cylindrical float placed within the container is acted upon by both centrifugal and gravitational forces, the competition of which causes fluid motion and, in certain parameter ranges, flow instability. The motion is determined, a criterion for separation is advanced, preliminary experiments and data are described and the relationship of this phenomenon to other examples of vortex shedding in rotating fluids is discussed.

On centrifugal separation of particles of two different sizes

Abstract We consider flow in a centrifugal force field of a non-dilute suspension with particles or droplets of two sizes. The volume fraction and the velocity fields are determined assuming small convection and shear terms. The resulting flow field is quite different from that in a gravitational settling of a similar mixture. In particular, the volume fraction is a function of time and radius in the sectors separated by kinematic shocks and the settling velocity is a non-monotonic function of the particle size.