Michael J. Miksis

SURFACE TENSION DRIVEN FLOWS.

Time-dependent potential flows of a liquid with a free surface are considered, with surface tension the force that drives them. Two types of configuration are analyzed, in each of which the flow and the free surface are self-similar at all times. One is a model of a breaking sheet of liquid. The other is a model of the flow near the intersection of the free surface of a liquid with a solid boundary. In both flows, the velocities are found to be proportional to

Dynamics of a drop in a constricted capillary tube

( \sigma /\rho t )^{1/3}

The effect of the contact line on droplet spreading

, where

Slip over rough and coated surfaces

\sigma

Periodic mass shedding of a retracting solid film step

is the surface tension,

Capillary instabilities in solid thin films: Lines

\rho

Experimental study of rivulet formation on an inclined plate by fluorescent imaging

is the liquid density and t is the time from the start of the motion. Each free surface is determined by converting the problem to an integrodifferential system of equations for the free surface and the potential on it. This system is discretized and solved numerically. On the resulting surfaces there are waves, which are also calculated analytically.

A level set projection model of lipid vesicles in general flows

Here we study the dynamics of a bubble or drop as it is driven by a pressure gradient through a capillary tube. For the case of a straight capillary, the drop can either approach a steady-state shape or the rear of the drop develops a re-entrant cavity. Also, depending on the initial conditions, the drop can break apart into smaller drops. For flow through a constricted capillary tube, depending on the physical parameters of the problem, the drop can either move through the constriction or break into two or more pieces as it moves past the constriction. We study this snap-off process numerically and determine the effect of the physical parameters on the dynamics of the drop.

Stability of a ridge of fluid

The motion of the free surface of a viscous droplet is investigated. By using lubrication theory a model is developed for the motion of the free surface which includes both the effect of slip and the dependence of the contact angle on the slip velocity. We solve the resulting nonlinear partial differential equation in several ways. First we investigate the initial motion of the drop at a non-equilibrium contact angle using the method of matched asymptotics. Then we develop a pseudo-spectral method to numerically solve the full nonlinear system. The dependence of the spreading rate of the drop on the various physical parameters and for different slip models is determined.

The effect of an electrostatic field on film flow down an inclined plane

We study the effect of surface roughness and coatings on fluid flow over a solid surface. In the limit of small-amplitude roughness and thin lubricating films we are able to derive asymptotically an effective slip boundary condition to replace the no-slip condition over the surface. When the film is absent, the result is a Navier slip condition in which the slip coefficient equals the average amplitude of the roughness. When a layer of a second fluid covers the surface and acts as a lubricating film, the slip coefficient contains a term which is proportional to the viscosity ratio of the two fluids and which depends on the dynamic interaction between the film and the fluid. Limiting cases are identified in which the film dynamics can be decoupled from the outer flow.