Abstract (click to view)
In this talk I shall describe two rather different, but not entirely
unrelated, problems involving thin-film flow of a viscous fluid which
I have found of interest and which may have some application to a
number of practical situations, including condensation in heat
exchangers and microfluidics.
The first problem, which is joint work
with Adam Leslie and Brian Duffy at the University of Strathclyde,
concerns the steady three-dimensional flow of a thin, slowly varying
ring of fluid on either the outside or the inside of a uniformly
rotating large horizontal cylinder. Specifically, we study "full-ring"
solutions, corresponding to a ring of continuous, finite and non-zero
thickness that extends all the way around the cylinder. These
full-ring solutions may be thought of as a three-dimensional
generalisation of the "full-film" solutions described by Moffatt
(1977) for the corresponding two-dimensional problem. We describe the
behaviour of both the critical and non-critical full-ring
solutions. In particular, we show that, while for most values of the
rotation speed and the load the azimuthal velocity is in the same
direction as the rotation of the cylinder, there is a region of
parameter space close to the critical solution for sufficiently small
rotation speed in which backflow occurs in a small region on the
upward-moving side of the cylinder.
The second problem, which is
joint work with Phil Trinh and Howard Stone at Princeton University,
concerns a rigid plate moving steadily on the free surface of a thin
film of fluid. Specifically, we study two problems involving a rigid
flat (but not, in general, horizontal) plate: the pinned problem, in
which the upstream end of plate is pinned at a fixed position, the
fluid pressure at the upstream end of the plate takes a prescribed
value and there is a free surface downstream of the plate, and the
free problem, in which the plate is freely floating and there are free
surfaces both upstream and downstream of the plate. For both problems,
the motion of the fluid and the position of the plate (and, in
particular, its angle of tilt to the horizontal) depend in a
non-trivial manner on the competing effects of the relative motion of
the plate and the substrate, the surface tension of the free surface,
and of the viscosity of the fluid, together with the value of the
prescribed pressure in the pinned case. Specifically, for the pinned
problem we show that, depending on the value of an appropriately
defined capillary number and on the value of the prescribed fluid
pressure, there can be either none, one, two or three equilibrium
solutions with non-zero tilt angle. Furthermore, for the free problem
we show that the solutions with a horizontal plate (i.e. zero tilt
angle) conjectured by Moriarty and Terrill (1996) do not, in general,
exist, and in fact there is a unique equilibrium solution with, in
general, a non-zero tilt angle for all values of the capillary
number. Finally, if time permits some preliminary results for an
elastic plate will be presented.