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.
Part of this work was undertaken while I was a
Visiting Fellow in the Department of Mechanical and Aerospace Engineering
in the School of Engineering and Applied Science at Princeton University, Princeton, USA.
Another part of this work was undertaken while I was a
Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM),
University of Oxford, United Kingdom.
This publication was based on work supported in part by Award No KUK-C1-013-04,
made by King Abdullah University of Science and Technology (KAUST).