Modelling phase change in the presence of a flowing thin liquid film
There are numerous physical phenomena that involve a melting solid
surrounded by a thin layer of liquid, or alternatively a solid
forming from a thin liquid layer. This talk will involve two such
problems, namely contact melting and the Leidenfrost phenomenon.
Contact melting occurs, for example, when a solid is placed on a
surface that is maintained at a temperature above the solid melting
temperature. Consequently the solid melts, while the melt layer is
squeezed out from under the solid due to its weight. This process
has applications in metallurgy, geology and nuclear technology, and
also describes a piece of ice melting on a table. Leidenfrost is
similar, but involves a liquid droplet evaporating after being
placed on a hot substrate. This has applications in cooling systems
and combustion of fuel or a drop of water on a hot frying pan.
The talk will begin with a brief introduction into one-dimensional
Stefan problems before moving on to the problem of melting coupled
to flow. Mathematical models will be developed, analysed and
compared with experimental results. Along the way the Heat Balance
Integral Method (HBIM) will be introduced. This is a well-known
method primarily used by engineers to approximate the solution of
thermal problems. However, it has not proved so popular with
mathematicians, due to the arbitrary choice of approximating
function and a lack of accuracy. The method will be demonstrated on
a simple example, then it will be shown how it may be modified to
significantly improve the accuracy. In fact, in the large Stefan
number limit the modified method can be shown to be more accurate
than the asymptotic solution to second order.