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Topics for PG Research

This section is for postgraduate students who are contemplating the
idea of becoming researchers in one of the areas of applied mathematics
and, ideally, would like not just to obtain a PhD and happily forget about
the whole ordeal but have long-term research plans or the desire to acquire
such plans. For such students the message is: The main outcome of the postgraduate
study period is not a PhD thesis; a PhD thesis is a natural by-product.
The main outcome, which is much less palpable and by far more difficult
to achieve, should be the development of the research ideology and the
understanding of what the meaning of science is going to be for your research
lifetime. In achieving these goals, your supervisor's views and expertise
can give you at most only the starting point. However, as you already know
from your numerical exercises, the starting point is important for the
subsequent iterations to converge...
Given the above remark, it is obvious that the direction of research
can be decided upon only after an interview with the candidate in
order to take into account both the candidate's views and plans for the
future as well as the supervisor's expertise. The topics listed below should
be seen as examples.

For convenience we will divide the topics into `regular' and `advanced'.
Clearly, there is no border between the two groups, and the distinction
has been made on the basis of the following, rather subjective, criterion.
In the case of a regular topic, one has a well-defined starting point for
research and several options as to how to proceed (and, possibly, by-pass
some difficulties of principle). Advanced topics are more challenging and
it is the difficulty of principle which is their main target. These projects
require deep understanding of/interest in the key issues of modelling
and the ultimate success depends more on the input of original ideas rather
than on command of standard mathematical techniques. If you need more information
either about any of the topics or would like to know more about any
general aspect of research, you are welcome to contact me at any time.

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(R1) Non-axisymmetric interactions of drops with solid boundaries

The project is dealing with the hysteresis of the (dynamic) contact angle
and its influence on the spreading of a finite volume of fluid on a solid
surface. The results could give a macroscopic way of testing mathematical
models of dynamic wetting in different regimes. A possible line of research
is also the rivulet formation and some stability issues.
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(R2) Oscillating liquid jet

The method of jet vibration proposed by Bohr in 1909 is used for measuring
the surface tension of a newly formed free surface. It has the highest
time resolution among the existing methods and became an indispensable
tool in colloid science. However, there is a significant gap in its mathematical
foundation which is currently `patched' by ad hoc `calibration'. This procedure
raises serious doubts about many results in the area. The aim of the project
is to remove the existing weakness of the method by providing a self-consistent
mathematical description of the jet near the orifice.
####
(R3) Topological transitions in bubble dynamics

The problem of capillary breakup of a bubble is one of the most important
in dynamics of bubbly liquids and some other multiphase flows (e.g. bubble
boiling). The topic is labelled `regular' here thanks to recent advances
in the general problem of coalescence/breakup, which give the framework
- and hence the `starting point' - for the study of bubble dynamics though
putting this topic among `advanced' would be equally justifiable.
####
(R-A) Flows with topological transitions of flow domains

This wide class of flows includes coalescence of drops and bubbles, breakup of liquid jets, rupture of free liquid sheets and films on a solid substrate and many other fluid motions. The difficulty here is that the topological transition treated in the framework of conventional fluid mechanics is associated with singularities in the corresponding solutions which makes these solutions unacceptable from the physical point of view and results in discrepancies between theoretical results and experimental observations. The goal of the project, which has industrial connections, is to develop a physically-satisfactory systematic way of modelling such flows.
####
(A1) Formation of free-surface cusps in unsteady flows

The problem of resolving singularities is essentially
that of incorporating `extra' physics missing in the original formulation.
The question of how the singularities evolve in a finite time and how this
extra physics should be `switched on' is of general interest, and the proposed
topic considers a particular case where this question is likely to be solvable.
####
(A2) Dynamic wetting of rough/inhomogeneous surfaces

The main difficulty is associated with describing randomly rough/inhomogeneous
solid surface from the viewpoint of the process of dynamic wetting and
developing a self-consistent way of modelling this process macroscopically.
The problem has much in common with the problem of averaging in mechanics
of multiphase systems. The topic is of considerable importance in a number
of industrial applications.