(1) first introduce the history of the subject, what the current status of this subject is ( so you need to read all classic literatures and all the most recent literatures on the topic).
(2) state clearly the problems that you are going to work on, are these problems open problems? Why are these problems important? and what results you expect to establish.
(3) what tools, methods, and theory you are going to employ to achieve the expected result.
I have some topics that I choose
1- Rigorous results in existence and selection of
Saffman–Taylor fingers by kinetic undercooling
2- ANALYTIC SOLUTION TO AN INTERFACIAL FLOW WITH
KINETIC UNDERCOOLING IN A TIME-DEPENDENT GAP
HELE-SHAW CELL
Please read the two topics and there are open problems mentioned in the papers. Those open problems could be my dissertation topics .
Alos, there are some classic literature on Hele-Shaw flows
1-Complex variable methods in Hele-Shaw moving
boundary problems.
2-The penetration of a fluid into a porous medium
or Hele-Shaw cell containing a more
viscous liquid.
3-The effect of surface tension on the shape
of fingers in a Hele-Shaw cell.
4-The effect of nonzero viscosity ratio on the stability of fingers and bubbles in a Hele-Shaw cell.
5-Rigorous Results in Steady Finger Selectionin Viscous Fingering.
6-Analyticity and Nonexistence of Classical Steady Hele-Shaw Fingers.