Nothing in mathematics of fluid mechanics is more challenging than the NS problem: http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation

Hi

http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation

You can find many references searching in internet for phd maths in the area of fluid mechanics

You will find it hard to be successful if you do not have a thesis advisor with experience in the area. If you do, you need to talk to him or her and trust their intuition and experience in what is a good problem in the thin area between the trivial and the impossible and that fits your background and skills.

I can tell you how simple maths solved a 200 year old problem in heat exchange between fluids. It is simple sets, but it worked.

The issue was to move heat into a pressure vessel to influence a working fluid (read gas). For 200/300 years, the heat is brought into the vessel by conduction through the wall. With high pressure, the wall can get thick and limit the possibilities.

So we put a common denominator both sides of the wall, liquid, we then have an instance of liquid inside the wall and an instance of liquid outside the wall.

All we needed to do then to move heat into the pressure vessel, is to swap hot liquid in for cold liquid out. But the two instances of liquid are at different pressures.

So using sets, we introduce a third set of liquid, and the third set swaps liquid with each of the other two to provide a nett result. There is a little decompression, but easily fixed.

So common denominator and sets solved the 200 year old problem. This short video shows it clearly. https://youtu.be/phGaFavF5GQ

enjoy.

Tnx a lot sir

i will try to study