Hi all! I defended my Ph.D. thesis back in 2019 and I also served as the creator and moderator for the subreddit r/FluidMechanics for a long time. I think with that I have gathered enough experience and courage to answer some of your queries. Some broad topics that I can answer questions on are:
computation fluid mechanics
scientific programming and HPC
nonlinear shallow water equations
statistical description of turbulence: spectra, energy budget etc.
It is a discreet thing if you consider pure materials. Phase transitions are sharp for all materials.
Non-newtonian fluids are usually a mixture of fine solid particles and liquid creating funky physics. Stress and shear rate are not linear as Newton's law dictates. They can be shear thickening like corn starch in water or shear thining like blood. Shear thinning fluids are considered pseudo plastics, which is also a property in some solids.