Not to pick a nit, but would you care to explain what you think the connection is between the "continuum hypothesis" and "the nature of a fluid such as water"?
I may have the wrong term, it's been a while. But it has to do with the (mathematical) definition of a "fluid" for the purposes of fluid mechanics (which is probably the relevant mathematical "theory" here, not just "PDEs" as I had said.)
Scientist-type people want to model the behavior of (say) water. Water is a collection of a humongous number of particles (molecules). To simplify their predictions scientists want to use physical laws (i.e. Newton's F=ma law, conservation of mass..) to derive some kind of equations, which they can then solve. The mathematical solutions will then predict the behavior of quantities such as the "density" or "velocity" of the water, at various physical points.
Let's stick with "density". How do you go about defining the "density" of water at such-and-such location in the Pacific Ocean? You take a big box, say ten miles on a side; you count the particles in it; and divide by the volume of the box. Too coarse, too inaccurate? Fine, make the box smaller, a mile on a side. Or a meter. It seems like you're getting somewhere. The smaller the box, the better; your "density" measurement seems to settle on a definite number; what's better, you can get usable "density" measurements at a higher and higher number of points which are closer and closer together.
The problem is, this can't continue forever, because water isn't actually a (let's call it) "continuum". It's just particles. So after some point, your box is so small that it contains only one molecule of water (if the box happens to be centered where a water molecule is), or zero molecules (more likely). One or the other. After this point your "density" measurement becomes junk; at each location of the ocean the "density" is either zero or infinite, it's just a bunch of delta function spikes. This makes it hard to write down equations for it and predict things.
So what is done is, some kind of assumption is (necessarily) made that it's valid to study an abstract "fluid" which has smoothly-varying properties such as density, velocity, etc at every point of space. This is not actually true of actual fluids in the real world, but it is assumed that it's valid to neglect the fluctuations etc at the molecular level. So, you imagine measuring "density" etc with boxes not vanishingly small, but which are "not too small" - in some intermediate range, and endowing a (mathematical) "fluid" with those properties. Then you can write down all the PDEs you wish, but you have to hope that the "continuum assumption" you've made in the process doesn't make your model fatally inaccurate, when you try to connect your predictions back to the real world. This is a thing that has to be checked, and sometimes the "continuum" assumption is not valid.
I probably haven't explained very well but that's about the best I can do for a 400-post thread ;-)