Archimedes of Syracuse is widely regarded as one of the three most brilliant men of history, with Sir Isaac Newton and Carl Friedrich Gauss as the others. All three would be considered applied mathematicians today, though all three are also claimed as physicists (by physicists) and pure mathematicians (by pure mathematicians). But I think we have the best claim, not that the modern distinction would have meant much to them.
Archimedes is perhaps most popularly known for his discovery, also involving fluids, of how to decide if his cousin the King's crown was made of pure gold or an imitation alloy. Whilst Archimedes was bathing, he realized that the volume of water displaced when an object is immersed is measurable, which allows one to measure the density of an irregularly shaped object, thus allowing its purity to be assessed. It is said that he leapt naked from the bath, shouting "Eureka!" (meaning `I have found it!' and not, as Terry Pratchett maintains, `Someone get me a towel!') and ran to tell of his discovery.
Another story told, again a success of applied mathematics, is that Archimedes told his cousin that using his newly-invented pulleys he could beach a loaded galley with one hand. The King decided to test this, and ordered Archimedes to prove it. He did, beaching a galley loaded with goods and slaves. Thereafter, his cousin the King passed a law requiring that whatever Archimedes said, he was to be believed.
But fluid mechanics has come a long way since Archimedes. His principle still keeps ships afloat, but now we have hypersonic flight to deal with, with space shuttles doing Mach 25 and other highly maneuverable craft scarcely less. A whole new science, of fluids in motion, was invented in the seventeen to eighteen hundreds, with refinements crucial to flight added in our own century. It is not widely appreciated that the Wright brothers used mathematics to design their wing---without it, they would not have got off the ground.
The problem of fluids in motion is still not solved; we believe we have good equations describing the motion of fluids, the so-called Navier-Stokes equations (though some would claim that the Ladyzhenskya equations are better). But we can't solve them, except using scientific computing, and even then the interesting and practical case of turbulence still poses daunting problems.
At least one of our department members, Paul Sullivan, is a recognized expert on turbulence, and in particular on the problem of turbulent dispersion of contaminants. See turbulent dispersion for details.