It is often difficult to see how the sequence of first and second year "core" courses fits in with upper year material. In the case of continuum and fluid mechanics this seems to be especially true. Much of this has to do with the fact that while the calculus sequence begins, for the sake of simplicity, with functions of one variable, and only moves onto the calculus on vectors in the course AMATH 231 (which a small percentage of math faculty students take), the study of fluids is unthinkable without vectors. Indeed anyone who has marveled at the eddies and bubbles at the bottom of a waterfall understands implicitly that the flow of fluids is generally irregular with flow at a variety of speeds and in a variety of directions, with changes from point to point in space as well as instant to instant in time. The fluid velocity is thus a vector function of both the three coordinates of space and time. Any statement (think of Newton's second law in particle mechanics) expressing the change in fluid properties (like temperature or density) at a point in space will have to include some way of accounting for the fact that a flowing fluid transports its properties downstream. Carrying this out in any mathematically reasonable sense requires bits and pieces from the entire calculus sequence. In some sense the story is complicated further by the fact that once the governing equations are derived they are nonlinear, and hence nearly impossible to solve. Thus fluid mechanics is in some sense the modern applied discipline, with progress inextricably tied to computation. With these thoughts in mind, we have put together a look at some general themes in the study of continuum and fluid mechanics.
Of course being so closely tied to computation both fluid and continuum mechanics are inherently visual fields. After all these are the branches of physics that cover all the length scales actually related to our lives. Some pictures and videos of different fluid experiments can be viewed here.