Navier-Stokes equation describes the evolutionary law of fluid velocity. Essentially, it is a special form of Newton's second law. But why not position here? People don't care about trajectories of particles or what? I have seen Wikipedia explain some things but it lacked many details. For example, it says velocity makes more sense for a fluid. Then in what sense?
It is more to the type of approach, either Eularian or Langrangian... If you understand these approaches, than for sure you might understand the answer.
If you use the particle finite element method (PFEM) then you can capture both. The issue hear lies within the boundary conditions and the number of particles that you will assume to discretize your fluid domain. Through this approach you can also include solid bodies within your fluid domain. Of course as all numerical methods it is approximative and has its disadvantages.
http://www.cimne.com/pfem/whatispfem.asp
It is a good observation by Chang. The reason might be as follows: if you start with a fluid element, then this fluid element will usually be stretched and compressed endlessly and what started as a small element will evolve into a something that gets contorted. It is then very difficult to say where the element is: it "diffused" (not to be taken too literally) over a large piece of space.
I see it in this way.
Positions are important in solid mechanics. If you load e.g. a beam, its various point move from a certain position to a new one (deformation) and you write the balance equations by using displacements (in Hook's law, displacements are involved).
In fluid mechanics, time comes into play, so you are not interested in deformation but rather in deformation rates (simply think of Newton's law that links shear rates to shear stresses) so to write the balance equations you have to deal with velocities.
Naturally, once you solve your equations obtaining velocities, you may integrate your velocity field to compute your position as a function of time.
marco's answer is very colourful. Indeed, paths of fluidelements are not usually important (if so, you use so-called marke rparticles in your numerical computations to see where they are going). But you are generally interested in the state of change of bulk quantities at given locations, hence concepts as density, pressure and.....velocity are the ones you want to know. With or without Italian or Dutch soccer :-)
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