Brouwer's fixed point theorem in a practical setting If we assume that a fluid is a continuum then if we have for example a cup of tea and we stir the fluid then there will be a point in the fluid that is on the same location before and after the stirring.
Now, a practical fluid is not a continuum although it is for many practical situation in fluid mechanics.
Does this influence the statement much? Is there for example a small box where the particle will be in (which is somewhat in the same magnitude as the size of the particles) or does the complete statement break down?
 A: You can substitute the discretized version of the Brouwer fixed point theorem (which can be used to prove the usual version): Sperner's lemma. This isn't a complete answer but it should point the way. I believe the idea is to use colors to indicate the possible directions a particle travels, although I haven't thought about the details. 
A: I imagine that after stirring a cup of ideal tea twenty or thirty times, some points that were initially less than an atomic radius apart will become separated by more than half the diameter of the cup. Which makes Brouwer's fixed-point theorem physically meaningless.
A: Brouwers fixed point theorem doesn't apply practically or even theoretically at all for the simple reason that stirring is not a continuous transformation.
For instance if you stirred vigourously and a drop of liquid jumped out and then fell back in, it's no longer continuous. If some liquid sticks to the spoon as you withdraw it then nope.
The cup of water is topologically equivalent to a sphere, and any transformation to a non-equivalent volume is discontinuous. If the spoon touches the bottom, then now the fluid is equivalent to a torus temporarily and nope. If you suck down a single air bubble, or as you swirl a vortex forms which collapses on itself then nope.
If you were to carefully perform some action in a way to guarantee continuity then I doubt anyone would identify that action with 'stirring'. 
