168 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
displacement ∆x(t)=ζ−^1
∫t
0 f(t′)dt′.Suppose we apply some forcef(t), moving the particle andall the surrounding fluid. We could bring the particle, and every other fluid element in the sample,
back to their original positions by any force whose integral is equal and opposite to the original
one. It doesn’t matter if the return stroke is hard and short, or gentle and long, as long as we stay
in the low Reynolds-number regime.
5.3.1′ The ratio of parallel to perpendicular drag is not a universal number, but instead depends
on the length of the rod relative to its diameter (the “aspect ratio”). The illustrative value 2/3
quoted above is for an ellipsoidal rod 20 times as long as its diameter. In the limit of an infinitely
long rod the ratio slowly falls to 1/2. The calculations can be found in (Happel & Brenner, 1983,
§§5-11).