5.3. Biological applications[[Student version, December 8, 2002]] 153
abc
Figure 5.5:(Schematic.) Three swimmers. (a)The flapper makes reciprocal motion. (b)The twirler cranks a stiff
helical rod. (c)The spinner swings a stiff, straight rod.
In short, for solids the stress is proportional to the strain (∆z)/d,while for fluids it’s proportional
to the strainrate,ddt
(∆z
d)
=vd.Asimple elastic solid doesn’t care about the rate; you can shift the
plates and then hold them stationary, and an elastic solid will continue resisting forever. Fluids, in
contrast, have no memory of their initial configuration; they only notice how fast you’rechanging
that configuration.
The difference is one of symmetry: In each case if we reverse the applied distortion spatially,
the opposing force also reverses. But for fluids, if wetime-reversethe distortion ∆z(t)then the
force reverses direction, whereas for solids it doesn’t. The equation of motion for distortion of an
elastic solid is time-reversal invariant, a signal that there’s no dissipation.
T 2 Section 5.2.3′on page 167 describes an extension of the above ideas to materials with both
viscous and elastic behavior.
5.3 Biological applications
Section 5.2.3 brought us close to the idea of entropy, promised in Chapter 1. Entropy will measure
preciselywhatis increasing irreversibly in a dissipative process like diffusion. Before we finally
define it in Chapter 6, the next section will give some immediate consequences of these ideas, in
the world of swimming bacteria.
5.3.1 Swimming and pumping
Wesaw in Section 5.2.1 on page 146 that in the low Reynolds-number world, applying a force to
fluid generates a motion that can be cancelled completely by applying minus the time-reversed force.
These results may be amusing to us, but they are matters of life and death to microorganisms.
An organism suspended in water may find it advantageous to swim about. It can only do so by
changing the shape of its body in some periodic way. It’s not as simple as it may seem. Suppose
youflap a paddle, then bring it back to its original position by the same path (Figure 5.5a). You
then look around and discover that you have made no net progress, just as every fluid element
returned to its original position in the unstirring experiment (Figure 5.2). A more detailed example
can help make this clearer.
Consider an imaginary microorganism, trying to swim by pushing a part of its body (“paddles”)
relative to the rest (“body”) (see Figure 5.6). To simplify the math, we’ll suppose that the creature
can only move in one direction, and the relative motion of paddles and body also lies in the same
direction. The surrounding fluid is at rest. We know that in low Reynolds-number motion, moving