5.3. Biological applications[[Student version, December 8, 2002]] 155
effective stroke recovery stroke123
45 10 9876Figure 5.7:(Schematic.) The ciliary cycle. The effective stroke (left) alternates with the recovery stroke (right).
The motion is not reciprocal, so it can make net progress in sweeping fluid past the surface.
must last twice as long as the power stroke in order to prepare the creature for
another cycle!So a strictly reciprocating motion won’t work for swimming in the low-Reynolds world. What
other options does a microorganism have? The required motion must be periodic, so that it can
berepeated. It just can’t be of the reciprocal (out-and-back) type described above. Here are two
examples.
Ciliary propulsion Many cells use cilia, whiplike appendages 5–10μmlong and 200nmin diam-
eter, to generate net thrust. Motile cells (likeParamecium)use cilia to move, while others use them
to pump fluid (like the cells lining our air passages) or sweep food to themselves (see Figure 2.12
on page 40).
Each cilium contains internal filaments and motors which can slide the filaments across each
other, creating an overall bend in the cilium. The motion in Figure 5.7 is typical. Indeed this
motion is periodic but not reciprocal. To see how it generates propulsion, we need one intuitive
result from low-Reynolds fluid mechanics, whose mathematical proof is however beyond the scope
of this book:The viscous friction coefficientζ‖for motion of a rod parallel to its axis is smaller
than the oneζ⊥for perpendicular motion.The exact factor depends on the length of the rod; this
book will use the illustrative value 2/3.
In other words, a rod dragged along its axis at velocityvfeels a resisting force proportional to
−v,that is, also directed along the axis. A rod dragged perpendicular to its axis feels a resisting
force also proportional to−v,that is, directed perpendicular to the axis, but with a somewhat
larger constant of proportionality.
Figure 5.7 shows a cilium initially lying parallel to the cell surface, pointing to the left. During
the power stroke (left panel) the entire cilium moves perpendicular to its axis, whereas during the
recovery stroke (right panel) most of it is moving nearly parallel to its axis. Thus the motion of the
fluid created by the power stroke gets only partly undone by the backflow created by the recovery
stroke. The difference between these flows is the net pumping of one cycle.
Bacterial flagella What if the speedvis neither parallel nor perpendicular to the axis, but in
between? In this case Figure 5.8 shows that the resulting drag force will also be somewhere in
between the parallel and normal directions, butnot alongv.Instead the force points closer to the
normal than does the velocity; the largerζ⊥“wins” over the smallerζ‖. The bacteriumE. coli