through water by large, dense objects and to swimming forward by larger, faster
animals. There is another source of drag, which is the requirement for force to
rearrange the intermolecular connections among water molecules in order to move
through them. That is called viscous drag. The relative importance of inertial and
viscous drag is expressed as a ratio, the Reynolds number, Re, which has the product
of factors proportional to inertial drag in the numerator and the water viscosity in the
denominator. Inertial drag is proportional to the linear size (l) of the moving body,
often best chosen as the longest dimension perpendicular to the path, to the velocity
(v) relative to the water, and to the water density (ρ): lvρ. The viscosity (as discussed
here, the dynamic viscosity) is the molecular resistance to shearing forces, symbolized
μ (or often η), with SI units of Pascal·s (Pa·s) = Newton·s m−2. Work with the units
here. After some conversions, it will become apparent that those of the Reynold’s
number numerator (m, m s−1 and kg m−3) and denominator cancel. Re is a
dimensionless number.
(^) Experimental work (also some theory) shows that at high Re, >∼100, viscosity can
be neglected in drag calculations, at least for processes like swimming, because
inertial effects are so dominant. At Re less than ∼1, inertial effects are small and
viscous effects dominate. Algal cells, other protists, and many smaller metazoans like
clam larvae or copepod nauplii, live in an apparently very viscous world, because
both their l and v values are small. This has important effects on the mechanics of
swimming and of approaching nearby food particles. The viscosity of water (and
seawater, the effect of salt is small) varies not quite linearly with temperature, from
∼0.65 mPa·s (milliPascal·s) at 40°C to ∼1.8 mPa·s at 0°C. This difference
approximately triples the work that ciliated or flagellated protists must do to move at
0°C compared to 40°C.
(^) So, how does swimming work? When drag is principally inertial, so is the force
exertion of an animal against water. A fin or tail sweeps through the water at an angle
to the intended trajectory, and pushes a mass of water backward. There is an equal
reaction on the mass of the fish or seal moving it forward. There are often elegant
details. For example, a tuna that can swim at ∼20 m s−1 has an ideal fusiform shape,
minimizing the distance that water must be accelerated to the side and then back to
the center line behind as it passes through the water. It has scales along its tail
peduncle that lie flat during initial acceleration, and then extend out to initiate
turbulence at intermediate speeds. That is useful because drag actually drops
substantially at the transition from smooth laminar flow along the skin surface to
turbulent churning. To avoid drag from laterally extended pectoral and dorsal fins,
tunas can pull them flat against the body into precisely fitted grooves.
(^) Swimming by ciliated and flagellated cells is substantially different (Purcell 1977).
Mass moved behind by a flagellum stroke is so small that the forward reaction is
ineffective. But motion is achieved from the differential in the amount of