extremely fast relative speeds (hundreds of body lengths s−1). However, when power
strokes are stopped, drag rapidly drops into the viscous range, with the advantage that
at rest the sinking rate is very slow despite a moderate excess of density over the
surrounding water. Little work is required to maintain a vertical location.
(^) That brings us to sinking rates. A cannonball that missed its target (most of them
did) would accelerate downward until inertial drag equaled the gravitational force
attracting it, and then it proceeded to the bottom at that substantial terminal velocity
(>100 m min−1). The effective mass would, of course, be reduced by buoyancy from
the water, the differential density determining the “effective” mass. Thus, a
sufficiently hollow, perhaps aluminum, cannonball might move up not down after
splashing in. The size of the cannonball makes only a miniscule difference. Sinking of
a tiny fecal pellet from a zooplankter, partly filled with dense opal from diatom shells,
will be affected primarily by viscous drag, and for a spherical fecal pellet the sinking
velocity, VS, is given by Stokes’s Law:
(^) in which g is gravitational acceleration, ρ
p and ρf are the densities respectively of the
pellet and the fluid, R is the pellet radius, and μ is the dynamic viscosity. A modest
difference from this depends upon the shape of the pellet, but use of an equivalent
diameter of a sphere of the same volume will give a decent approximation. Care with
units is required (!), but left to your attention. Notice that the larger the particle, the
faster it sinks, with VS varying with the square of the linear dimensions. If ρp is less
than ρf, then the particle will rise. Consider the impact, mentioned above, of
temperature on μ: a particle of ρp will sink about three times faster at 40°C than at
0°C, despite the effect of T on ρf. Stokes’s Law is a simplified (viscous drag only)
version of the Navier–Stokes’s equation, the version of Newton’s acceleration law F =
ma, to which hydrodynamicists have given lifetimes of thought and a googol (10^100 )
of computer calculations.
(^) Drag effects take on special characteristics at boundaries between water and solid
surfaces, even soft ones like jellyfish skin or algal cell membranes. The fluid
alongside, except in shear regimes strong enough to induce cavitation, remains stuck
to the surface, so that exactly at the surface there is no relative motion. This is the
“no-slip” condition. Velocity relative to the solid surface, the scales of a fish, say,
increases away from the surface, reaching the full background relative velocity
asymptotically at a considerable distance out. The zone of outward acceleration (Fig.
1.4) is called the fluid “boundary layer”. As the figure shows, very close to a surface,
up to about 1 cm but varying with the distant velocity, the local velocity increases
linearly outward because viscous effects are dominant, and in this range viscosity