A model microorganism swims at Reynolds number using a single rigid paddle hinged to…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluid Dynamics, Ch Stokes flow, lubrication and viscous wetting

Problem

A model microorganism swims at Reynolds number

Re\sim10^{-4}

using a single rigid paddle hinged to its body. One stroke consists of the paddle sweeping through the same set of angles

\phi\in[0,\phi_{\max}]

, first opening (taking time

t_1

) and then closing by exactly retracing the same sequence of shapes in reverse (taking a different time

t_2\neq t_1

). ((a)) Using the linearity and instantaneity of the Stokes equations, show that the body's instantaneous swimming speed can be written

U(t) = A(\phi(t))\,\dot\phi(t)

for some function

A

that depends only on the shape

\phi

, and use this to prove that the net displacement of the body over one complete stroke (opening plus closing) is zero, whatever the values of

t_1

and

t_2

. ((b)) State what feature of the stroke, not merely its timing, would have to change for the organism to swim.

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