= Scallop theorem
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= Purcell scallop theorem
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A <force-free>, <torque-free> <microswimmer> in a stationary <Newtonian fluid>, with negligible fluid and body <inertia> and steady <Stokes flow> at each instant, gains no net displacement from a reciprocal stroke: one whose sequence of shapes is exactly retraced in reverse. The environment and boundary conditions must also be unchanged during that retracing. Opening a two-shell swimmer and then closing it follows the same sequence backwards; doing the closure faster does not help, because <Linearity of Stokes flow> changes the instantaneous <velocity> by the same factor as the stroke rate.
A single real shape coordinate that travels out and back along an interval gives this cancellation, as computed in the <force-free two-sphere stroke>. Two independently controlled shape coordinates can instead execute a loop that is not retraced, avoiding this obstruction. Continuous rotation also need not retrace an interval: the <helical microswimmer with a spherical head> provides another route to propulsion. The <scallop theorem> is a restriction on a stroke under stated conditions, not a claim that small organisms cannot swim. Purcell presented the opening-and-closing scallop and a swimmer with two hinges in https://www.math.uwaterloo.ca/~mdunphy/purcell.pdf[Life at low Reynolds number (1977), figures 6–7].
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