Scallop theorem

The Scallop theorem states that to achieve propulsion at low Reynolds number in Newtonian fluids a swimmer must deform in a way that is not invariant under time-reversal. Edward Mills Purcell stated this theorem in his 1977 paper Life at Low Reynolds Number explaining physical principles of aquatic locomotion. The theorem is named for the motion of a scallop - an opening and closing of a simple hinge - which is not sufficient to create migration at low Reynolds numbers.

Although the movement of animal cells is usually studied as they migrate, it seems likely that many motile cells can also swim. Thus, human granulocytes are able to migrate towards a source of a chemoattractant, the tripeptide FMLP, whilst suspended in a uniformly-dense (isodense) medium. They swim at the same speed as they would crawl on a solid surface. Likewise, Dictyostelium discoideum amoebae swim towards a chemical attractant, in this case cyclic AMP. The actual mechanism that these neutrophils or amoebae use to produce a thrust against the medium to propel themselves is uncertain; however, how they do so must be consistent with physical principles. To swim they must transmit a force against the viscous fluid in order to propel themselves forward. Different mechanisms by which they might do so were presented by Ed Purcell in a famous talk he gave celebrating the 80th birthday of his friend Viki Weisskopf.

In this he developed his “scallop theorem”: a normal scallop moves by opening its shells slowly and shutting them quickly. In the latter step it quickly squeezes the fluid between the shells backwards and, using the momentum of the water, pushes itself forward. Purcell realised that a microorganism trying to do the same would simply move forwards on shutting its shells and move backwards to its original position on opening them. The set of movements is “reciprocal”: it appears the same if viewed forwards or backwards in time. He concluded that microorganisms cannot move by a reciprocal mechanism: to move, they must exert some thrust against the medium and do so in a non-reciprocal manner. He suggested various ways in which an organism could swim:

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