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GEOS 240: Solving the Scallop Theorem.

In the previous post it was mentioned that organisms living within a low Reynolds number (Re) should not be able to swim. This is because the viscous forces dominate the fluid, and inertia is reduced to practically nothing. Any reciprocal motion results in no net motion. If an organism is to swim in these conditions, then their method of locomotion must be different than the motions humans are used to. Purcell imagines an animal in the shape of a torus, which resembles a doughnut (Purcell, 1977). The animal would move by rolling towards the inside; this is known are the “smoking ring” method (Swan et. al, 2011). Another imaginary organism involves two cells stuck together which would roll along provided there was some attraction between the two (Fig. 1). In actually, these animals are not found in nature.

Two common forms are the flexible oar and the corkscrew (Purcell, 1977). A stiff oar in a low Re fluid will never work because it will always result in reciprocal motion. Flexible oars will bend one way during the first part of the stroke, and will bend in the opposite direction for the second half. The corkscrew is certainly easier to envision. The rotation of a corkscrew is in only one direction, resulting in no reciprocal motion (Fig. 2). This will generate propulsion, as seen in this paper. One well-known organism that uses the corkscrew method is Escherichia coli (de Lima Bernardo, 2011). By rotating its flagella it can propel itself through the water. Purcell’s hypothetical organism, the torodial swimmer (Fig. 1), has been the subject of many studies in the field of nanotechnology and microswimmers (Leshansky, 2008). Purcell suggests that the surface layers of the torus would rotate inwards, labeling the process as “tank treading”. It turns out that the torodial form is more efficient than the corkscrewing method that E. coli uses, upwards of 13% (Leshansky, 2008).

Figures

Figure 1: Purcell's drawings of his hypothetical organisms that can swim in low Reynolds number fluids.

Figure 2: Two common solutions found in the natural world that solve the problem of swimming in a low Reynolds number fluid.

References

de Lima Bernardo, B., Moraes, F., 2011, Simplified model for the dynamics of a

helical flagellum: American Journal of Physics, v. 79, p. 736-740.

Leshansky, A.M., Kenneth, O., 2008, Surface tank treading: Propulsion of Purcell’s

toroidal swimmer: Physics of Fluids, v. 20, p. 1-15.

Purcell, E.M., 1977, Life at low Reynolds number: American Journal of Physics, v. 1,

p. 3-11.

Swan, J.W., Brady, J.F., Moore, R.S., 2011, Modeling hydrodynamic self-propulsion

with Stokesian Dynamics. Or teaching Stokesian Dynamics to swim: Physics of Fluids, v. 23, p. 1-20.

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