A research group proposes two designs for a micro-swimmer to move through a fluid of…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Biophysics & Medical Physics · Physical foundations of the living cell
Forces, Machines and Signals, Ch Life at low Reynolds number
Problem
A research group proposes two designs for a micro-swimmer to move through a fluid of viscosity comparable to water at low Reynolds number. Design A: a spherical body carries a single stiff rod attached by one hinge. A motor swings the rod back and forth through the same angle, ±25∘ from the body axis, so the whole stroke is described by one shape variable, the hinge angle, which runs forward and then retraces exactly the same values on the way back. Design B: the rod is replaced by a rigid left-handed helical filament, attached to the body through a rotary motor that always turns it about its own axis in the same rotational sense.
((a)) Using the scallop theorem, decide whether Design A can produce a net translation of the swimmer over one cycle of the hinge, and explain why in terms of the number of shape variables and the path they trace.
((b)) Explain, using the drag anisotropy of the helical filament (ζ⊥≈2ζ∥ for a slender rod) and a torque balance on the whole swimmer (body plus filament), why Design B does propel the body, and why the body must rotate opposite to the filament.