Swimming at low Reynolds number in fluids with odd, or Hall, viscosity.
We apply the geometric theory of swimming at low Reynolds number to the study of nearly circular swimmers in two-dimensional fluids with nonvanishing "odd," or Hall, viscosity. The odd viscosity gives an off-diagonal contribution to the fluid stress tensor, which results in a number of striking effects. In particular, we find that a swimmer whose area is changing will experience a torque proportional to the rate of change of the area, with the constant of proportionality given by the coefficient ηo of odd viscosity. After working out the general theory of swimming in fluids with odd viscosity for a class of simple swimmers, we give a number of example swimming strokes which clearly demonstrate the differences between swimming in a fluid with conventional viscosity and a fluid which also has an odd viscosity. We also include a discussion of the extension of the famous Scallop theorem of low Reynolds number swimming to the case where the fluid has a nonzero odd viscosity. A number of more technical results, including a proof of the torque-area relation for swimmers of more general shape, are explained in a set of Appendixes.
Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA.
Matthew F
MF
Lapa
Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA.
Taylor L
TL
Hughes
D000818:Animals; D001683:Biological Clocks / Q000502:physiology*; D003198:Computer Simulation; D017276:Friction / Q000502:physiology; D006801:Humans; D008954:Models, Biological; D012212:Rheology / Q000379:methods*; D033081:Shear Strength / Q000502:physiology; D013550:Swimming / Q000502:physiology*; D014783:Viscosity; D014867:Water / Q000737:chemistry*
D014867:Water