Lorentz Reciprocal Theorem in Fluids with Odd Viscosity

The Lorentz reciprocal theorem---that is used to study various transport phenomena in hydrodynamics---is violated in chiral active fluids that feature odd viscosity with broken time-reversal and parity symmetries. Here we show that the theorem can be generalized to fluids with odd viscosity by choosing an auxiliary problem with the opposite sign of the odd viscosity [1]. To demonstrate its applicability, we use the thorem to determine the swimming velocity of two categories of microswimmers in a Stokes fluid with odd viscosity: first to swimmers with an imposed slip velocity on their surface and then to swimmers with a prescribed tangential force density. We solve both problems for spherical swimmers in a 3D fluid as well as disk-shaped swimmers in 2D. In both cases, we show that odd viscosity does not affect the motion of swimmers with prescribed surface velocity, but it does affect those with prescribed propulsive forces. A particularly interesting case is a "twister" which propels itself by exerting only a torque dipole on the fluid.

[1] Y. Hosaka, R. Golestanian, and A. Vilfan, Phys. Rev. Lett. 131, 178303 (2023).