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We discuss hydrodynamic forces acting on a two-dimensional liquid domain thatmoves laterally within a supported fluid membrane in the presence of oddviscosity. Since active rotating proteins can accumulate inside the domain, wefocus on the difference in odd viscosity between the inside and outside of thedomain. Taking into account the momentum leakage from a two-dimensionalincompressible fluid to the underlying substrate, we analytically obtain thefluid flow induced by the lateral domain motion, and calculate the drag andlift forces acting on the moving liquid domain. In contrast to the passive casewithout odd viscosity, the lateral lift arises in the active case only when thein/out odd viscosities are different. The in/out contrast in the odd viscosityleads to nonreciprocal hydrodynamic responses of an active liquid domain.

Hydrodynamic lift of a two-dimensional liquid domain with odd viscosity