  We theoretically and computationally study the low-Reynolds-number
hydrodynamics of a linear active microswimmer surfing on a compressible thin
fluid layer characterized by an odd viscosity. Since the underlying
three-dimensional fluid is assumed to be very thin compared to any lateral size
of the fluid layer, the model is effectively two-dimensional. In the limit of
small odd viscosity compared to the even viscosities of the fluid layer, we
obtain analytical expressions for the self-induced flow field, which includes
non-reciprocal components due to the odd viscosity. On this basis, we fully
analyze the behavior of a single linear swimmer, finding that it follows a
circular path, the radius of which is, to leading order, inversely proportional
to the magnitude of the odd viscosity. In addition, we show that a pair of
swimmers exhibits a wealth of two-body dynamics that depends on the initial
relative orientation angles as well as on the propulsion mechanism adopted by
each swimmer. In particular, the pusher-pusher and pusher-puller-type swimmer
pairs exhibit a generic spiral motion, while the puller-puller pair is found to
either co-rotate in the steady state along a circular trajectory or exhibit a
more complex chaotic behavior resulting from the interplay between hydrodynamic
and steric interactions. Our theoretical predictions may pave the way toward a
better understanding of active transport in active chiral fluids with odd
viscosity, and may find potential applications in the quantitative
microrheological characterization of odd-viscous fluids.
