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Inspired by the recent realization of a 2D chiral fluid as an activemonolayer droplet moving atop a 3D Stokesian fluid, we formulate mathematicallyits free-boundary dynamics. The surface droplet is described as a general 2Dlinear, incompressible, and isotropic fluid, having a viscous shear stress, anactive chiral driving stress, and a Hall stress allowed by the lack oftime-reversal symmetry. The droplet interacts with itself through its driveninternal mechanics and by driving flows in the underlying 3D Stokes phase. Wepose the dynamics as the solution to a singular integral-differential equation,over the droplet surface, using the mapping from surface stress to surfacevelocity for the 3D Stokes equations. Specializing to the case of axisymmetricdroplets, exact representations for the chiral surface flow are given in termsof solutions to a singular integral equation, solved using both analytical andnumerical techniques. For a disc-shaped monolayer, we additionally employ asemi-analytical solution that hinges on an orthogonal basis of Bessel functionsand allows for efficient computation of the monolayer velocity field, whichranges from a nearly solid-body rotation to a unidirectional edge currentdepending on the subphase depth and the Saffman-Delbruck length. Except in thenear-wall limit, these solutions have divergent surface shear stresses atdroplet boundaries, a signature of systems with codimension one domainsembedded in a three-dimensional medium. We further investigate the effect of aHall viscosity, which couples radial and transverse surface velocitycomponents, on the dynamics of a closing cavity.

Incompressible active phases at an interface. I. Formulation and axisymmetric odd flows