Exact results for dissipation and steady creeping flow in three-dimensional chiral active fluids

Chiral active fluids consist of self-spinning particles that rotate as a result of a continuous injection of energy on the microscopic scale (e.g., by activity or an external field). The hydrodynamics of such fluids is described by antisymmetric contributions in the viscosity tensor -called odd viscosity-, which are allowed by symmetry due to the presence of a non-trivial spin angular momentum density. By generalising the Helmholtz minimum dissipation theorem to systems with odd viscosity, we show that incompressible three-dimensional odd fluids in the presence of sources that induce flow (e.g. surfaces that impose boundary conditions) admit a unique solution for their steady flow fields at low Reynolds number. Furthermore, we prove that such flows dissipate more energy than ordinary Stokes flow, provided that the flow field is affected by odd viscosity. As an example, we consider a model fluid described by one shear viscosity and one odd viscosity in the creeping flow regime. We explicitly compute the stress tensor when such a fluid is subjected to a point force density. Finally, we compute exact results for the pressure and flow fields around a translating and rotating spherical particle from their singularity representations. From these solutions and our extended Helmholtz theorem, we explain why a translating sphere dissipates more energy when odd viscosity is present, whereas a rotating sphere does not.