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

TL;DR

This paper derives exact solutions for steady creeping flows in three-dimensional chiral active fluids with odd viscosity, revealing how such fluids dissipate more energy and providing explicit flow and pressure fields around particles.

Contribution

It generalizes the Helmholtz minimum dissipation theorem to include odd viscosity and computes exact flow and stress fields in chiral active fluids.

Findings

Steady flows in odd fluids are unique and dissipate more energy than in ordinary fluids.

Explicit solutions for flow and pressure fields around particles are provided.

Translating spheres dissipate more energy with odd viscosity, rotating spheres do not.

Abstract

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.