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

TL;DR

This work extends the steady creeping-flow theory to incompressible 3D chiral active fluids with odd viscosity, proving a unique solution exists and that odd viscosity typically increases energy dissipation under fixed boundary data. It develops a generalized Helmholtz dissipation framework and provides explicit Green's-function-based representations for the fluid response to localized forcing. Using a singularity-method, the authors derive exact velocity and pressure fields for translating and rotating spheres in an odd-viscosity fluid, revealing that translating spheres dissipate more energy while rotating spheres retain the same dissipation as in Stokes flow for their model. The results yield closed-form mobility/friction tensors and pave the way for accurate simulations of microswimmers and colloidal particles in odd fluids, with potential topological and flow-pattern implications due to azimuthal components induced by odd viscosity.

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.

Figures (2)

• Figure 1: Representative streamlines of the fluid velocity field $\hbox{\boldmath $v$}(\hbox{\boldmath $r$})$ around a spherical particle translating with velocity $\hbox{\boldmath $U$}$ in the absence of ambient flow. All plots are generated using the exact analytical solution Eq. \ref{['eq:v']}. (a) Stokes flow without odd viscosity ($\gamma =\eta_\mathrm{o}/\eta_\mathrm{s}=0$). (b)-(d) Odd viscous flow for $\gamma =3$ at different relative orientations of $\hbox{\boldmath $U$}$ and $\hbox{\boldmath $\hat{\ell}$}$, with (b) $\hbox{\boldmath $U$}\parallel\hbox{\boldmath $\hat{\ell}$}$, (c) $\sphericalangle(\hbox{\boldmath $U$},\hbox{\boldmath $\hat{\ell}$})=45^\circ$, and (d) $\hbox{\boldmath $U$}\perp\hbox{\boldmath $\hat{\ell}$}.$ Note that the flows in (a) and (b) are cylindrically symmetric around the $\hbox{\boldmath $U$}$ axis. All streamlines are directed from bottom to top.
• Figure 2: Projections of $\hbox{\boldmath $v$}(\hbox{\boldmath $r$})$ around a spherical particle translating with velocity $\hbox{\boldmath $U$}$ in an odd viscous fluid. The anisotropy axis $\hbox{\boldmath $\hat{\ell}$}$ is fixed in the $+z$ direction with the orientation of $\hbox{\boldmath $U$}$ being varied with respect to $\hbox{\boldmath $\hat{\ell}$}$. (a,b) Projections of the streamlines of $\hbox{\boldmath $v$}(\hbox{\boldmath $r$})$ onto the $xz$-plane. The colours are a measure for the out-of-plane component of $\hbox{\boldmath $v$}(\hbox{\boldmath $r$})$ in the $y$ direction. (c,d) Projections of the streamlines of $\hbox{\boldmath $v$}(\hbox{\boldmath $r$})$ onto the $xy$-plane, with the colours being a measure for $|\hbox{\boldmath $v$}(\hbox{\boldmath $r$})|$. Panels (a) and (c) are for $\gamma=1$, whereas panels (b) and (d) are for $\gamma=3$.