Lorentz Reciprocal Theorem in Fluids with Odd Viscosity

The Lorentz reciprocal theorem—that is used to study various transport phenomena in hydrodynamics—is violated in chiral active fluids that feature odd viscosity with broken time-reversal and parity symmetries. Here, we show that the theorem can be generalized to fluids with odd viscosity by choosing an auxiliary problem with the opposite sign of the odd viscosity. We demonstrate the application of the theorem to two categories of microswimmers. Swimmers with prescribed surface velocity are not affected by odd viscosity, while those with prescribed active forces are. In particular, a torque dipole can lead to directed motion.

Figure 1

The Lorentz reciprocal theorem links the main problem (a) of a body moving with velocity ${\mathbf{V}}_{\mathrm{A}}$, fluid velocity ${\mathbf{v}}_{\mathrm{A}}$ in the comoving frame, and fluid viscosity $\mathit{\eta }$ with an auxiliary problem (b),(c) of a body with the same shape and the velocity $\widehat{\mathbf{V}}$, fluid velocity $\widehat{\mathbf{v}}$, and viscosity $\widehat{\mathit{\eta }}$. Here, the main problem represents an active swimmer and the auxiliary problems in panels (b) and (c) represent bodies with no-slip (NS) and perfect-slip (PS) boundary conditions, respectively.

Figure 2

The 2D flow fields (a) around a PS disk for ${\widehat{\eta }}^{\mathrm{o}}=-{\widehat{\eta }}^{\mathrm{e}}$, which is used as an auxiliary problem, and (b) generated by a swimmer with prescribed active force density $f_{\mathrm{A}}^{\parallel }=-f_0\sin \varphi$ for ${\eta }^{\mathrm{o}}={\eta }^{\mathrm{e}}$, moving at an angle ${\varphi }_{\mathrm{H}}=-\pi /4$.

Figure 3

Swimming behavior of swimmers with (a)–(c) prescribed velocity ${\mathbf{v}}_{\mathrm{A}}$ and (d)–(f) prescribed traction ${\mathbf{f}}_{\mathrm{A}}^{\parallel }$. The tangential velocity or traction has the meridional (a),(d), orthogonal meridional (b),(e), or azimuthal (c),(f) direction [52]. The twister (f) moves along the axis defined by odd viscosity by exerting a torque dipole on the fluid.