Taylor columns and inertial-like waves in a three-dimensional odd viscous liquid

Abstract

Odd viscous liquids are endowed with an intrinsic mechanism that tends to restore a displaced particle back to its original position. Since the odd viscous stress does not dissipate energy, inertial oscillations and inertial-like waves can become prominent in such a liquid. In this paper, we show that an odd viscous liquid in three dimensions may give rise to such axially symmetric waves and also to plane-polarized waves. We assume tacitly that an anisotropy axis giving rise to odd viscous effects has already been established, and proceed to investigate the effects of odd viscosity on fluid flow behaviour. Numerical simulations of the full Navier–Stokes equations show the existence of inertial-like waves downstream of a body that moves slowly along the axis of an odd viscous liquid filled cylinder. The wavelength of the numerically determined oscillations agrees well with the developed theoretical framework. When odd viscosity is the dominant effect in steady motions, a modified Taylor–Proudman theorem leads to the existence of Taylor columns inside such a liquid. Formation of the Taylor column can be understood as a consequence of helicity segregation and energy transfer along the cylinder axis at group velocity, by the accompanying inertial-like waves, whenever the reflection symmetry of the system is lost. A number of Taylor column characteristics known from rigidly rotating liquids are recovered here for a non-rotating odd viscous liquid. These include counter-rotating swirling liquid flow above and below a body moving slowly along the anisotropy axis. Thus in steady motions, odd viscosity acts to suppress variations of liquid velocity in a direction parallel to the anisotropy axis, inhibiting vortex stretching and vortex twisting. In unsteady and nonlinear motions, odd viscosity enhances the vorticity along the same axis, thus affecting both vortex stretching and vortex twisting.