Wave-crests around obstacles in odd viscous liquids

Abstract

The values of liquid odd-viscosity coefficients remain largely unknown, with only a single experimental measurement reported to date [Nature Physics 15, 1188 (2019)]. In this work, inspired by the well-known consequences of dispersion surfaces in classical liquids from the work of Lighthill, we theoretically determine the shapes of constant-phase wave crests formed around obstacles moving at constant velocity in two- and three-dimensional odd viscous liquids, which may or may not undergo rigid rotation. From this analysis, we derive parametric relations that the odd-viscosity coefficients must satisfy, providing a framework for their experimental determination.

Figures (4)

• Figure 1: Wave-crests of constant phase (Eq. (\ref{['x']}) with dispersion equation (\ref{['P']}), see Appendix for the parametric representation of the curves) around a body moving with velocity $U \hat{\mathbf{z}}$, in the frame rotating with angular velocity $\Omega \hat{\mathbf{z}}$ of a three-dimensional odd viscous liquid characterized by the constitutive law (\ref{['sigma1']}). Arrows denote the direction of motion of the body and $r = (x^2 + y^2)^{1/2}$. Compare with the wave-crests in figure \ref{['dispersion_surfaces_non_odd']} determined from the same theory for a classical (non-odd) Newtonian liquid, which agreed exceedingly well with observation and experiment. Curve parametric forms are delegated to Appendix \ref{['sec: parametric']}.
• Figure 2: Squared radius (\ref{['radius2']}) of spherical fronts (second panel from the left in figure \ref{['wave_crests_odd']}) for the determination of the value of the odd viscosity coefficients in an experiment.
• Figure 3: Left panel: Dispersion surfaces (\ref{['P2D']}) where the number on the curve denotes the forcing frequency $\omega_0$ and the thin arrows denote the direction of increasing $\omega_0$. Thus long waves trail behind the obstacle and away from its path while short waves point in all directions and are concentrated close to the obstacle's path. Right panel: Wave-crests of constant phase (Eq. (\ref{['x']}) with dispersion equation (\ref{['P2D']}), see Appendix for the parametric representation of the curves). Thick arrow denotes the direction of motion of the traveling force effect (whose frequency $\omega_0$ has been set to zero).
• Figure 4: Known shapes of wave-crests of constant phase, formed behind an obstacle moving with constant velocity (black arrow) in a classical Newtonian liquid theoretically predicted by Lighthill1967Lighthill1978. Surface gravity waves observed to form behind ships (ship Kelvin waves Lighthill1978). Internal gravity waves experimentally observed to form behind an ascending sphere in stratified liquid Mowbray1967. The spherical wave-crests form around an obstacle moving with constant speed in a rigidly-rotating liquid.