On the Motion of Gravity–Capillary Waves with Odd Viscosity

Abstract

We develop three asymptotic models of surface waves in a non-Newtonian fluid with odd viscosity. This viscosity is also known as Hall viscosity and appears in a number of applications such as quantum Hall fluids or chiral active fluids. Besides the odd viscosity effects, these models capture both gravity and capillary forces up to quadratic interactions and take the form of nonlinear and nonlocal wave equations. Two of these models describe bidirectional waves, while the third PDE studies the case of unidirectional propagation. We also prove the well-posedness of these asymptotic models in spaces of analytic functions and in Sobolev spaces. Finally, we present a number of numerical simulations for the unidirectional model.

Instability of Liquid Film with Odd Viscosity over a Non-Uniformly Heated and Corrugated Substrate

“…In recent years, odd viscosity has emerged as a prominent subject of investigation in the study of thin film flow stability. Odd viscosity [25,26] is the non-dissipative component of the viscosity tensor and is contained in its antisymmetric part. Avron [27,28] made a break-through discovery by demonstrating that in a classical fluid, when time-reversal symmetries are broken, either spontaneously or due to an external magnetic field or rotation, the viscosity tensor can have a non-zero odd part that gives rise to a dissipationless linear response coefficient known as odd or Hall viscosity.…”

Nonlinear shallow-water waves with vertical odd viscosity

“…In boundary layers at fluid surfaces, odd viscosity can lead to effects akin to a surface tension, but with broken detailed balance [1,3]. In the nonlinear regime, these boundary layers interact with capillary effects [16] or compressibility [2], and modify the coefficients of the Korteweg-de Vries (KdV) equation in shallow water [29]. In all of these two-dimensional cases, odd viscosity has been assumed pointing out-of-plane and tangentially to the surface.…”