Interfacial behaviour in two-fluid Taylor–Couette flow

Abstract

The flow of a system of two viscous fluids between two concentric counter-rotating cylinders is discussed. A simple theory is presented that describes the evolution of shape of the interface between the fluids when they have near equal densities and identical viscosities. This suggests that the interface is neutrally stable, but that after sufficient time there are nevertheless points on the profile at which the curvature becomes very large. As a consequence, the interface develops cusp-like portions in its profile. A novel spectral method is developed for this problem in which the interface is represented as a region of finite width and over which the density changes rapidly but smoothly. The results confirm the general predictions of the asymptotic theory for rotation in a horizontal plane but when the rotation occurs vertically additional features develop in the flow.

The Rayleigh–Taylor instability in a porous medium

“…It might be expected that the inclusion of viscosity in the fluids would avoid a curvature singularity from forming at the interface, but Forbes et al [7] and Forbes and Bassom [8] have indicated that the situation is a little more nuanced than this. They showed that very large curvature spikes still form at the interface, even when viscosity is included, and to avoid them, a certain amount of fluid mixing is required, at an interface that must be of finite width.…”

Instability of a dense seepage layer on a sloping boundary

“…To suppress such points of infinite curvature forming at the interface, we have introduced viscosity into our models of the fluid behaviour. Nevertheless, viscosity alone has been shown by Forbes et al (2015) and Forbes & Bassom (2018) to be insufficient to prevent points of arbitrarily large curvature forming on an infinitesimally thin interface, and an appealing feature of the Boussinesq theory presented in § 4 is that it allows the interface to be modelled as a region of narrow -but finitewidth, across which the fluid density or salinity increases rapidly but smoothly. This bypasses the need to apply the difficult kinematic and dynamic boundary conditions (Batchelor 1972, p. 150) between viscous fluids on a mathematical line of density discontinuity, and perhaps of more importance in the present context, it is also entirely consistent with Boussinesq theory for the interface to have finite width, so eliminating the possibility of curvature singularity.…”

Triskeles and Symmetries of Mean Global Sea-Level Pressure

“…Forbes et al [85] show that a simple solution for Taylor-Couette flow of a viscous fluid between two concentric cylinders with radii a and b rotating in opposite directions and submitted to an azimuthal perturbation with angle θ can be written as:…”