Steep capillary-gravity waves in oscillatory shear-driven flows

Abstract

We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W , square root of the ratio of vibrational to gravitational forces). At a critical value W c , a transition to a linear variation in W is observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but 'finger'-like waves form for W > W c . In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas for W < W c , the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically with W to reach the capillary length for W = W c , i.e. the lengthscale for which surface tension forces balance gravitational forces. For W < W c , gravitational restoring forces dominate, but for W > W c , the wave development is increasingly defined by localized surface tension effects.

Dynamics of the interface between miscible liquids subjected to horizontal vibration

“…It is worth highlighting that it is the first experimental evidence of spatially modulated waves on an interface caused by horizontal vibrations. Previous experimental studies of wave patterns under horizontal vibrations with immiscible fluids (Jalikop & Juel 2009;Yoshikawa & Wesfreid 2011b;Gandikota et al 2014b) did not report wave modulation, although it was theoretically predicted by Varas & Vega (2007) in a nearly inviscid fluid subjected to horizontal vibration. We will return to this pattern in § 4.3.…”

“…The effect of horizontal vibrations on two superposed immiscible liquids has been examined both experimentally (Wolf 1969(Wolf , 1970Ivanova, Kozlov & Evesque 2001;Talib, Jalikop & Juel 2007;Mialdun et al 2008;Jalikop & Juel 2009;Yoshikawa & Wesfreid 2011b) and theoretically (Lyubimov & Cherepanov 1987;Khenner et al 1999), where it was shown that if the state of the flat contact line is unstable, nonpropagating periodic waves may occur on the interface. Following Wunenburger et al (1999), these waves, which are stationary in the reference frame of the vibrated cell, are referred to as 'frozen waves'.…”

“…Ivanova et al (2001) and Gandikota et al (2014a) reported a favourable agreement of measurements for the onset of interfacial instability with the inviscid theory. Another group of authors Jalikop & Juel 2009) carried out experiments and a linear stability analysis of two immiscible liquids for a very large range of viscosity contrasts between layers. They reported that Kelvin-Helmholz mode instability was hardly influenced by the viscosity for fluid layers with a small viscosity contrast, while a substantial decrease of the instability threshold was found in fluid layers with a large viscosity contrast.…”

Oscillatory Kelvin–Helmholtz instability. Part 2. An experiment in fluids with a large viscosity contrast

“…Only the tips of the oil fingers oscillate with a small amplitude. Jalikop & Juel (2009) investigated saturated waves generated in oscillatory two-layer flows, by experiments in a horizontally vibrated parallelepiped container. The considered upper-layer fluids were silicone oils of viscosities 114 mm 2 /s and 210 mm 2 /s.…”

Two-scale wave patterns on a periodically excited miscible liquid–liquid interface

“…The behaviour of an interface of immiscible liquids under horizontal vibrations in a container with end walls demonstrates the emergence of spatially periodic waves on the interface, which are often referred to as 'frozen waves' (Lyubimov & Cherepanov 410 V. Shevtsova and others 1987;Khenner et al 1999;Ivanova, Kozlov & Evesque 2001;Jalikop & Juel 2009;Gandikota et al 2014). The frozen waves are caused by a shear driven mechanism similar to the Kelvin-Helmholtz instability.…”