Yuriko Renardy

Instability at the interface between two shearing fluids in a channel

The linear stability of plane Couette flow composed of two immiscible fluids in layers is considered. The fluids have different viscosities and densities. For the case of equal densities, there is a critical Reynolds number above which the interfacial mode of the bounded problem is approximated by that of the unbounded problem for wavelengths that are not short enough to be in the asymptotic short‐wavelength range, as well as for short waves. The full linear analysis reveals unstable situations missed out by the long‐ and short‐wavelength asymptotic analyses, but which have comparable orders of magnitudes for the growth rates. For the case of unequal densities, it is found that the arrangement with the heavier fluid on top can be linearly stable if the viscosity stratification, volume ratio, surface tension, Reynolds number, and Froude number are favorable.

Couette flow of two fluids between concentric cylinders

We consider the flow of two immiscible fluids lying between concentric cylinders when the outer cylinder is fixed and the inner one rotates. The interface is assumed to be concentric with the cylinders, and gravitational effects are neglected. We present a numerical study of the effect of different viscosities, different densities and surface tension on the linear stability of the Couette flow. Our results indicate that, with surface tension, a thin layer of the less-viscous fluid next to either cylinder is linearly stable and that it is possible to have stability with the less dense fluid lying outside. The stable configuration with the less-viscous fluid next to the inner cylinder is more stable than the one with the less-viscous fluid next to the outer cylinder. The onset of Taylor instability for one-fluid flow may be delayed by the addition of a thin layer of less-viscous fluid on the inner wall and promoted by a thin layer of more-viscous fluid on the inner wall.

Oscillatory instability in a Bénard problem of two fluids

A linear stability analysis for a Benard problem with two layers is considered. The equations are not self‐adjoint. The system can lose stability to time‐periodic disturbances. For example, it is shown numerically that when the viscosities and coefficients of cubical expansion of the fluids are different, a Hopf bifurcation can occur, resulting in a pair of traveling waves or a standing wave. This may have application in the modeling of convection in the Earth’s mantle.

Perturbation analysis of steady and oscillatory onset in a Bénard problem with two similar liquids

In a recent paper, Renardy and Joseph [Phys. Fluids 28, 788 (1985)] studied the Benard problem for two layers of different fluids lying on top of each other and bounded by walls. Their study shows that, in contrast to the Benard problem for one fluid, the onset of instability can be oscillatory. The number of parameters involved in the problem is large, and there is as yet no comprehensive picture of when the instability is oscillatory and when it is not. The study of limiting cases, accessible by perturbation methods, may be helpful in this respect. In this paper, an analysis is given for the case when the properties of the two fluids are nearly equal and the fluids are allowed to slip at the boundaries.

Interfacial Stability in a Two-Layer Benard Problem.

A linear stability analysis of the Benard problem for two layers of different fluids lying on top of each other and bounded by free surfaces is considered. The fluids are assumed to be similar and perturbation methods are used to calculate the interfacial eigenvalue in closed form. The case of the Rayleigh number and wavenumber of the disturbance being close to the first criticality of the one‐fluid Benard problem has been investigated in a previous paper [Phys. Fluids 28, 2699 (1985)], and was found to exhibit both overstability and convective instability. In this paper, the Rayleigh number is assumed to be less than that of the first criticality of the one‐fluid problem, and in this situation, overstability does not occur. An unexpected result is that by an appropriate choice of parameters, it is possible to find linearly stable arrangements with the more dense fluid on top.

Remarks on the stability of viscometric flow

We study the stability of viscometric flow using the type of short memory introduced by Akbay, Becker, Krozer and Sponagel. The instability found by these researchers is recognized as a “change of type” leading to non-evolutionary character of the governing equations. We also address the question of justification for the short memory assumption and find that it cannot be justified for some of the more popular rheological models.

Trapping of water waves above a round sill

The three-dimensional problem of wave trapping above a submerged round sill was first analysed by Longuet-Higgins on the basis of a linear shallow-water theory. The large responses predicted by the theory were, however, not well borne out by the experiments of Barnard, Pritchard & Provis, and this has motivated a more detailed study of the problem. A full linear theory for both inviscid and weakly viscous fluid, without any shallow-water assumptions, is presented here. It reveals important limitations on the use of shallow-water theory and the reasons for them. In particular, while the qualitative features of wave trapping are similar to those of shallow-water theory, the nearly resonant frequencies differ significantly, and, since the resonances are narrow, the observed amplitudes at a given frequency differ greatly. The geometry is strongly indicative of long waves, and the dispersion relation appears quite consistent with that, but the part of the motion at wavenumbers that are not small has, despite the small amplitude, a substantial effect on the response to excitation. A theoretical study of the trapping that results when a train of small-amplitude plane waves of a fixed frequency is incident on a submerged steep-sided round sill (figure 1) was made by Longuet-Higgins (hereinafter referred to as LH) in 1967. His investigation was motivated by wave records taken at Macquarie Island showing the occurrence of regular oscillations of unusually large amplitudes. In view of these observations, LH considered a simplified geometry in which the island shelf was represented by a round sill, with a circumference of 80 km, submerged to a depth of 100 m. He based his calculation on linear inviscid shallow-water theory, and used separation of variables in cylindrical coordinates to determine the expressions for the surface displacements for each of the two regions of constant depths. Because of the depth independence of the velocity field in shallow-water theory, the velocity components could not be made continuous at the sill edge, and two approximate matching conditions were used : the continuity of surface elevation and the continuity of the horizontal component of the mass flux. LH’s analysis showed the existence of eigenfrequencies with very small imaginary parts. A train of plane waves with a frequency near such an eigenfrequency could theoretically excite ‘ nearly trapped ’ modes over the sill, and the response at such modes was determined. The largest responses were found to occur at the higher angular modes and at smaller ratios of the depths. These calculations have been confirmed and extended by Summerfield (1969), who applied shallow-water theory to a ‘shelf-island’ model consisting of a steep-sided round island rising from the top of a round sill of larger radius. He showed the eigenfrequencies for his system to be closely related to those of LH’s sill geometry.

Chapter 7 - Stability and Instability in Viscous Fluids

The systematic study of hydrodynamic instability began roughly a century ago, with the experimental investigations of Reynolds on parallel shear flows and Benard on convection, and the theoretical studies of Lord Rayleigh. Since then, instabilities in fluid motion and the new flows arising as a result of such instabilities have continued to pose a challenge to mathematicians, which has influenced and inspired progress in several fields of mathematics—for example, dynamical systems, partial differential equations, functional analysis, and asymptotics. This chapter begins with a general introduction to mathematical issues arising in the study of stability and bifurcation and latter discusses the connection between linear stability and spectrum, nonlinear stability, the derivation of reduced equations near the onset of instability, and the analysis of bifurcations. The chapter also discusses hydrodynamic applications grouped around four general topics: thermal convection, flow between rotating cylinders, parallel shear flows, and capillary instability of jets.

Weakly nonlinear interactions and wave trapping

When the flow over a submerged, round, upright cylinder, situated in a large ocean, is forced by a train of plane waves, linear theory (Yamamuro 1981) shows that the response can be abnormally large for certain forcing frequencies. The aim of this paper is to present a weakly nonlinear theory, where wave interactions, arising from the quadratic terms in the free-surface boundary conditions, can yield abnormally large responses. A specific interaction will be considered between a flow at a subharmonic frequency and a flow at the driving frequency. The reason for considering such an interaction derived from a consideration of some experimental results of Barnard, Pritchard & Provis (1981).