Linear stability of two-layer Couette flows

The stability of two-layer Couette flow is investigated under variations in viscosity ratio, thickness ratio, interfacial tension and density ratio. The effects of the base flow on eigenvalue spectra are explained. A new type of interfacial mode is discovered at low viscosity ratio with properties that are different from Yih’s original interfacial mode (Yih, J. Fluid Mech., vol. 27, 1967, pp. 337–352). No unstable Tollmien–Schlichting waves were found over the range of parameters considered in this work. The results for thin films with different thicknesses can be collapsed onto a single curve if the Reynolds number and wavenumber are suitably defined based on the parameters of the thin layer. Interfacial tension always has a stabilizing effect, but the effects of density ratio cannot be so easily generalized. Neutral stability curves for water–alkane and water–air systems are presented as an initial step towards better understanding the effects of flow stability on the longevity and performance of liquid-infused surfaces and superhydrophobic surfaces.

A note on the similarity between the normal-field instability in ferrofluids and the thermocapillary instability

A striking resemblance between the normal-field instability in ferromagnetic fluids and the interfacial mode of the thermocapillary instability in viscous fluids is presented. A nonlinear evolution equation describing the dynamics of the free surface for a ferrofluid layer subject to a uniform normal magnetic field is derived, and compared to that for a thin viscous layer heated from below. Their similarity predicts the possibility of mutual nonlinear stability control.

On the patterns of interaction between shear and interfacial modes in plane air–water Poiseuille flow

The current work deals with the numerical analysis of linear stability problems in a stratified plain Poiseuille flow of air over water with equal layer heights. The interaction and branch exchange between Tollmien–Schlichting instability in air and interfacial instability is discovered and investigated. This effect is shown to stabilize disturbances with wavelengths of the order of channel height for interfacial waves and to produce a closed stable region inside the neutral curve of the interfacial mode. The behaviour of three unstable modes in the problem, corresponding to Tollmien–Schlichting type instability in air and water layers and interfacial instability respectively, has been studied in detail. Neutral conditions for all three modes and the stable region have been calculated.

Linear growth in two-fluid plane Poiseuille flow

In recent years a new paradigm has emerged in linear stability theory due to the recognition of the importance of non-normality in the Orr–Sommerfeld equation as derived from the method of normal modes. For single-fluid flows it has been shown that it is possible for the kinetic energy of certain stable mode combinations to grow transiently before decaying to zero. We look again at the linear stability of two-fluid plane Poiseuille flow in two dimensions, concentrating on transient growth and its dependence on the viscosity and depth ratio. The procedure is to solve the stability equations numerically and consider disturbances defined as a sum of the least stable eigenmodes (not just the least stable interfacial mode). It is found that the variational method used to find maximum growth cannot be based upon the kinetic energy of the flow only and that interface deflection must be included in the formulation. We show which modes are necessary for inclusion in the disturbance expression and find that the interfacial mode does not make a significant contribution to possible energy growth. We examine the magnitude of maximum growth and the nature of the disturbances that lead to this growth. The linear energy rate equation shows that at moderate Reynolds numbers the mechanism responsible for the largest two-fluid growth is transfer of energy from the basic flow via the Reynolds stresses. The energy transfer is facilitated by streamline tilting that can be seen at the channel walls or at the interface. A similar effect has been found in single-fluid plane Poiseuille flow.

Stability of two-layer stratified flow in inclined channels: applications to air entrainment in coating systems

To understand the physics of air entrainment in thin-film liquid coating and other applications, the stability characteristics of general stratified two-layer Poiseuille-Couette flow are examined in inclined channels. Only one mode of instability, the interfacial mode, is obtained in the long-wave asymptotic limit. The generalized eigenvalue problem, formed by spectral decomposition and solution of the general two-layer Orr-Sommerfeld equation, is solved to obtain all of the critical modes. Analysis of the air/liquid interface corresponding to experiments reveals that because of the large density variation between the two layers, the interfacial mode is the only mode of instability in air entrainment. Results from the stability analysis of the flow near the contact line where air entrainment occurs are consistent with previous experimental observations.