Session AF: Interfacial Instabilities

Convective and absolute instabilities in stratified flow of Newtonian and Bingham-like layers.

The stability of an interface separating either two Newtonian fluids or a Newtonian and a Bingham-like fluid in pressure-driven channel flow at moderate Reynolds numbers is analysed both theoretically and numerically. Inertia, interfacial tension and gravity are also accounted for in the study. In the linear regime, our theoretical analysis reveals that the interface is absolutely unstable over an intermediate range of Reynolds numbers and interfacial depths; convectively unstable regimes are present in the complimentary ranges of these parameters. Increasing the viscosity ratio and/or the yield stress of the Bingham layer promotes the absolute nature of the interfacial instability. Results obtained from numerical simulations elucidate~the nonlinear evolution of the interface which is accompanied by ligament formation leading to pinchoff. The transition point from a convective to an absolute regime predicted by simulations also agrees well with the theoretical analysis. Simulations with Bingham layers (although initiated from a fully-yielded base state) show that~unyielded regions~appear in the wave troughs during late stages of wave evolution.   

Instability in turbulent stratified channel flow

We determine the stability properties of a deformable interface separating a fully-developed turbulent gas flow in a channel from a thin laminar liquid layer. To do this, we derive a linear model to describe the interactions between the turbulent gas flow and the interfacial waves. This model involves two steps. First, we derive a flat-interface base-state velocity. This takes account of the laminar sublayer present in the near-interfacial region of the gas, and provides a way of determining the wall and interfacial shear stresses as a function of the applied pressure gradient. Next, we perform an Orr-Sommerfeld analysis on the Reynolds-averaged Navier-Stokes equations. This necessitates the selection of a turbulent-stress closure scheme. This approach gives the growth rate of the wave amplitude, as a function of the relevant dimensionless system parameters and turbulence closure relations. It also extends previous work by accounting for the effects of the thin liquid layer on the dynamics.   

Viscosity stratification in miscible channel flow

The stability of two-layer miscible flows in planar channels, focusing on the neutrally-buoyant displacement of a highly viscous fluid by a less viscous one, is studied. The flow dynamics are governed by the continuity and Navier-Stokes equations coupled to a convective-diffusion equation for the concentration of the more viscous fluid through a concentration-dependent viscosity. A generalized linear stability analysis (in which both the spatial wavenumber and temporal frequency are complex) is carried out, which allows the demarcation of the boundaries between convectively and absolutely unstable flows in the space of relevant parameters: the Reynolds and Schmidt numbers, and a viscosity ratio. The flow in the linear regime delineates the presence of convective and absolute instabilities and identifies the vertical gradients of viscosity perturbations as the main destabilizing influence. Our transient numerical simulations demonstrate the development of complex dynamics in the nonlinear regime, characterized by roll-up phenomena and intense convective mixing; these become pronounced with increasing flow rate and viscosity ratio, as well as weak diffusion.   

Numerical simulations of immiscible two-fluid channel flow in the presence of phase changes

We study the interaction between a fast-flowing fluid over a highly viscous layer in a two-dimensional channel using direct numerical simulations; the two fluids are immiscible. The flow regime varies from stratified-wavy to dispersed for moderate to high Reynolds numbers, respectively. The equations of mass, momentum and energy conservation in both fluids are solved using a procedure based on the diffuse interface method. This equation set is complemented by the Cahn-Hilliard equation for the volume fraction. No-slip and no-penetration conditions are imposed at the walls, and constant flow rate and outflow conditions are prescribed at the inlet and outlet, respectively. Our model accounts for the formation of the highly viscous fluid due to phase change in the bulk fluid, which is ultimately deposited at the wall. This is driven by thermal instability, which is taken into account using a chemical equilibrium model based on the Gibbs free energy. We present results showing typical flow dynamics and the effect of system parameters on the average deposit thickness. This work is of direct relevance to `fouling' in oil-and-gas refineries.   

Pressure-driven miscible two-fluid channel flow with density gradients

We study the effect of buoyancy on pressure-driven flow of two miscible fluids in inclined channels via direct numerical simulations. The flow dynamics are governed by the continuity and Navier-Stokes equations, without the Boussinesq approximation, coupled to a convective-diffusion equation for the concentration of the more viscous fluid through a concentration-dependent viscosity and density. The effect of density ratio, Richardson number, and channel inclination on the flow dynamics is examined, for moderate Reynolds numbers and viscosity ratios. We present results showing the spatio-temporal evolution of the flow together with an integral measure of mixing.   

Effect of solubility on the interfacial-surfactant instability of shear flows

A slow flow of a two-layer system with a soluble surfactant in the film and on the interface is considered. The linear stability theory of the plane Couette flow is developed for the liquid film adjoining a thicker layer of fluid. For the (Frenkel and Halpern 2002) non-inertial longwave instability which results from an interplay between the interfacial surfactant and flow shear, the effect of the surfactant solubility is studied, assuming a Langmuir-type adsorption-desorption kinetics at the interface and the advection-diffusion dynamics of the bulk surfactant. It is not a priori clear that the instability persists for the non-zero values of the surfactant solubility: The limit of vanishing solubility might be different from the case of zero solubility (= an insoluble surfactant case). For certain parametric regimes, this work analytically demonstrates that the instability does persist for non-zero surfactant solubilities; however, no matter how weak the surfactant solubility, its mitigating effect on the instability is found to be arbitrarily large for sufficiently long waves. Thus, the insoluble surfactant approximation fails for such waves.   

Wetting failure and contact line dynamics in a Couette flow

Liquid-liquid wetting failure is investigated in a two-dimensional Couette system with two immiscible fluids of arbitrary viscosity. The problem is solved exactly using a sharp interface treatment of hydrodynamics (lubrication theory) as a function of the control parameters: capillary number, viscous ratio and separation of scale. The transition at a critical Capillary number, from a stationary to a non-stationary interface, is studied at changing the control parameters. Comparisons with similar existing analysis for other geometries, as for the Landau-Levich problem, are also carried out. A numerical method of analysis is also presented, based on diffuse interface models obtained from multiphase extensions of the lattice Boltzmann equation (LBE). Sharp interface and diffuse interface models are quantitatively compared, face to face, indicating the correct limit of applicability of the diffuse interface models.   

Linear and nonlinear stability of a two-fluid interface in channel flow under the influence of parallel and normal electric fields

The use of an electric field on a two-fluid interface has been shown to be an efficient way to trigger an interfacial instability which, in turn, can enhance mixing or lead to droplet formation in a microfluidic channel. Keeping the latter applications in mind, the instability of a flat interface between two liquids confined in a channel and subjected to Poiseuille flow is studied in the presence of an electric field either parallel or normal to the flat interface. The liquids are considered to be viscous, incompressible and leaky dielectric. We have analyzed the effect of various parameters, as well as compared the role played by a parallel versus normal electric field on the dispersion curve, i.e., the growth rate as a function of wavenumber. While the flow is found to have no direct effect on the linear stability of the interface, its effect can be clearly observed in the nonlinear regime.   

Dynamics of a two-fluid interface in a channel in the presence of electric fields

We study the stability of the interface between two superposed fluids in a channel in the presence of a uniform electric field acting horizontally with respect to the undisturbed configuration. The two fluids are taken to be inviscid, incompressible and nonconducting, but can have different densities and permittivities. We consider the physical effects of surface tension, gravity and electrically induced forces. The approach involves the derivation of a set of nonlinear evolution equations for the interfacial shape, horizontal velocity and electric potential of the upper layer. The electric field effects enter nonlocally. Linear stability analysis reveals that the presence of the electric field causes a stabilization of the flow in the sense that it can compete with the unstable density stratification. In particular, it is shown that for given physical parameters, there exists a critical value of the voltage potential difference between electrodes, above which the electric field suppresses the Rayleigh-Taylor instability. Traveling waves are calculated and their behavior studied as the electric field increases.   

Interfacial Instabilites in Two-Layer Flows of Viscoelastic Fluids

We consider interfacial instability of pressure-driven, two-layer flows with one of the fluids being viscoelastic (like a polymeric solution or a polymer melt). We build on the work of Ganpule and Khomami (JNNFM, v.81, pp.27-69, 1999) by including finite extensibility in a complete way (as appropriate for the strong elongational flows that drive our present interest), and by extending the parameter space considered by them. While from the complete model they find only a Yih mode, and while by matching viscosities they find an elastic mode, we find the simultaneous existence of both modes. The parameter space for this occurrence is a subset of conditions from 1$<$De$<$5, 5$<$n$<$30, k$>$4.5, and viscoelastic layer thicknesses less than the Newtonian one (less viscous). In the above, n is the viscosity ratio (solution divided by the solvent) and k is the wave number. For Re$>$40, the elastic mode disappears, and the shear (T-S) becomes dominant mode.