Session S20: Interfacial/Thin Film Instability VI

Irrotational analysis of the early stages of break-up of a viscous drop in a high-speed gas stream

The early stages of the break-up process of a liquid drop suddenly exposed to a high-speed gas stream behind a shock wave are considered. A linear analysis of the stability of the gas-liquid interface is conducted including the tangential component of the gas velocity near the interface and drop acceleration. The interfacial dynamics is thus governed by the combined mechanisms of Kelvin-Helmholtz and Rayleigh-Taylor instabilities. Visualizations of drop break-up by a gas stream at high Weber numbers reported in the literature reveal that in this regime instabilities driven by the shearing action of the gas play a role in the disintegration of the drop; this feature is central in developing the theory presented here. The dispersion relation for the growth rate and wave speed resulting from the stability analysis is written in terms of the density ratio, gas Weber and Reynolds numbers, and the liquid Ohnesorge number, which are typically used to specify an experimental run, and the Bond number, which contains the drop acceleration. Predictions from the stability analysis are discussed in the context of observations of experimental runs described in the literature for different values of the governing dimensionless parameters.   

The velocity of ``large'' viscous drops falling on a coated vertical fiber

Kalliadasis \& Chang (1994) showed within lubrication theory that if the coating is thicker than a critical value, drops accelerate and grow, whereas on thinner coatings, drops fall at a constant velocity. As the thickness of the coating increases to the critical value, the drops move faster and are bigger. We revisit their asymptotic analysis of these large drops.   

Solidification of molten metallic foams

High-porosity metallic solids can be formed by solidification of the corresponding molten gas-liquid foam. However, molten metallic foams are thermally and dynamically unstable, so in the absence of solidification the thin liquid films drain rapidly toward the bubble vertices and eventually become unstable to interfacial instabilities, leading to film rupture and bubble coalescence. To explore the competition between coarsening and freezing we have constructed a large-scale network model to describe the dynamics and stability of a planar foam with low liquid fraction, incorporating a coupling between pressure and volume in the gas bubbles, surface tension forces on the gas- liquid interfaces, draining flow in the films, a criterion for film rupture, thermal fluctuations and a solidification front. Initially, the foam is arranged in a regular array of approximately polygonal bubbles, held at a uniform temperature above the melting point of the material. The walls of the container are then cooled to a temperature well below the melting point, driving a solidification front inwards; numerical simulations of the model predict the structure of the resulting porous metal solid.   

Biaxial extensional motion of an inertially driven liquid disk

We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equations that describes the biaxial extensional motion of a viscous disk driven by inertia. The linear stability of the exact solution to axisymmetric and two-dimensional perturbations is examined in the inviscid limit within the framework of the long-wave approximation. Both transient growth and long-time asymptotic stability are considered.   

Stabilizing the Interface in a Saffman-Taylor Problem by Heating

The interface in a Saffman Taylor problem can be stabilized to perturbations of \textit{any wave length} by simply heating from above. The same is true for the Rayleigh Taylor instability. We present simple formulas for estimating the temperature difference required to do this and find that more reasonable temperature differences obtain in the Saffman-Taylor problem because the temperature dependence of the viscosity is ordinarily much stronger than the temperature dependence of the density.   

Electrohydrodynamic instabilities of biomimetic bilayer membranes

A DC electric pulse can destabilize a fluid bilayer membrane, causing a transient bending of the interface [1]. The membrane deformation relaxes as the membrane capacitor charges. We investigate the possibility of using AC electric fields and charged lipids to control the instability. In AC fields, the dynamics of a charge-free membrane depends on the relative magnitudes of the inverse of the field frequency and the capacitor charging time. At low frequencies, where the capacitor is fully charged and draws no current, the membrane is stable. As the frequency increases and the capacitor becomes short-circuited, the membrane behaves as a simple interface separating leaky dielectric fluids and can become unstable depending on the fluid electric properties (conductivities and dielectric constants). In the presence of native charge, the charge redistribution leads to even more complex membrane dynamics. \\[4pt] [1] Schwalbe et al. Physics of Fluids 23: 041701 (2011)   

The effect of permeability gradients on immiscible displacement in Hele-Shaw flows

In heterogeneous media, it is well known that when a fluid of high viscosity displaces a less viscous fluid, the interface can still be unstable and exhibit finger-like patterns due to capillary fingering. Motivated by porous media flows in natural geological formations, we consider homogeneous displacement in a Hele-Shaw cell subjected to a permeability gradient. The permeability gradient is introduced by linearly varying the Hele-Shaw cell depth. We study how capillary forces can affect interfacial stability in the presence of the gradient via linear stability analysis. Depending on the system, we find that surface tension can either have a stabilizing or a destabilizing role. We report the emergence of an important dimensionless parameter--the ratio of the permeability gradient to the capillary number--that determines the stability of the interface along with the well-studied viscosity ratio. Experiments testing the theoretical findings will also be presented.   

Contact-line dynamics, bifurcation and bi-stability of droplets driven by thermal gradients

We consider the quasi-static spreading of a sessile drop on a radially-heated, partially-wetting substrate.~ In general, for a non-thermally simple fluid, this type of heating can generate both axial and radial temperature gradients along the drop interface, which produce thermo-capillary forces.~ These generate flows that affect the spreading process through the contact-line dynamics. Our model employs lubrication theory together with a constitutive law that relates~contact-angle to contact-line speed. We identify parameter regimes in which the droplets continue to spread indefinitely, and compute spreading exponents. In regions of parameter space in which the droplets converge to an equilibrium shape, we show that competition between surface chemistry (wetting conditions) and flow induced by thermal gradients can give rise to bi-stability. The path to equilibrium is then typically complex and the droplet may evolve through a number of intermediate states, such as a capillary ridge or a shape whose bulk and contact-line regions have essentially de-coupled.   

Stability of Liquid Rivulets on Liquid Substrate

The human tear-film lipid layer is a thin (100nm) oily film on water. Such films are unstable and dewet into lenses surrounded by a monolayer (Harkins, 1941). Dewetting has four stages: initial rupture, hole growth, hole coalescence, and retraction into lenses. The human lipid layer is shown to behave similarly. Brochard-Wyart has addressed the first two stages (1993); here we focus on the third. As adjacent holes grow into each other, the oil between them takes the shape of a long, thin rivulet with a lens cross-section. Eventually this rivulet undergoes an instability and the holes coalesce. We perform a linear stability analysis on a thin symmetric lens rivulet on a horizontal liquid substrate at low Bond number (Davis, 1980; Schiaffino, 1997), accounting for the first time for the liquid-substrate hydrodynamics. Analytical expressions are derived for the wavelength and breakup times associated with the maximum growth rate of the instability for multiple substrate flow conditions. We show that for negligible thin-film forces, a liquid rivulet on an immiscible liquid substrate is unstable at a critical wavelength disturbance. The instability growth rate varies by orders of magnitude depending on the lens contact angle.   

Contact line instability of a liquid rivulet partially wetting an inclined plane

We analyze the stability of a liquid rivulet of cross section, $A$, positioned across a plane with inclination angle, $\alpha$. The liquid partially wets the substrate with a static contact angle, $\theta_0$, when the substrate is horizontal. The contact line stability is studied using the lubrication approximation and with a slip model. Both normal and parallel components of gravity are included A static solution exists for small $\alpha$'s and its linear stability is considered. We use an pseudo-spectral Chebyshev method with a combination of basis functions that automatically satisfies the conditions at the contact lines. We analyze the effects of $A$, $\theta_0$ and $\alpha$ on the predictions of the model, such as stability regions, the maximum growth rate and the behavior of most unstable perturbation. Experiments with silicone oils spreading on a coated glass substrate are considered for a number of different fluid volumes and $\alpha$'s. We find a good agreement between the wavelength of maximum growth predicted by the model and the experimental average distance between drops.