On the Hamiltonian structure and three-dimensional instabilities of rotating liquid bridges

We consider a rotating inviscid liquid drop trapped between two parallel plates. The liquid–air interface is a free surface and the boundaries of the wetted regions in the plates are also free. We assume that the two contact angles at the plates are equal. We present drop shapes that generalize the catenoids, nodoids and unduloids in the presence of rotation. We describe profile curves of these drops and investigate their stability to three-dimensional perturbations. The instabilities are associated with degeneracies of eigenvalues of the corresponding Hamiltonian linear stability problem. We observe that these instabilities are present even in the case when the analogue of the Rayleigh criterion for two-dimensional stability is satisfie

Instabilities of volatile films and drops

We report on instabilities during spreading of volatile liquids, with emphasis on the novel instability observed when isopropyl alcohol (IPA) is deposited on a monocrystalline silicon (Si) wafer. This instability is characterized by emission of drops ahead of the expanding front, with each drop followed by smaller, satellite droplets, forming the structures which we nickname “octopi” due to their appearance. A less volatile liquid, or a substrate of larger heat conductivity, suppress this instability. In addition, we examine the spreading of drops of water (DJW)-JPA mixtures on both Si wafers and plain glass slides, and describe the variety of contact line instabilities which appear. We find that the decrease of IPA concentration in mixtures leads to transition from “octopi” to mushroom-like instabilities. Through manipulation of our experimental set up, we also find that the mechanism responsible for these instabilities appears to be mostly insensitive to both the external application of convection to the gas phase, and the doping of the gas phase with vapor in order to create the saturated environment. In order to better understand the “octopi” instability, we develop a theoretical model for evaporation of a pure liquid drop on a thermally conductive solid substrate. This model includes all relevant physical effects, including evaporation, thermal conductivity in both liquid and solid, (thermocapillary) Marangoni effect, vapor recoil, disjoining pressure, and gravity. The crucial ingredient in this problem is the evaporation model, since it influences both the motion of the drop contact line, and the temperature profiles along the liquid-solid and liquid-gas interfaces. We consider two evaporation models: the equilibrium “lens” model and the non-equilibrium one-sided (NEOS) model. Along with the assumption of equilibrium at the liquid-gas interface, the “lens” model also assumes that evaporation proceeds in a (vapor) diffusion-limited regime, therefore bringing the focus to the gas phase, where the problem of vapor mass diffusion is to be solved, which invokes analogy with the problem of lens-shaped conductor from electrostatics. On the other hand, NEOS model assumes non-equilibrium at the liquid-gas interface and a reaction-limited regime of evaporation; the liquid and gas phases are decoupled using the one-sided assumption, and hence, the problem is to be solved in the liquid phase only. We use lubrication approximation and derive a single governing equation for the evolution of drop thickness, which includes both models. An experimental procedure is described next, which we use in order to estimate the volatility parameter corresponding to each model. We also describe the numerical code, which we use to solve the governing equation for drop thickness, and show how this equation can be used to predict which evaporation model is more appropriate for a particular physical problem. Next, we perform linear stability analysis (LSA) of perturbed thin film configuration. We find excellent agreement between our numerical results and LSA predictions. Furthermore, these results indicate that the IPA/Si configuration is the most unstable one, in direct agreement with experimental results. We perform numerical simulations in the simplified 2d geometry (cross section of the drop) for both planar and radial symmetry and show that our theoretical model reproduces the main features of the experiment, namely, the formation of “octopus” -like features ahead of the contact line of an evaporating drop. Finally, we perform quasi-3d numerical simulations of evaporating drops, where stability to azimuthal perturbations of the contact line is examined. We recover the “octopi” instability for IPA/Si configuration, similarly as seen in the experiments

Thin films flowing down inverted substrates: Three dimensional flow

We study contact line induced instabilities for a thin film of fluid under destabilizing gravitational force in three dimensional setting. In the previous work (Phys. Fluids, {\bf 22}, 052105 (2010)), we considered two dimensional flow, finding formation of surface waves whose properties within the implemented long wave model depend on a single parameter, $D=(3Ca)^{1/3}\cot\alpha$, where $Ca$ is the capillary number and $\alpha$ is the inclination angle. In the present work we consider fully 3D setting and discuss the influence of the additional dimension on stability properties of the flow. In particular, we concentrate on the coupling between the surface instability and the transverse (fingering) instabilities of the film front. We furthermore consider these instabilities in the setting where fluid viscosity varies in the transverse direction. It is found that the flow pattern strongly depends on the inclination angle and the viscosity gradient

Numerical simulation of the shape of charged drops over a solid surface.

In this work we study the static shape of charged drops of a conducting fluid placed over a solid substrate, surrounded by a gas, and in absence of gravitational forces. The problem can be posed, since Gauss, in a variational setting consisting of obtaining the configurations of a given mass of fluid that minimize (or in general make extremal) a certain energy involving the areas of the solid-liquid interface and of the liquid-gas interface, as well as the electric capacity of the drop. In [6] we have found, as a function of two parameters, Young's angle θY and the potential at the drop's surface V0, the axisymmetric minimizers of the energy. In the same article we have also described their shape and showed the existence of symmetry-breaking bifurcations such that, for given values of θY and V0, configurations for which the axial symmetry is lost are energetically more favorable than axially symmetric configurations. We have proved the existence of such bifurcations in the limits of very flat and almost spherical equilibrium shapes. In this work we study all other cases numerically. When dealing with radially perturbed equilibrium shapes we lose the axially symmetric properties and need to do a full three-dimensional approximation in order to compute area and capacity and hence the energy. We use a boundary element method that we have already implemented in [3] to compute the surface charge density. From the surface charge density we can obtain the capacity of the body. One conclusion of this study is that axisymmetric drops cannot spread indefinitely by introducing sufficient amount of electric charges, but can reach only a limiting (saturation) size, after which the axial symmetry would be lost and finger-like shapes energetically preferre