Session EF: Drops and Bubbles III*

Strongly nonlinear deformation and instability of a viscous droplet in an electric field

We consider a neutrally buoyant and initially uncharged drop in a second liquid subjected to an electric field. The liquids are taken to be leaky dielectrics, with the jump in electrical properties creating an electric stress that is balanced by hydrodynamic and capillary stresses. Creeping flow conditions are assumed to prevail, and the problem is solved numerically with the boundary integral method. The system is characterized by the physical property ratios $R$ (resistivities), $Q$ (permitivities) and $\lambda$ (dynamic viscosities). The relative importance of the electric stress and of the drop/ medium interfacial tension is measured by a dimensionless parameter called electric capillary number ($C_E$). We present a survey of the various behaviours obtained for a wide range of $R$, $Q$, and $\lambda$. When $\lambda=1$, we delineate the regions in the $(R,Q)$ plane, in which the deformation either asymptotes to a steady value or reaches a limit point past a critical $C_E$. We identify the latter with linear instability of the steady shape to axisymmetric disturbances. Other break up modes are also identified, as well as more complex behaviours such as subcritical bifurcations and transition from unstable to stable solution branches. We also show how the viscosity contrast $\lambda$ can stabilize the drop or advance break-up.   

Deformation, Breakup and transport of drops in MEMS devices

In recent approaches developed for laboratory-on-a-chip applications of microfluidic devices the fluids to be analyzed/manipulated are transported as drops. One of the possible techniques for transporting drops is by applying a non uniform electric field which has been successfully used in the past to transport rigid particles. A drop, however, not only experiences a net force which transports it, but also an electric stress on its surface which deforms it and can even break it into one or more droplets if the applied electric is sufficiently strong. We use the direct simulation approach to study this problem of deformation and transport of a dielectric drop in a non uniform electric field. Our technique is based on a finite element scheme in which the droplet and its surrounding fluid are moved using the fundamental equations of motion. The interface is tracked by the level set method and the electric forces are computed using the Maxwell stress tensor. The drop is assumed to be immiscible with the ambient fluid and its dielectric constant is different from that of the ambient fluid. The electric field is generated by placing electrodes at the bottom of the MEMS device.   

Electrowetting-induced oil film entrapment and instability

Electrowetting is a classic example of interaction between fluids and electric fields: by applying a voltage between a drop of conductive liquid and an insulator-covered hydrophobic electrode, the contact angle $\theta$ of the drop can be reduced by several tens of degrees compared to Young's angle (F. Mugele and J.-C. Baret, J. Phys.: Condens. Matter 17, R705, 2005). We investigate the spreading at variable rate of a water drop on a smooth hydrophobic substrate in an ambient oil bath driven by electrowetting. We find that a thin film of oil is entrapped under the drop. Its thickness is described by an extension of the Landau-Levich law of dip coating that includes the electrostatic pressure contribution. Once trapped, the thin film becomes unstable under the competing effects of the electrostatic pressure and surface tension and dewets into microscopic droplets, in agreement with a linear stability analysis. By varying the thickness of the dielectric layer, we expect to be able to tune the relative importance of the electrostatic contribution in future experiments. Our results recommend electrowetting as an efficient experimental approach to the problem of dynamic wetting in the presence of a tunable substrate-liquid interaction.   

Dynamics of Oscillating and Rotating Liquid Drop using Electrostatic Levitator

In order to understand the nonlinear behavior of liquid drop with oscillatory and/or rotational motions, an experimental study was performed. The electrostatic levitator was employed to achieve liquid drop formation on ground. A liquid drop with about 3 mm in diameter was levitated. The oscillation of mode $n$=2 along the vertical axis was induced by an external electrostatic force. The oscillatory motions were observed to clarify the nonlinearities of oscillatory behavior. A relationship between amplitude and frequency shift was made clear and the effect of frequency shift on amplitude agreed well with the theory. The frequency shift became larger with increasing the amplitude of oscillation. To confirm the nonlinear effects, we modeled the oscillation by employing the mass-spring-damper system included the nonlinear term. The result indicates that the large-amplitude oscillation includes the effect of nonlinear oscillation. The sound pressure was imposed to rotate the liquid drop along a vertical axis by using a pair of acoustic transducers. The drop transited to the two lobed shape due to centrifugal force when nondimensional angular velocity exceeded to 0.58.   

Thermocapillary migration of a droplet along a free surface

The proximity of a free surface can have a significant impact on the thermocapillary migration of droplets (or bubbles) submerged in a layer of fluid. For instance, since the thermocapillary effect produces forces on both the surface of the droplet and the surface of the fluid layer, the droplet can be forced to migrate in, or opposite to, the direction of the temperature gradient. To quantify the migration velocity we use the method of reflections to construct an explicit Lamb's series solution for the velocity field inside and outside the droplet in the presence of a uniform as well as a nonuniform temperature gradient.   

Thermocapillary migration of a drop at a fluid interface

Employing the thermocapillary effect to manipulate a liquid droplet trapped at a fluid-fluid interface has been proposed as a foundation for an optically controlled microfluidic device. We solve the Stokes equations for such a system, subject to a linear temperature gradient at infinity. The presence of a temperature gradient will induce surface tension variations along the fluid-fluid interfaces. These gradients, in turn, will give rise to a flow inside and outside of the droplet. The velocity and temperature fields are calculated numerically using a spectral collection scheme. We analyze the dependence of the flow structure and the drop migration velocity on the dimensionless parameters characterizing the problem.   

The Effect of Substrate Conductivity on the Evaporation of a Fluid Droplet

The evaporation of a fluid droplet is of fundamental importance in a wide range of practical applications, including ink-jet printing, spray cooling and DNA mapping, and has been the subject of growing research activity in recent years. A mathematical model for the evaporation of an axisymmetric sessile droplet whose contact line is pinned by surface roughness (or other) effects is developed and analysed. In particular, our model generalises the work of earlier authors to include the effect of substrate conductivity, and the theoretical predictions we obtain are in excellent agreement with the results of recent physical experiments performed using a variety of substrates.   

Sharp-interface conditions for fluid-fluid systems undergoing phase transformation

In the simplest models for the solidification of a pure material, the interface conditions at the solid-liquid interface include a balance of energy -- i.e. the jump in heat flux across the interface is balanced by the release of latent heat at the interface -- and a statement that the interface temperature is the equilibrium melting temperature of the substance. Such is the case in the classical Stefan problem. More sophisticated models of solidification adopt the Gibbs-Thomson equation, which recognizes that the interface temperature may depart from this equilibrium melting temperature due to curvature of the interface. Through the consideration of configurational forces in a thermodynamic framework, we obtain interfacial conditions for a fluid-fluid system undergoing phase transformation. In particular, we identify the counterpart of the Gibbs-Thomson equation. Our model equations include the effects of density difference between the phases, applied pressure, interfacial energy and kinetics, bulk viscosity, and interfacial viscosity. We discuss solutions to these equations for a problem involving the condensation of a liquid droplet within the vapor phase.