Session MS: Drops XI

Numerical studies of the deformation of an initially rotating droplet

An initially rotating droplet subjected to an impulsive acceleration by the uniform free stream is studied numerically using the moving mesh interface tracking method (Quan, \textit{et}. \textit{al}., J. Comp. Phys, \textbf{221}, 2007) at \textit{Re}$_{i}$=40, \textit{We}$_{i}$=40, \textit{$\eta $}=\textit{$\lambda $}=50. The rotation axis is aligned in the transverse direction and the dimensionless rotation rate, \textit{$\Omega $}$^{\ast }$, from 0--1 is considered. For \textit{$\Omega $}$^{\ast }\le $0.2, the droplet deforms in a similar fashion to the stationary droplet except the droplet is tilted. At higher \textit{$\Omega $}$^{\ast }$, the centrifugal force acting on the droplet increases and the droplet is spun radially away from the rotation axis. As a result, the surface area normal to the free stream decreases and this leads to a significant reduction in drag coefficient. The droplet deformation also has a substantial effect on the lift coefficient. As the droplet deforms, the kinetic energy of the rotation is mainly transferred to the surface energy on the interface, which results in a decline in lift coefficient after the initial jump as the surrounding flow field becomes symmetric again.   

The effect of surfactant redistribution on interactions of deformable drops in gravity and a temperature gradient

Trajectories are calculated by the boundary-integral method for two contaminated deformable drops under the combined influence of buoyancy and a constant temperature gradient at low Reynolds number and with negligible thermal convection. The surfactant is bulk-insoluble, and its coverage is determined by solution of the time-dependent convective-diffusion equation. Two limits are considered. For small drops, the deformation is small, and thermocapillary and buoyant effects are of the same order of magnitude. In this case, comparison is made with incompressible surfactant results to determine when surfactant redistribution becomes important. Convection of surfactant can lead to elimination of saddle points in the relative-trajectory phase plane and can increase the difference between the drops' velocities. For larger drops, deformation can be significant, leading to breakup or capture, and buoyant motion dominates thermocapillary migration. In this case, convection of surfactant can increase deformation and offset previously observed inhibition of breakup for clean drops when the driving forces are opposed.   

Inertial effects on the dynamics of a drop in a shear flow and the dispersed stresses

We numerically simulate the flow field and the streamlines around a drop in shear using a front tracking finite difference method. Inertia destroys the closed streamlines found in a Stokes flow, and gives rise to spiraling and reversed streamlines. Orientation angle of the drop increases with inertia and becomes larger than 45 degrees with increasing inertia at low capillary numbers. However, at higher capillary numbers, it becomes nonmonotonic--it first increases with Reynolds number and then decreases as the drop deformation becomes large. Inertia also introduces transient overshoot and oscillation in the drop deformation before it achieves a steady shape. We provide a simple model for the oscillation that shows the correct scaling with Reynolds and capillary numbers. We also investigate the interfacial tensor that determines the dispersed stress in an emulsion of such drops. We show that the higher than 45 degree inclination leads to a change in sign of the dispersed normal stress differences. The effects of viscosity ratio on the streamline pattern around a drop are also studied.   

Effect of drop shape on thermally-induced drop motion at the free interface of immiscible liquid layers

Drops at the air interface of immiscible liquids (water on oil) usually form partially-submerged lens shapes. When a lateral thermal gradient is maintained along the surface, such drops move in the direction of decreasing temperatures, as reported in earlier studies. We show that in addition to the lens configuration, it is possible to create spherical (ball-shaped) drops at the interface. Unlike lens-shaped drops, such spherical drops migrate towards warmer regions; i.e. direction of increasing temperatures. Opposite direction of thermally induced motion for drops at the free surface of immiscible liquids is explained based on drops shape and the dynamics of the underlying liquid film subject to a thermal gradient; mainly deformation of the free surface, and the development of an outward moving (hot to cold) flow at the free interface. The proposed physical models predict experimental results satisfactorily. Thermocapillary motion of drops on liquid platforms is ideal for biochemical Microsystems and Lab-on-chip applications where droplets can be transported faster, with higher level of controllability and with less thermal loading of drops as compared to using solid substrates. In addition, other disadvantages of using dry surfaces such as drop evaporation, contamination, and surface pinning are avoided.   

Transient drop dynamics in convergent-divergent tubes filled with liquids

The transient dynamics of a deformable drop with initial momentum moving in convergent-divergent liquid-filled tubes is numerically studied by the moving mesh interface tracking (MMIT)/finite volume method. The geometry effects of the tube on the drop deformation and on the drag coefficient are investigated by varying the radius of the tube neck, and the smallest neck radius is 1.25 times of the initial droplet radius. The deformability effects on the droplet dynamics are examined by simulating cases with three Weber numbers. The drop experiences a dramatical deceleration as it approaches and enters the narrow region of the tube, and the drag coefficient increases with the decrease of the radius of the neck. As the Weber number increases, the droplet deforms more, and for the largest Weber number, the initially spherical droplet deforms to a Taylor-drop like shape, especially for the tube with the narrowest neck. After the drop exits the neck, the drop experiences oscillations. The thin film between the drop and the tube wall in the narrow region is resolved by local mesh adaptations. The quantitative analysis will be presented.   

Effects of Deformation on Drag and Lift Forces Acting on a Droplet in a Shear Flow

The droplet behavior in a linear shear flow is studied numerically to investigate the effect of deformation on the drag and lift acting on droplet. The droplet shape is calculated by a level set method which is improved by incorporating a sharp-interface modeling technique for accurately enforcing the matching conditions at the liquid- gas interface. By adopting the feedback forces which can maintain the droplet at a fixed position, we determine the acting force on a droplet in shear flow field with efficient handling of deformation. Based on the numerical results, drag and lift forces acting on a droplet are observed to depend strongly on the deformation. Droplet shapes are observed to be spherical, deformed, and oscillating depending on the Reynolds number. Also, the present method is proven to be applicable to a three- dimensional deformation of droplet in the shear flow, which cannot be properly analyzed by the previous studies. Comparisons of the calculated results by the current method with those obtained from body-fitted methods [Dandy and Leal, \emph{J. Fluid Mech.} 208, 161 (1989)] and empirical models [Feng and Beard, \emph{J. Atmos. Sci.} 48, 1856 (1991)] show good agreement.   

Effect of relative humidity on contact angle of inkjet-printed evaporating colloidal drops

The deposition behavior of inkjet-printed aqueous colloidal drops onto glass and polymer (PEN and PET) substrates has been investigated by using fluorescence microscopy, a high-resolution CCD camera, and scanning electron microscopy. Real-time side-view images show that the contact angle of an evaporating colloidal drop is a function of the ambient humidity. The relative humidity also affects the extent to which the drop is able to spread after impacting a substrate, the evaporation rate at the drop surface, and the evaporatively-driven flow inside the drop that drives the suspended particles towards the contact line. The difference between the contact line velocity and liquid velocity at the drop contact line induced by evaporation creates a larger contact angle compared to that of the case without evaporation. This increase in contact angle becomes more significant for a low ambient humidity. Results also show that the particle deposition area and pattern change with the ambient humidity.   

The electro-hydrodynamic flow about and the shape of strongly deformed drops

A liquid drop is suspended in another liquid and is exposed to an otherwise uniform electric field. For strong fields, the drop can elongate significantly. We analyze the strong deformation problem using slender-body analyses, the solution being obtained via expansion in the small slenderness. This parameter is not a priori prescribed, and must be found throughout the course of the asymptotic solution. We employ matched asymptotic expansions to calculate the electric and flow fields. The fields within the drop are continued into the `inner' region outside the drop, at the drop cross-sectional scale, and are then matched into a singularity representation in the `outer' region, at the drop longitudinal scale. For both dielectric and leaky dielectric liquids we obtain the scaling of Sherwood (1991), where the aspect-ratio is proportional to the 6/7-power of the electric field.   

Contact line dynamics of a liquid meniscus advancing into a microchannel with chemical heterogeneities

We examine the motion of a liquid meniscus and associated contact lines advancing into a two-/three-dimensional (2D/3D) microchannel with chemically heterogeneous inner walls. Our study is based on a phase field model of the Cahn-Hilliard type, appropriately modified to take into account the interaction between the fluid and the walls. By solving this model numerically, we characterize the influence of the chemical disorder of the walls on both the interface and contact line dynamics. We perform a detailed statistical analysis of our numerical results by generating several chemical disorder realisations. Examination of the advancing and receding motion of the contact lines in the 2D case reveals that the apparent contact angle suffers a hysteresis behaviour induced by the wall disorder and enhanced as the disorder strength is increased. For the 3D system it is shown that the surface chemical disorder makes the interface and contact line undergo a kinetic roughening process, characterised by a scaling growth with pinning-depinning effects and associated avalanche dynamics.   

Dynamics of Acoustically Vaporized Microdroplets

A combined theoretical and computational approach is utilized to understand the bubble evolution dynamics resulting by vaporizing the superheated dodecafluoropentane (DDFP, C$_{5}$F$_{12})$ microdroplets via an acoustic perturbation. This work is inspired by a developmental gas embolotherapy technique for cancer treatment by infarcting tumors using selectively formed gas bubbles. The evolution process comprises of three regimes; an initial linear rapid spherical growth followed by a linear compressed oval shaped growth and finally a slow asymptotic non-linear spherical growth. The bubble evolution process compares quite well with the ultra high-speed experiments. The final bubble radius scales linearly with the initial droplet radius and is approximately five times the initial droplet radius. A pressure pulse with amplitude approximately twice as that of ambient conditions is observed. The pressure pulse wavelength increases with an increasing droplet size whereas the pulse amplitude is weakly dependent on droplet size. This work is supported by NIH grant R01EB006476.