Session KF: Drops and Bubbles VII

Drops in Taylor-Couette flow: from toroidal rings to liquid sheaths

We study the deformation of a water drop in Taylor-Couette flow using numerical simulations and direct experimentation. Our experimental results demonstrate that either toroidal rings or sheath-like configurations are formed when the inner shaft is rotated. This is in agreement with results from axisymmetrical direct numerical simulations, obtained with a level-set method to track the deforming interface. The simulations show that ridges observed in experiments at the ends of spreading sheaths contain swirling vortex rings, which are accompanied by rings in the outer fluid. The simulations further elucidate the onset of sheath formation. An analytical model based on lubrication theory and the integral method has also been developed to investigate the dynamics of a thin sheath, especially at later stages of the evolution of sheaths. Results from this model are compared with those from direct numerical simulations and experiments. A parametric study is carried out to investigate the influence of the the viscosity ratio, interfacial tension and droplet size on the dynamics. Comparisons between the three approaches yield favourable agreement.   

Navier-Stokes computations for water drops falling in air: intermediate Reynolds numbers

We report Navier-Stokes computations, using a finite-element technique, of the steady descent of a deformable water drop through air. Variations in drop shape, internal circulation, and drag coefficient with Reynolds number are discussed in terms of simple physical mechanisms. Discrepancies between the computational results and the experimental data of Beard and Pruppacher, obtained in an open-return vertical wind tunnel, are interpreted in terms of scavenging from the polluted atmosphere of surfactants, as measured by Kawamura et al. in collected rainwater samples several hundred meters southeast of the wind-tunnel intake.   

Lattice Boltzmann Method for Interfacial Flows with High Density Ratio

A computational model using Lattice Boltzmann Equation (LBE) method for two-phase flow with sharp interface and high density ratio is developed. The Lattice Boltzmann scheme simulates incompressible two-phase flow by solving two distribution functions simultaneously. The interfacial dynamics are modeled by incorporating the intermolecular interaction force (He et al., JCP, pp642-663, 1999). By using a new surface tension formulation to eliminate oscillations in surface tension profile, numerical stability is significantly improved. Sharp interface can be maintained for flows with density ratio at $O$(10$^{2})$ or higher with little oscillation in velocity and pressure across interface. The detailed numerical assessment on the performance of the scheme based on the simulations of static bubble and rising bubble will be presented. The motion of a droplet on a wall is also studied using this improved method. The wetting boundary conditions on the wall are implemented to minimize the total free energy of the system. The motion of contact line, the contact angle, the surface tension, the velocity field and the pressure distribution are analyzed.   

Drop detachment in the presence of a diffusion-controlled surfactant at finite Peclet

When a buoyant drop is injected into a viscous external surfactant solution, the drop deforms to form an elongated shape, and detaches by the rapid formation of a neck. The thinning of this neck is driven by the local surface-tension related stresses, and resisted by the outflow of viscous liquid from the neck. Surfactant accumulates in the region of the neck as the interface contracts rapidly. The surfactant desorbs to establish a sublayer concentration adjacent to the interface in equilibrium with the local surface concentration, and diffuses away from the surface in an attempt to restore equilibrium. In the event that the diffusion flux is slow compared to the local rate of surface contraction, the local surface tension is reduced, slowing the rate of neck thinning and changing the neck shape. The occurrence of these non-equilibrium effects for a system of fixed physical chemistry as a function of surfactant concentration is studied numerically at finite Peclet number and compared to prior results in the sorption controlled limit.   

Characteristics and Dynamics of drops emitted from a Taylor cone

We report results from optical characterization of the stream of droplets emitted from the tip of a Taylor cone. We demonstrate the ability to image and track individual droplets emitted from a Taylor cone using fluorescence microscopy, pulsed laser illumination and statistical particle tracking velocimetry (SPTV). Single-exposed images are used to study the dependence of droplet size on flow rate, conductivity and field. Velocity distributions are obtained from analysis of double-exposure images using SPTV. In addition to droplet size and velocity distributions, we present data on the instability of the tip stream and on the formation and subsequent evolution of the spray cone.   

Motion and deformation of droplets through circular tubes

The motion and deformation of a droplet immersed in an immiscible fluid and flowing through a circular tube of comparable diameter is investigated. A parametric study is performed experimentally, varying the drop volume and capillary number. When increasing the parameters, the drop evolves from a circular shape to a more elongated one. A first transition occurs as the rear curvature of the drop inverses. When increasing further the parameters, the rear inwards curvature evolves into a cavity, an annular liquid sheet being formed at the back of the drop. A second transition takes place at a critical set of parameters, above which the drop no longer reaches a steady-state configuration. The annular sheet lengthens over time until break-up occurs. A numerical simulation has been conducted using the volume-of-fluid method. A comparison of the numerical results with the experimental observations will be given.   

Drop deformation in shear flow between parallel plates

Studying the nature of flow in confined geometries has become increasingly important due to downsizing of equipment. Examples include microfluidic devices as lab-on-a-chip and flow through porous media. Here, we focus on the flow of a single drop in a matrix fluid confined between two parallel walls, where the distance between the walls is in the order of the drop diameter. To model this system a 3-dimensional boundary integral method is used with the inclusion of the two parallel walls in the free-space kernels of the boundary integral method. The deformation of a drop in shear flow as function of the capillary number and the distance between the walls is studied. The drop shapes found in the presence of the walls substantially differ from the typical ellipsoidal shaped drops found in unbounded flows. Overall deformation, expressed in the Taylor deformation parameter, increases when reducing the distance between the walls. Furthermore, the angle of the major drop axis with the velocity direction also decreases. Finally, the effect of an insoluble surfactant is shown on deformation.   

Rheology of an emulsion of viscoelastic drops in steady shear

Rheology of dilute emulsions with viscoelastic inclusions in steady shear flow is numerically investigated using direct numerical simulations. A new mathematical formulation following Batchelor's work for purely viscous components is developed. Viscoelasticity is modeled using the Oldroyd-B constitutive equation. A front tracking finite difference code is used to numerically determine the drop shape, and solve for the velocity and stress fields. The effective stresses have three different components due to viscosity difference, interfacial tension and the drop phase viscoelasticity. The interfacial stresses---first and second normal stress differences and shear stresses---show behavior similar to a Newtonian emulsion. The normal stress difference due to the drop phase viscoelasticity is quadratic in shear rate and depends also on the relaxation time of the Oldroyd-B model. Drop phase viscoelasticity does not contribute significantly to effective shear viscosity of the emulsion.