Migration Of Drop Research Articles

Thermocapillary migration of a compound drop inside a spherical cavity

Thermocapillary migration of a compound drop inside a spherical cavity

Thermocapillary migration of a compound drop in an arbitrary viscous flow

The thermocapillary migration of a concentric compound drop in an arbitrary viscous flow under the consideration of negligible Reynolds number is investigated. The thermocapillary effect refers to the migration of a drop under the influence of a temperature gradient. The thermal and hydrodynamic problems are examined. The thermal field is governed by the heat conduction equation whereas the hydrodynamic fluid velocities are governed by the linearized Navier–Stokes equations. Presence of temperature gradient results in variation of the interfacial tension which is assumed to depend on temperature linearly. Variation of interfacial gradient leads to the coupling of the hydrodynamic problem with the thermal problem through the boundary condition. A complete general solution of Stokes equations is utilized to obtain closed-form expressions for the velocity vector and pressure. The hydrodynamic forces acting on the compound drop are obtained and expressed in terms of Fax́en’s law. Some important asymptotic limiting cases of hydrodynamic drag are also derived. The hydrodynamic drag for cases of uniform flow, shear flow, and heat source with the known ambient flow are derived and it is found that in the case of shear flow, the hydrodynamic drag is contributed only by the thermal component and the shear flow has no effect on it. The obtained results for drag and torque in the limiting cases are in agreement with the existing results in the literature. Furthermore, the migration velocity of the compound drop is obtained by equating the hydrodynamic drag force to zero. The results obtained for migration velocity are explained with the aid of graphs. The migration velocity is found to be a monotonic function of the Marangoni number and the radius of the innermost drop.

A model for thermocapillary driven motion of liquid drops on an inclined plane

Thermocapillary drop migration is driven by surface tension gradient. On an inclined plane, gravity acts as an additional force that can have a promoting or resisting effect depending on the thermal gradient direction. The present 2-D model extends Brochard and Chebbi models for thermocapillary migration of small drops on horizontal plates by accounting for Marangoni and Poiseuille flow components, while including the effect of gravity, acting in the case of inclines. The present treatment is based on fundamentals of conservation of mass and momentum transport, and uses the lubrication approximation and extended boundary conditions at the receding and advancing contact lines. A model for the dynamics of migration is provided, and an analytical solution is derived for drop migration dynamics and the critical angle of inclination.

Thermocapillary migration of a drop with a thermally conducting stagnant cap

Hypothesis The thermocapillary migration of a spherical drop with a stagnant cap in the presence of a constant applied temperature gradient can be strongly affected by the finite thermal conductivity of the stagnant cap.Numerics The heat conduction of the stagnant cap is analytically modeled. The effects of the additional interfacial stresses generated by the disturbances to the local temperature field due to the presence of the cap at the fluid-fluid interface and the corresponding velocity of migration of the drop are evaluated by solving for the temperature and hydrodynamic field equations in and around the drop. An asymptotic model is derived to predict the terminal velocity in the presence of an infinitely conducting stagnant cap.Findings The effects of the surface conductivity and size of the stagnation region alongside the bulk thermal conductivities and viscosities of the drop and surrounding media are evaluated. The terminal velocity of the drop is shown to have a monotonic dependence on the conductivity of the stagnant cap. The bounds to the terminal velocity increment due to the stagnant cap are derived. These bounds can be of significance to multiphysics problems involving particle laden drops, Pickering emulsions and other multi-phase technologies where the conductivity of the surface adsorbents is non-negligible.

Eccentricity-induced dielectrophoretic migration of a compound drop in a uniform external electric field

Eccentric compound drops, which are ubiquitous in many naturally inspired and engineering systems, can migrate under the sole presence of a uniform electric field, unlike the case of isolated single drops. Here, we report the migration of eccentric compound drops under a uniform electric field, imposed parallel to the line of centres of the constituting drops, by developing an approximate analytical model that applies to low Reynolds number limits under negligible droplet deformation, following axisymmetric considerations. In contrast to the sole influence of the electrohydrodynamic forces that has thus far been established to be emphatic for the eccentric configuration, here we report the additional effects induced by the dielectrophoretic forces to result in decisive manipulation in the drop migration. We show that the relative velocity between the inner and outer drops, which is a function of the eccentricity itself, dictates the dynamical evolution of the eccentricity variation under the competing electrohydrodynamic and dielectrophoretic interactions. This brings out four distinct regimes of the migration characteristics of the two drops based on their relative electro-physical properties. Our results reveal that an increase in eccentricity and the size ratio of the inner and outer droplets may induce monotonic or non-monotonic variation in the drop velocities, depending on the operating regime. We show how the interplay of various properties holds the control of selectively increasing or suppressing the eccentricity with time. These findings open up various avenues of electrically manipulative motion of encapsulated fluidic phases in various applications encompassing engineering and biology.

Electrically modulated cross-stream migration of a compound drop in micro-confined oscillatory flow

In this work, we present a means of controlling the cross-stream trajectory of a compound drop under the combined action of a transverse electric field and an oscillatory axial pressure gradient in a microfluidic channel. We bring out the decisive role of the flow pulsation in delaying the drop's attainment of a steady-state transverse position. With an enhancement in the frequency of oscillation, both the axial and transverse movement of the drop may be preferentially attenuated, with its dynamic traversal occurring in a locus offset to the central axis as precisely set in by the action of transverse electric forcing, to bring in exclusive controllability on the drop trajectory along with its eventual position of settlement. Moreover, our results also decipher that the value of the viscosity ratio between inner-to-outer droplet phases having less than unity delays the compound drop migration, whereas the converse enhances the same. In addition, we observe that a leaky dielectric compound drop having the electrical permittivity ratio of the inner-to-outer droplet phase surpassing their respective electrical conductivity ratio is not only capable of selectively altering the direction of the resulting drop trajectory from toward the channel centerline to away from the same but at the same time facilitates a precise settling of the same at an intermediate transverse location by harnessing the interplay of electrical and hydrodynamic shear. We further identify the key dimensionless parameters along with their desirable ranges accountable for the directional switching of the drop trajectory with high specificity. These findings open up novel perspectives of controllable maneuvering of the double emulsion system in a confined microenvironment bearing decisive implications in engineering and biology.

Study of the trapping mechanism of merging drops moving under thermocapillary migration on a surface with wettability contrast

Trapping of drops due to wettability contrast in porous materials emerges in a variety of subsurface and manufacturing applications. In carbon capture and storage, carbon dioxide must be trapped in order to avoid its release into the atmosphere, while trapped oil must be displaced. Both carbon dioxide and oil can exist as drops in porous media. The migration of a single drop attached to a wall can be hindered if the wall surface has a different wettability. However, the trapping condition becomes more complicated in the presence of two or more drops. In this work, we aim to study the trapping mechanism during thermocapillary migration of a merging drop on a heterogeneous surface. To do so, numerical simulations have been performed using the Front-Tracking/Finite Volume Method, where the Navier-Stokes equations are coupled with the conservation equations. The generalized Navier boundary condition (GNBC) has been used as the slip model to remove the viscous singularity. The combined finite-volume and tracking method is able to deal with different types of discontinuities in compressible or incompressible fluid flows, as e.g., interfaces. The material properties of the drop and the ambient fluid are different, and surface tension depends on the temperature. The results have shown that there exist three regimes characterizing the motion of merging drops, including the passage of the leading drop, the trapping of the merging drop, and the partial trapping, which correspond to the trapping of the leading drop but the passage of the merging drop. We show that there is a critical wettability contrast at which the merging drop gets trapped. The effects of Marangoni number (Ma) and viscosity ratio are investigated. The critical wettability contrast decreases with the increase of Ma number and the regime shift as a function of the viscosity and Marangoni number. These findings have implications for the design of geological carbon dioxide storage, improvement of oil recovery and microfluidic device development.

Migration of a viscoelastic drop in a ratchet microchannel

We numerically investigate the deformation and migration of a viscoelastic droplet in a ratchet microchannel, a phenomenon that occurs in many practical applications and involves rich physics due to the associated complex geometry and non-Newtonian nature of the fluid. The numerical method required to handle viscoelastic fluid is also extremely challenging, especially when the Weissenberg number, a dimensionless relaxation time related to the viscoelastic fluids, is high. In the present study, we incorporate a log-conformation stabilization method that overcomes this numerical problem and examine the shape transition dynamics of a viscoelastic droplet for a wide range of parameters, including the wavelength and type of the ratchet, Weissenberg number, and Reynolds number. The droplet dynamics in a ratchet microchannel is compared to that in a straight channel. The presence of the ratchet section causes a distinct stress distribution in the viscoelastic droplet, which significantly influences the residence period, the shape of the droplet in the microchannel, and shape recovery dynamics when flowing out of the channel. • Deformation and migration of a viscoelastic droplet in a ratchet microchannel is investigated. • A log-conformation stabilization method is incorporated. • The wavelength and type of the ratchet play an important role in the dynamics and residence period. • The roles of Reynolds, Capillary, and Weissenberg numbers have been investigated. • The study may help understand the migration of biological cells in complex geometries.

Shear-induced migration of a viscous drop in a viscoelastic liquid near a wall at high viscosity ratio: Reverse migration

• A drop under shear migrates laterally away from a wall in shear. • Medium viscoelasticity and viscosity ratio affects the migration. • High viscosity ratio and high Deborah number leads to migration towards the wall. • The critical viscosity ratio varies inversely with the capillary number. • The critical Deborah number also varies inversely with the capillary number. Wall-induced migration of a viscous drop in a viscoelastic fluid subjected to a plane shear is numerically simulated to investigate the effects of drop/matrix viscosity ratio. In a Newtonian system, drop migration away from the wall is inhibited as the viscosity ratio is increased. Here, we show that the introduction of the matrix viscoelasticity further decreases the migration and can even reverse its direction ‘from away’ to ‘towards the wall’, a phenomenon not seen in Newtonian systems. The migration towards or away from the wall eventually settles in a quasi-steady state that only depends on the instantaneous wall separation independent of the initial position of the drop. Drops migrating towards the wall initially increase their velocity, but as they approach the wall, they decelerate, showing a non-monotonic variation. The migration direction depends on the viscosity ratio, viscoelasticity (Deborah number), and the capillary number. We compute phase diagrams in the parameter space showing boundaries where migration changes direction. The critical Deborah number (at a fixed viscosity ratio) and the critical viscosity ratio (at a fixed Deborah number) for direction reversal approximately scales with the inverse of the capillary number.

The ejection of large non-oscillating droplets from a hydrophobic wedge in microgravity

When confined within containers or conduits, drops and bubbles migrate to regions of minimum energy by the combined effects of surface tension, surface wetting, system geometry, and initial conditions. Such capillary phenomena are exploited for passive phase separation operations in micro-fluidic devices on earth and macro-fluidic devices aboard spacecraft. Our study focuses on the migration and ejection of large inertial-capillary drops confined between tilted planar hydrophobic substrates (a.k.a., wedges). In our experiments, the brief nearly weightless environment of a 2.1 s drop tower allows for the study of such capillary dominated behavior for up to 10 mL water drops with migration velocities up to 12 cm/s. We control ejection velocities as a function of drop volume, substrate tilt angle, initial confinement, and fluid properties. We then demonstrate how such geometries may be employed as passive no-moving-parts droplet generators for very large drop dynamics investigations. The method is ideal for hand-held non-oscillatory ‘droplet’ generation in low-gravity environments.

Motion and deformation of migrating compound droplets in shear-thinning fluids in a microcapillary tube

The deformation dynamics of a moving compound droplet inside a microcapillary tube is investigated in Newtonian and non-Newtonian ambient fluids. The numerical method used is a ternary phase-field model to determine the temporal evolution and the subsequent hydrodynamics of compound droplets traveling inside a microcapillary tube. By solving the axisymmetric momentum equations, the compound droplet movement is identified by tracing the spatiotemporal evolution of a pair of two deformable interfaces. Results indicate that the rheological properties of the continuous phase significantly influence the compound droplet characteristics. The position of the center of the mass of both inner and outer drops along the tube axis increases linearly in a Newtonian ambient fluid while it is nonlinear in a non-Newtonian medium, and the deviation from the linear behavior increases as the concentration of the polymer solution increases. These distinct behaviors occur after a particular normalized time (t*, which is normalized with R/Uavg in which R is the radius of the tube and Uavg is the average inlet velocity) during the early stages of droplet formation (t*∼1). The migration of the compound drop inside a Newtonian fluid approaches a relatively constant velocity after a specific time of the initial movement. Conversely, the movement velocity of both drops increases gradually until the pinch-off moment in non-Newtonian fluids, which means that a moving compound drop has an accelerative motion in all stages of movement in non-Newtonian fluids. The findings presented here are essential for employing compound droplets in droplet microfluidic systems for biological applications.

Hydrodynamics of a compound drop in plane Poiseuille flow

We numerically investigate the hydrodynamics of a two-dimensional compound drop in a plane Poiseuille flow under Stokes regime. A neutrally buoyant, initially concentric compound drop is released into a fully developed flow, where it migrates to its equilibrium position. Based on the results, we find that the core–shell interaction affects the dynamics of both the core and the compound drop. During the initial transient period, the core revolves about the center of the compound drop due to the internal circulation inside the shell. At equilibrium, depending upon the nature of the flow field inside the shell, we identify two distinct core behaviors: stable state and limit-cycle state. In the stable state, the core stops revolving and moves outward very slowly. The core in the limit-cycle state continues to revolve in a nearly fixed orbit with no further inward motion. The presence of the core affects both deformation and migration dynamics of the compound drop. A comparison with the simple drop reveals that the core enhances the deformation of the compound drop. The outward moving core in the stable state pushes the compound drop toward the walls, while the revolving core in the limit-cycle state causes the compound drop to oscillate at its equilibrium position. The migration of the compound drop also affects the eccentricity of the core significantly. From the parametric study, we find that the core affects the compound drop dynamics only at intermediate sizes, and an increase in any parameter sufficiently causes a transition from the limit-cycle state to the stable state.

Flotation mechanism of sulphide melt on vapour bubbles in partially molten magmatic systems

The motion of dense Fe-rich immiscible sulphide liquids is generally supposed to be dominated by gravitational settling in crustal magma chambers, however they may become buoyant by attachment to low-density vapour bubbles to form compound drops, possibly contributing to the upward transfer of sulphur and transition metals in degassing magma bodies. Here, using numerical models, we consider constraints on the flotation of compound drops, and find a wide range of morphologies that would be expected to be capable of migrating through magmatic systems, including both melt- and mush-dominated domains, but also some situations where the compound drops will spontaneously separate to allow the vapour to rise while the sulphide phase is sequestered in the mush. The stability of compound drops is empirically related to a function f(Bo)=|σMV−σMS|σMS⋅1/BoS+|σVS−σMS|σMS⋅1/BoV, where M, V, and S denote silicate melt, vapour, and sulphide liquid, σ is surface tension (N⋅m−1), and Bo is the Bond number. Over a range of plausible conditions in magmas, f(Bo) must exceed values of 3 to 4 for capillary forces to overcome buoyancy forces tending to pull them apart. Using published thermodynamic models for vapour and sulphide solubility in silicate melts, we show that in many magmas the second boiling of vapour occurs synchronously with the saturation of sulphide liquid, increasing the potential for coupling of vapour and sulphide droplets. Diffusive coarsening produces mm-sized bubbles within 100-500 years, generating enough buoyancy to trigger the migration of compound drops. During compaction-driven expulsion of interstitial melt, compound drops can pass through constrictions between crystals in mush. The flotation of sulphide-vapour aggregates is likely to occur in crustal magma reservoirs and provides a feasible mechanism for the removal of sulphide liquid from crystal mushes, promoting its ability to participate in the generation of magmatic sulphide and porphyry copper deposits, emissions of the metals during volcanic eruptions, and even the remobilization of chalcophile metals sequestered in deep arc cumulates to generate porphyry Cu-(Au) deposits.

Single recalcitrant bubble simulation using a hybrid lattice boltzmann finite difference model

In this paper, 2D simulations of a single recalcitrant bubble using a hybrid finite difference lattice Boltzmann method are presented. To solve the flow field, the conservative phase field lattice Boltzmann model of Geier et al. (2015) is employed with a multi-relaxation-time (MRT) scheme which makes it capable of handling multiphase flows with high density and viscosity contrasts. A finite difference approach is applied to obtain the temperature field with a second-order spatial discretization (central) which marches in time with the 2nd order Runge-Kutta marching technique. The well-known continuum surface tension force (CSF) is implemented to define the thermocapillary force which depends on the temperature field. Before delving into the main aim of this work, we first assess the accuracy of the present model by verifying the Laplace law for a static bubble with four different values of surface tension. To further validate the model, thermocapillary migration of a deformable drop, and thermocapillary flow with two superimposed planar fluids are studied as our second and third benchmarks. Comparing the analytical solutions for both velocity and temperature fields with the simulation ones, good agreements are found. We then propose a simple theory with some simplification for the study of the migration of a single recalcitrant bubble, and after that weperform a series of 2D simulations to analyze the effects of density ratios, viscosity ratios, entrance average velocities of the flow, and the values of second derivative of surface tension with respect to the temperature.

Lateral migration of viscoelastic droplets in a viscoelastic confined flow: role of discrete phase viscoelasticity.

The cross-stream motion of viscoelastic droplets in viscoelastic fluids has received little attention since the classical study of migration of drops in a second order fluid. In this work, going beyond the existing classical theory, we experimentally elucidate the effect of drop-to-medium viscosity ratio k and elasticity ratio ξ on wall and center migration of viscoelastic droplets in a Poiseuille flow of a viscoelastic medium (PVP) at low Reynolds numbers (Re ≪ 1). We observed a contrasting migration behavior of Newtonian and viscoelastic droplets having the same viscosity ratios and propose the presence of a lift force FVD due to the viscoelasticity of the droplet phase. We use analytical scaling and empirical modelling to show that the force FVD scales with a prefactor that depends upon the Weissenberg number WiD and drop-to-medium viscosity ratio k and elasticity ratio ξ. Further, we utilize the proposed force for sorting of viscoelastic and Newtonian droplets.

Thermally modulated cross-stream migration of a surfactant-laden deformable drop in a Poiseuille flow

In the present study, we investigate the cross-stream migration of a deformable droplet suspended in a non-isothermal Poiseuille flow in the presence of bulk-insoluble surfactants. Owing to the non-linearity present in the system of governing equations, an asymptotic approach is adopted, in an effort to capture the intricate and non-trivial coupling between the various influencing parameters. With the assumption of negligible inertia in fluid flow and convective transport of thermal energy, we obtain the droplet migration velocity through small-deformation perturbation analysis for two different limiting cases, namely, convection-driven-surfactant transport and surface-diffusion-dominated surfactant transport. Under each of these limiting cases, the cross-stream migration of droplet is studied for a constant temperature gradient applied in the same direction as well as in a direction opposite to the imposed flow. For the former limiting case, the droplet is always migrates towards the centerline of flow. For a highly viscous droplet, the direction of its cross-stream migration reverses. When the temperature decreases in the direction of the imposed flow, cross-stream migration velocity reduces with increase in the applied temperature gradient till a critical point is reached at which there is no cross-stream migration. Beyond the critical point, there is a gradual increase in the magnitude of the cross-stream velocity. The droplet, below the critical temperature gradient, migrates towards the flow centerline; however, above it the droplet moves away from the centerline. For the other limiting case of surfactant transport dominated by surface convection, the magnitude of the cross-stream velocity is found to be significantly larger and at the same time independent of the droplet-carrier phase viscosity ratio.

An efficient front-tracking solver for thermocapillary migration simulations

An efficient numerical scheme with the front-tracking strategy is presented to investigate the thermocapillary migration of drops. The speed bottleneck of current simulations, especially when the Marangoni number (Ma) is large, is caused by the non-separable elliptic partial differential equations (PDE). In this paper, Fast Fourier Transform is adopted to solve the standard Poisson equation (Tri-FFT), and the non-separable elliptic PDE is solved by the iterative usage of Tri-FFT with high efficiency. In general, the computational cost of the whole system is very low with a hybrid package of Tri-FFT and the Successive Over-Relaxation methods. Finally, the impacts of Ma and Reynolds numbers on thermocapillary migration simulations when Ma>100 are studied.

Cross-stream migration of drops suspended in Poiseuille flow in the presence of an electric field.

The present study focuses on the cross-stream migration of a neutrally buoyant two-dimensional drop in a Poiseuille flow in a channel under the influence of an electric field. In the absence of an electric field, the important nondimensional parameters describing this problem are the viscosity ratio (λ) between the drop fluid and the surrounding medium, the ratio of drop diameter to channel height (a^{*}), and the capillary number (Ca). The influence of all these parameters on drop migration is investigated. It is observed that a large drop moves slowly as compared to a smaller drop, but attains a steady shape at the center line of the channel. The increase in value of the capillary number enhances the cross-stream migration rate, while the increase in viscosity ratio reduces the tendency of the drops to move towards the channel center line. The presence of an electric field introduces additional interfacial stresses at the drop interface, which in turn alters the dynamics observed in the absence of an electric field. Extensive computations are carried out to analyze the combined effect of the electric field and the shear flow on the cross-stream migration of the drop. The computational results for a perfect dielectric indicate that the droplet migration enhances in the presence of an electric field. The permittivity ratio (S) and the electric field strength (E) play major roles in drop migration and deformation. Computations using the leaky dielectric model also show that for certain combinations of electrical properties the drop undergoes immense elongation along the direction of the electric field. The conductivity ratio (R) is again a vital parameter in such a system of fluids. It is further observed that for certain conditions the leaky dielectric drops exhibit rotation together with translation.

Study of the motion of a spheroidal drop in a linear shear flow

The motion of a spheroidal deformable drop in a simple shear flow is simulated using a finite-difference/front-tracking method. The effect of surface tension coefficient, viscosity ratio and inertia on lateral migration and deformation of the drop is investigated. It is revealed that the deformation of a spheroidal drop is directly related to the capillary and Reynolds numbers. In the limit of finite Reynolds numbers, the equilibrium position of prolate drops depends strongly on the viscosity ratio; the final position of more viscous drops is closer to the wall in contrast with the spherical ones. As the deformability of drops increases and the inertial force decreases, the rate of migration of the prolate drops increases. Although the steady-state position does not depend on the capillary and Reynolds numbers, the migration rate depends considerably on these dimensionless parameters. In addition, the rate of migration is a decreasing function of the aspect ratio due to different direction of the lift force acting on the drop.

Experimental research on thermocapillary migration of drops by using digital holographic interferometry

The thermocapillary migration of drops in a rectangular cell, with a heated top wall and a cooled bottom wall, was investigated experimentally on the ground. The rectangular test cell was 70 mm high, with a horizontal cross section of 40 mm × 40 mm. In the present experiment, 30 cSt silicon oil was used as the continuous phase, and a water–ethanol mixture was used as the drop phase, respectively. The drops ranged in size from 1.87 to 6.94 mm in diameter and were injected into the continuous phase, where the temperature gradients ranged from 0.193 to 0.484 °C mm−1. In order to measure the temperature distribution of the liquid, a digital holographic interferometry was used, which was non-contact, full-field, and in-situ. The holograms were recorded, and then the corresponding wrapped phase distributions images were numerically reconstructed. The temperature distribution of the continuous phase liquid in the cell had been obtained following the unwrapping. Also, through an algebra layer analysis, the temperature distribution around the drop during the thermocapillary migration was obtained. As a result, the drop was colder than the continuous phase liquid, and a thermal wake existed behind the drop. The influence of convective transport on the drop migration was also investigated for the Marangoni number in the range of 7–174. With the increasing of the Marangoni number, the dimensionless interface temperature difference decreased, which was caused by the convective transport enhanced results in the drop thermocapillary migration velocity becoming decreased. The data were compared with previous space experiments to explain the phenomena of the drop migration. Finally, with the increasing Marangoni numbers, the length of the thermal wake region increased, and the thermal wake region became extended.