Drops In Simple Shear Flow Research Articles

A lattice Boltzmann modeling of viscoelastic drops’ deformation and breakup in simple shear flows

The deformation and breakup of viscoelastic drops in simple shear flows of Newtonian liquids are studied numerically. Our three-dimensional numerical scheme, extended from our previous two-dimensional algorithm, employs a diffusive-interface lattice Boltzmann method together with a lattice advection–diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids and the latter to describe the polymer dynamics modeled by the Oldroyd-B constitutive model. A block-structured adaptive mesh refinement technique is implemented to reduce the computational cost. The multiphase model is validated by a simulation of Newtonian drop deformation and breakup under an unconfined steady shear, while the coupled algorithm is validated by simulating viscoelastic drop deformation in the shear flow of a Newtonian matrix. The results agree with the available numerical and experimental results from the literature. We quantify the drop response by changing the polymer relaxation time λ and the concentration of the polymer c. The viscoelasticity in the drop phase suppresses the drop deformation, and the steady-state drop deformation parameter D exhibits a non-monotonic behavior with the increase in Deborah number De (increase in λ) at a fixed capillary number Ca. This is explained by the two distribution modes of the polymeric elastic stresses that depend on the polymer relaxation time. As the concentration of the polymer c increases, the degree of suppression of deformation becomes stronger and the transient result of D displays an overshoot. The critical capillary number for unconfined drop breakup increases due to the inhibitive effects of viscoelasticity. Different distribution modes of elastic stresses are reported for different De.

Cox modified model to describe the time evolution of a drop deformation for high viscosity drops in simple shear flow

In this work a modification of Cox's model for the deformation of a highly viscous drop is presented, which describes accurately the period of the oscillations during transient deformations. It is demonstrated that the oscillation period depends not only on the shear rate as predicted by Cox model but also on the viscosity ratio. This modified model is obtained from the analysis of numerical simulations of drop deformation using the boundary element method, while the experimental results are those obtained in quasi-simple shear flow generated in a two roll mill (2TRMs) device, as well as experimental data available in the literature. The predictions of the periods match very well with experimental data.

Motion of a spherical liquid drop in simple shear flow next to an infinite plane wall or fluid interface

The velocity of translation of a spherical liquid drop under the influence of simple shear flow near an infinite plane wall or fluid interface is computed with high accuracy using a boundary-integral method for Stokes flow based on Fourier decomposition in the absence of significant gravitational effects. Accurate results are obtained for very small gaps between the drop surface and the wall or interface, several orders of magnitude smaller than the drop diameter. As the gap decreases, a liquid drop tends to stick to a solid wall but is able to slide over an interface irrespective of the drop and ambient fluid viscosities. In the case of flow near an interface, the dependence of the drop velocity on the drop position and viscosity ratio is not necessarily monotonic. The interfacial deformation is computed to leading order with respect to the capillary number and is shown to exhibit an asymmetry due to the presence of the wall or interface. The predictions of the asymptotic analysis for nearly spherical drops are in agreement with boundary-element simulations of three-dimensional flow.

Breakup of drops in simple shear flows with high-confinement geometry

Abstract The problem of a drop subject to a simple shear flow in high constriction geometry is addressed numerically for different flow conditions. Wall effect on the critical capillary number and drop deformation is analyzed. Under uniform condition, drops in low and moderate Reynolds flows are more stable when the confinement is increased. The critical capillary number is shown to increase for drops more viscous than the medium (viscosity ratio λ = 0.3 ) and decreases when the medium is more viscous ( λ = 1.9 ) or when Reynolds number is increased. A discussion on the accuracy of the numerical method and solutions to typical problems are included for comparison. The drop interface is reconstructed using the piecewise linear interface calculation (PLIC) and transported with the volume-of-fluid (VOF) method, which follows unsplit case-by-case schemes based on the basic donating region (BDR) or the defined donating region (DDR). Surface tension is included with the continuum-surface-force (CSF) model. A high-resolution (SMART) semi-implicit finite-volume discretization is employed in the linear momentum equations. Mass is conserved by following an implicit pressure-correction method (SIMPLEC). The normal vector of the interface is computed from height functions using least squares fitting. The advantage of the DDR scheme lies in its volume-conserving capabilities which have not been exploited in recent investigations.

Production of Particle‐Stabilized Nonspherical Emulsion Drops in Simple Shear Flow

AbstractNonspherical drops are of interest in the formation of microcapsules in life sciences like food, pharmacy, and cosmetics, agro and fine chemicals as well as material sciences. Out of many systems, particle‐stabilized emulsion drops, so‐called Pickering emulsions, exhibit an interesting formulation. Systems with Pickering particles applied in an excess amount were investigated. During deformation, the particles fully covered the enlarged drop interfaces and prevented its relaxation to a spherical drop shape. Nonspherical drops could be produced in simple shear flow using an adequate process routine. The production in a simple device is a promising high‐throughput alternative to microfluidic devices.

Emulsification of particle loaded drops in simple shear flow

Abstract We investigated the influence of a particle load of disperse phase drops on their emulsification. Newtonian liquids were chosen for this investigation and a surfactant was present in all cases. A Couette-like shear cell was used enabling us to apply simple shear flow. The drop size distribution and drop morphology of the resulting oil-in-water emulsion were characterized. The influence of the wetting characteristics of the particles and the influence of the volume fraction of solids were investigated by comparing the breakup of silicone oil drops containing two types of particles, which showed different wetting behavior with that of pure silicone oils. One type of particles was situated in the liquid-liquid interface, the other type remained mainly wetted in the oil drops. With increasing volume fraction both suspension types showed shear rate dependent rheological behavior due to particle-particle interactions. Emulsification results were interpreted by means of apparent critical capillary numbers and correlated with the measured drop-to-matrix viscosity ratio. To calculate the viscosity ratio, the viscosities of the disperse phases were measured for each applied shear stress. Depending on the wetting characteristics, the particle load of the disperse phase showed different influence on the emulsification result. Particles situated in the liquid-liquid interface did not affect the apparent critical capillary number and therefore the drop breakup in the investigated volume fraction range. This result was explained by the reduction of particle loading during drop deformation and breakup due to the shearing-off of highly filled drops. Therefore the matrix fluid of the disperse phase governs the overall emulsification result. In contrast, particles located inside the disperse phase hindered the drop breakup. In comparison to Newtonian liquids with comparable bulk viscosities, the apparent critical capillary number was higher. It increased with increasing particle volume fraction. Additionally the apparent critical capillary number showed a dependence on the applied stress in case of the particle containing drops. This behavior is assumed to be due to an inhomogeneous viscosity inside the particle-loaded drop. We suggest using an effective drop viscosity to calculate the viscosity ratio in case of this type of system.

On the diffuse interface method using a dual-resolution Cartesian grid

We investigate the applicability and performance of diffuse interface methods on a dual-resolution grid in solving two-phase flows. In the diffuse interface methods, the interface thickness represents a cut-off length scale in resolving the interfacial dynamics, and it was found that an apparent loss of mass occurs when the interface thickness is comparable to the length scale of flows [24]. From the accuracy and mass conservation point of view, it is desirable to have a thin interface in simulations. We propose to use a dual-resolution Cartesian grid, on which a finer resolution is applied to the volume fraction C than that for the velocity and pressure fields. Because the computation of C field is rather inexpensive compared to that required by velocity and pressure fields, dual-resolution grids can significantly increase the resolution of the interface with only a slight increase of computational cost, as compared to the single-resolution grid. The solution couplings between the fine grid for C and the coarse grid (for velocity and pressure) are delicately designed, to make sure that the interpolated velocity is divergence-free at a discrete level and that the mass and surface tension force are conserved. A variety of numerical tests have been performed to validate the method and check its performance. The dual-resolution grid appears to save nearly 70% of the computational time in two-dimensional simulations and 80% in three-dimensional simulations, and produces nearly the same results as the single-resolution grid. Quantitative comparisons are made with previous studies, including Rayleigh Taylor instability, steadily rising bubble, and partial coalescence of a drop into a pool, and good agreement has been achieved. Finally, results are presented for the deformation and breakup of three-dimensional drops in simple shear flows.

Three-dimensional numerical simulation of drops suspended in simple shear flow at finite Reynolds numbers

A three-dimensional study of suspension of drops in simple shear flow has been performed at finite Reynolds numbers. Results are obtained using a finite difference/front tracking method in a periodic domain. The effects of the Reynolds number and the Capillary number are addressed at two volume fractions: 0.195 and 0.34. It is observed that suspensions of deformable drops exhibit a shear-thinning behavior. Similar to the motion of a single drop, drops migrate away from the walls. The effective viscosity, the first and the second normal stress differences oscillate around a mean value in all cases. The first normal stress difference increases with the Capillary number, the Reynolds number and the volume fraction. Results show that drops deform more and orient more in the flow direction as the Capillary number or the volume fraction is increased. Also, the average size of clusters is smaller than for suspension of rigid particles. The radial dependence of the pair distribution function across the channel has been studied. This dependency shows that the tendency to form clusters is reduced as the Capillary number increases or the volume fraction decreases.

Binary collision of drops in simple shear flow at finite Reynolds numbers: Geometry and viscosity ratio effects

The collision of two equal-size drops in an immiscible phase undergoing a shear flow is simulated over a range of viscosity ratios (@l) and different geometries. The full Navier-Stokes equations are solved by a finite difference/front tracking method. Based on experimental data, different cases were simulated by changing the offset, size of drops, and viscosity ratio. The distance between drop centres along the velocity gradient direction (z) was measured as a function of time. It was found that @Dz increases after collision and reaches a new steady-state value after separation. The values of @Dz, during the interaction, increases with increasing initial offset. Our results show that the time of approaching of drops at low initial offset is greater than the other cases, but the maximum deformation is the same for equal drop sizes. The deformation decreases with decreasing the size of drops. As the initial offset increases, the drops rotate more quickly and the available contact time for film drainage decreases. We found that the trajectories of drops in the approaching stage are different owing to the different initial offsets. However, after the drops come into contact, it observed that they follow the same trajectories. As @l increases, the drops rotate more slowly, and the point at which the drops separate is delayed. The trajectories of drops become more symmetric with the increased @l.

A finite-element method for interfacial surfactant transport, with application to the flow-induced deformation of a viscous drop

A Galerkin finite-element method is developed for solving the transport equation governing the evolution of the surface concentration of an insoluble surfactant over a stationary or evolving fluid interface. The numerical procedure is implemented on an unstructured three-dimensional surface grid consisting of six-node curved triangular elements. Numerical investigations show that the finite-element method is superior to a previously developed finite-volume method for both convection- and diffusion-dominated transport, and especially when the interfacial grid is coarse and steep gradients arise due to local accumulation. The numerical methods for surface transport are combined with a boundary-element method for Stokes flow, and dynamical simulations are performed to illustrate the possibly significant effect of the surface equation of state relating the surface tension to the surfactant concentration on the deformation of a viscous drop in simple shear flow.

Numerical Simulation of Motion of Drops in a Shear Flow Using a Statistical-Thermodynamic Interface Model

For simulating immiscible two-phase flow and easily adjusting surface tension and thickness of diffused interface to desired values, a lattice-Boltzmann model based on the van-der-Waals-Cahn-Hilliard theory is improved by (a) introducing two parameters in a surface free energy and (b) adopting a particle number density function at a local equilibrium state for the convection term of the lattice Boltzmann equation. From the simulations of neutrally-buoyant drops in simple shear flows, it is confirmed that (1) the improved model can give good predictions for deformation and breakup of single drops, (2) the critical Reynolds number at which drop breakup takes place depends not only on the capillary number and the drop number density but also on the initial spatial arrangement of drops, (3) the pressure at the throat section of dumbbell-like drop immediately before breakup is higher than that at the edge of ellipsoidal-shaped drop.

Mesoscopic Simulation of Drops in Gravitational and Shear Fields

In seeking validation of Dissipative Particle Dynamics (DPD) for the mesoscopic modeling of multiphase fluid−fluid systems in external fields, simulations of a pendant drop and a drop in simple shear flow have been performed. The shape profile of the simulated pendant drop was found to be in perfect agreement with that computed by solving the Laplace equation. At increased values of the gravitational force (g), the drop underwent considerable elongation, developing a “neck” between the solid support and its bulk part. Further increases in g resulted in thinning of the neck, which ruptured as g exceeded a certain value, leading to the detachment of the drop. This picture of the detachment process is consistent with the experimental observations published in the literature. Also, the simulations reproduced the drop volume experiment quantitatively. For the drop in shear flow, the degree of deformation was found to be a linear function of the capillary number (Ca) in the region Ca ≤ 0.11, in good agreement with Taylor's theory; this is despite the fact that the hydrodynamic regime in the simulations (Re ∼ 1−10) is quite different from that assumed in the theory (Re ≪ 1). At increased shear rates the results showed positive departure from linearity, in agreement with theory and experiment. Further increases in Ca resulted in the drop assuming a dumbbell like shape, the middle part of the “dumbbell” gradually stretching to form a thin neck. The rupture of the neck was occasioned by the instabilities manifested in the form of stochastic oscillations which magnified as the critical point was reached. The time evolution of the shape of the drop as it underwent the breakup process in our simulations bears remarkable similarity to the experimental observations of Torza et al. The critical value of the capillary number obtained in the simulations is in reasonable agreement with the experimental figure.

Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method

A spherical drop, placed in a second liquid of the same density, is subjected to shearing between parallel plates. The subsequent flow is investigated numerically with a volume-of-fluid (VOF) method. The scheme incorporates a semi-implicit Stokes solver to enable computations at low Reynolds number. Our simulations compare well with previous theoretical, numerical, and experimental results. For capillary numbers greater than the critical value, the drop deforms to a dumbbell shape and daughter drops detach via an end-pinching mechanism. The number of daughter drops increases with the capillary number. The breakup can also be initiated by increasing the Reynolds number.

Three-dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three-dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary-only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two-dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of Boundary Elements 1997; 19: 279), the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two-phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 John Wiley & Sons, Ltd.

Numerical Study of Flows of Two Immiscible Liquids at Low Reynolds Number

We review our recent results on fingering, bamboo waves, and drop breakup in low Reynolds number flows composed of two viscous liquids under shear.

Significance of the dispersed-phase viscosity on the simple shear flow of suspensions of two-dimensional liquid drops

The significance of drop-fluid viscosity on the effective rheological properties and on the dynamics of the microstructure of mono-disperse suspensions of two-dimensional liquid drops with constant interfacial tension is investigated by means of numerical simulations at vanishing Reynolds number, using the boundary integral method for Stokes flow. Three important features of the numerical method are the computation of the doubly-periodic Green's function and associated stress tensor by tabulation and interpolation, the iterative solution of a deflated integral equation for the interfacial velocity, and the repositioning of the drop interfaces at close proximity to avoid artificial coalescence. In the first part of the simulations, the interaction of two intercepting drops in simple shear flow is studied in an extended range of conditions, and the results are used to quantify the behaviour and develop insights into the physics of dilute systems. In the second part of the simulations, the motion of a random suspension of 25 drops repeated periodically in the two spatial directions is studied at the areal fraction Φ = 0.30, drop fluid to ambient fluid viscosity ratio λ = 1 or 10, and drop capillary number Ca = 0.10 or 0.30, a total of four combinations

Binary collision of drops in simple shear flow by computer-assisted video optical microscopy

The collision of two equi-sized drops immersed in an immiscible liquid phase undergoing a shear flow in a parallel plate apparatus has been investigated over a range of capillary numbers. The drops were observed along the vorticity direction of shear flow by video enhanced contrast optical microscopy. Images of the colliding drops were processed by image analysis techniques. The distance Δy between the drop centres along the velocity gradient direction was measured as a function of time during approach, collision and separation of the two drops. It was found that Δy increases irreversibly after collision, thus providing a mechanism for drop dispersion in a concentrated system. Drop shape evolution during collision was characterized by measuring a deformation parameter and the angle made by the drop major axis with respect to the velocity gradient direction. The extent of the near-contact region when the drops are sliding on each other was also estimated. Coalescence was a rare event and was observed in the extensional quadrant of the shear flow. The experimental results show good agreement with numerical simulations recently reported in the literature.

The collision rate of small drops in linear flow fields

A dilute dispersion containing small, force-free drops of one fluid dispersed in a second, immiscible in a linear flow field is considered for small Reynolds numbers and large Péclet numbers under isothermal conditions. The emphasis of our analysis is on the effects of pairwise drop interactions on their collision rate, as described by the collision efficiency, using a trajectory analysis. Simple shear flow and uniaxial extensional or compressional flow are considered. For both flows, the collision efficiency decreases with increasing drop viscosity due to the effects of hydrodynamic interactions. It also decreases as the ratio of the smaller drop radius to the larger radius decreases. For uniaxial flow, finite collision rates are predicted in the absence of interdroplet forces for all finite values of the drop size ratio and the ratio of the viscosities of the drop and suspending medium. In contrast, several kinds of relative trajectories exist for a pair of drops in simple shear flow, including open trajectories, collision trajectories, and closed and semi-closed trajectories, in the absence of interdroplet forces. When the ratio of small to large drop diameters is smaller than a critical value, which increases with increasing drop viscosity, all of the relative trajectories that start with the two drops far apart remain open (no collisions), unless in the presence of attractive forces. Attractive van der Walls forces are shown to increase the collision rates.

Tipstreaming of drops in simple shear flows

The tipstreaming phenomenon, as observed in simple shear flows, has been analysed. Tipstreaming in simple shear flows is a mode of droplet break-up in which the droplet develops a sigmoidal shape and a stream of tiny droplets is ruptured off the tips of the drop. Suggested causes of the tipstreaming phenomenon have been tested experimentally. It was found that tipstreaming can occur if interfacial tension gradients can develop, resulting in low interfacial tension at the tips and a higher tension elsewhere. Tipstreaming will not occur at extremely low surface active material levels, when the interfacial tension can not even be lowered locally, nor at high levels where there is so much surface active material present that the interfacial tension will be low all over the droplet.