Drop oscillations under simple shear in a highly viscoelastic matrix

Recent two-dimensional numerical simulations and experiments have shown that, when a drop undergoes shear in a viscoelastic matrix liquid, the deformation can undergo an overshoot. I implement a volume-of-fluid algorithm with a paraboloid reconstruction of the interface for the calculation of the surface tension force for three-dimensional direct numerical simulations for a Newtonian drop in an Oldroyd-B liquid near criticalities. Weissenberg numbers up to 1 at viscosity ratio 1 and retardation parameter 0.5 are examined. Critical capillary numbers rise with the Weissenberg number. Just below criticality, drop deformation begins to undergo an overshoot when the Weissenberg number is sufficiently high. The overshoot becomes more pronounced, and at higher matrix Weissenberg numbers, such as 0.8, drop deformation undergoes novel oscillations before settling to a stationary shape. Breakup simulations are also described.     

Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations

We implement a volume-of-fluid algorithm with a parabolic re-construction of the interface for the calculation of the surface tension force (VOF-PROST). This achieves higher accuracy for drop deformation simulations in comparison with existing VOF methods based on a piecewise linear interface re-construction. The algorithm is formulated for the Giesekus constitutive law. The evolution of a drop suspended in a second liquid and undergoing simple shear is simulated. Numerical results are first checked against two cases in the literature: the small deformation theory for second-order liquids, and an Oldroyd-B extensional flow simulation. We then address the experimental data of Guido et al. (2003) for a Newtonian drop in a viscoelastic matrix liquid. The data deviate from existing theories as the capillary number increases, and reasons for this are explored here with the Oldroyd-B and Giesekus models.     

Numerical simulation of a drop undergoing large amplitude oscillatory shear

Direct numerical simulations are conducted for a Newtonian drop in a Newtonian matrix subjected to large amplitude oscillatory shear flows. In the experimental study of Guido et al. (in Rheol Acta 43:575–583, 2004), the drop shape is found to oscillate at higher harmonics of the forcing frequency when the capillary number is increased. Their phenomenological model requires a much smaller capillary number for predicting the harmonic nature of the experimental data. In this paper, computational results on the evolution of drop length and inclination angle are obtained at the same fluid and flow properties as the experiments, and are shown to reasonably reproduce the experimental data. In particular, the computed velocity fields around the drop are shown to elucidate the over-rotation, which is a mechanism for the experimentally observed harmonics.     

Inertia dominated flow and heat transfer in liquid drop spreading on a hot substrate

The present work deals with computational modeling of the fluid flow and heat transfer taking place in the process of impact of a cold liquid drop (T_d = 20-25 °C) onto a dry heated substrate characterized by different thermophysical properties. The computational model, based on the volume-of-fluid method for the free-surface capturing, is validated by simulating the configurations accounting for the conjugate heat transfer. The simulations were performed in a range of impact Reynolds numbers (Re = 2000-4500), Weber numbers (We = 27-110) and substrate temperatures (T_s = 100-120 °C). The considered temperature range of the drop-surface, i.e. liquid-solid system does not account for the phase change, that is boiling and evaporation. The model performances are assessed by contrasting the results to the reference database originating from the experimental and complementary numerical investigations by Pasandideh-Fard et al. [Pasandideh-Fard, M., Aziz, S., Chandra, S., Mostaghimi, J., 2001. Cooling effectiveness of a water drop impinging on a hot surface. International Journal of Heat and Fluid Flow, 22, 201-210] and Healy et al. [Healy, W., Hartley, J., Abdel-Khalik, S., 2001. On the validity of the adiabatic spreading assumption in droplet impact cooling. International Journal of Heat and Mass Transfer, 44, 3869-3881]. In addition, the thermal field obtained is analyzed along with the corresponding asymptotic analytical solution proposed by Roisman [Roisman, I.V., 2010. Fast forced liquid film spreading on a substrate: flow, heat transfer and phase transition. Journal of Fluid Mechanics, 656, 189-204]. Contrary to some previous numerical studies, the present computational model accounts for the air flow surrounding the liquid drop. This model feature enables a small air bubble to be resolved in the region of the impact point. The reported results agree reasonably well with experimental and theoretical findings with respect to the drop spreading pattern and associated heat flux and temperature distribution.     

Viscosity of a Newtonian fluid calculated from the deformation of droplets covered with a surfactant under a linear shear flow

The viscosity of small fluid droplets covered with a surfactant is determined using drop deformation techniques. This method, proposed by Hu and Lips, is here extended to the case of the presence of a surface-active adsorpted at the liquid–liquid interface, to consider more general scenarios. In these experiments, a droplet is sheared by another immiscible fluid of known viscosity, both Newtonian liquids. From the steady-state deformation and retraction mechanisms, the droplet viscosity is calculated using an equation derived from the theories of Taylor and Rallison. Although these theories were expressed for surfactant-free interfaces, they can be applied when a surfactant is present in the system if the sheared droplet reaches reliable steady-state deformations and the surfactant attains its equilibrium adsorption concentration. These determinations are compared to bulk viscosities measured in a rheometer for systems with different viscosity ratios and surfactant concentrations. Very good agreement between both determinations is found for drops more viscous than the continuous phase.     

Diffusion effects on the interfacial tension of immiscible polymer blends

Most methods of measuring the interfacial tension between two immiscible polymers are based on the analysis of the shape that a drop of one polymer immersed in the other one exhibits under the action of flow or gravity. In such a situation, the small, yet nonzero mutual solubility between the two polymers acts toward mass transfer between the drop and the surrounding fluid. In this work, diffusion effects on the interfacial tension of the pair polyisobutylene/polydimethylsiloxane have been investigated by drop deformation under shear flow. When the drop was made of polyisobutylene, drop size decreased with time due to diffusion. Drop shrinkage was associated with a significant increase in interfacial tension, until an apparent plateau value was reached. The effect was attributed to a selective migration of molecular weights, which would act to enrich the drop with higher molar mass material. To support such an interpretation, drop viscosity was evaluated by drop shape analysis and it was actually found to increase with time. In some cases, the ratio between drop and continuous phase viscosity became higher than the critical value for drop breakup in shear flow. Upon inverting the phases (i.e., when the drop was made of polydimethylsiloxane), no significant transient effects were observed. In the light of these results, the problem of what are the correct values of interfacial tension and viscosity ratio for a polymer blend of a certain composition will also be discussed. Received: 25 January 1999 Accepted: 24 May 1999     

Deformation and orientation of single droplets during shear flow: combined effects of confinement and compatibilization

Combined effects of geometrical confinement and compatibilization on the deformation and orientation of single droplets during steady-state shear flow are investigated in a counter-rotating cell by means of microscopic observations. The model system consists of polydimethylsiloxane droplets of varying sizes and viscosities in a polyisobutylene matrix. To this system, a premade polyisobutylene–polydimethylsiloxane block copolymer is added as compatibilizer in different concentrations. For each droplet, the equilibrium interfacial tension is determined by comparing droplet axes with the predictions of the confined Minale model for uncompatibilized droplets at the appropriate degree of confinement. Although large reductions in interfacial tension are seen for all compatibilized droplets, it is shown that the effect of compatibilization on droplet deformation and orientation can efficiently be taken into account in the equilibrium capillary number. This way, for all viscosity ratios and confinement ratios, steady-state data for compatibilized and uncompatibilized droplets coincide, and agree well with the predictions of the confined Minale model at sub-critical conditions. For near-critical capillary numbers, compatibilization slightly reduces droplet deformation and postpones breakup, irrespective of the degree of confinement.     

Motion of non-wetting drop in constricted geometry

This work is a contribution to the study of deformation of a non-wetting drop transported under the combined effect of gravity and permanent fluid motion in a vertical channel. The deformation being caused during passage of the drop through a constriction formed by two spherical obstacles placed opposite in a vertical channel. For this purpose a three-dimensional computation is conducted in order to illustrate the behavior of the drop in the condition of non-wettability. The flow based on Navier–Stokes equation is solved numerically with volume of fluid (VOF) method. The corresponding simulations are carried out in view to analyse the behavior of the drop when it is forced to move between the obstacles for different values gap size until the breakup is obtained.     

Study of the shock-induced acceleration of hexane droplets

An experimental study of the interaction of a shock wave with a hexane droplet is presented. The main goal of the experiments was to record images of the process and measure basic parameters describing movement, dispersion and evaporation of the droplets engulfed by a shock wave propagating in air. A shock tube with a visualization section was used for this research. Photography of the process allowed one to measure the positions, velocities and sizes of mist clouds created by the interaction processes. Analysis of the pictures shows that there is no qualitative difference between cases for different size droplets, but shock Mach number had a significant effect on the process. Quantitative analysis shows that under certain conditions, a catastrophic breakup mechanism of dispersion occurred. The droplets are shattered into a mist cloud before they achieve mechanical equilibrium with the surrounding gas. The approximate time for the complete dispersion and acceleration of the fuel droplet varies from 300 to 500 μs, and depends both on the droplet diameter and shock velocity. The dispersion time is controlled principally by the droplet diameter, and to a lesser extent, the shock Mach number. This paper is based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005.     

Strain recovery of model immiscible blends: effects of added compatibilizer

The effect of added compatibilizer on the strain recovery of model immiscible blends after cessation of shear was studied. Blends were composed of polyisobutylene drops (up to 30% by weight) in a polydimethylsiloxane matrix, with viscosity ratio (viscosity of the drops relative to the matrix viscosity) ranging from 0.3 to 1.7. Up to 1% by weight of a PIB-PDMS diblock copolymer was added as compatibilizer. The ultimate recovery recorded after reaching steady-shear conditions increased significantly due to added compatibilizer. Furthermore, the compatibilizer also slowed down the kinetics of the recovery; however, unlike uncompatibilized blends, the recovery could no longer be captured by a single retardation time. The largest increase in ultimate recovery due to compatibilizer occurred at the lowest viscosity ratio. In contrast, the greatest slowing down of the recovery due to compatibilizer occurred at the highest viscosity ratio. The rheological data by themselves are insufficient to reach a definitive conclusion about the mechanism of compatibilizer action. The results are consistent with the effects of flow-induced gradients in compatibilizer concentration. An alternative constitutive modeling approach that captures compatibilizer effects in terms of an interfacial dilational elasticity can reproduce the recovery curves qualitatively, but some predictions of the model contradict experimental results.     

Dynamic pre-strain and inertia effects on the fracture of metals

A combined experimental and theoretical study is described which examines the influence of strain-rate and dynamic pre-strain on the ductile fracture of thin cylinders. The thin-cylinder configuration is particularly important in this case because it allows inertia terms to be directly incorporated into the theory of plastic instability. A series of quasi-static and dynamic tests is conducted on three materials with differing degrees of strain-rate sensitivity and strain-hardening. The experimental observation that fracture is inhibited at high strain-rates is in accord with the theory when inertia can no longer be considered insignificant. It is also shown that dynamic pre-strain has little or no effect on the flow stress or the strain at fracture in materials which-are essentially strain-rate insensitive, but does reduce the fracture strain in the strain-rate sensitive materials.     

Determination of interfacial tension by the retraction method of highly deformed drop

The traditional retraction of the deformed drop method (DDRM) to determine the interfacial tension is reformulated to relax the limit the small deformation assumption. The kernel of the new formalism is the calculation of the velocity gradient on the vertex of the ellipsoidal drop. Two models were used for such calculations: the Jackson and Tucker model [J Rheol 47:659–682] and the Yu and Bousmina model [J Rheol 47:1011–1039]. The method can be used either in the retraction of shear deformed drop, or in the retraction of elongated drops produced by the breakup of a long thread. Comparison with experimental results of the literature showed that conversely to the classical DDRM, good accuracy is obtained when the new modeling for the determination of interfacial tension is used both under small and large deformations.     

Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

Diffuse interface model of surfactant adsorption onto flat and droplet interfaces

For applications where droplet breakup and surfactant adsorption are strongly coupled, a diffuse interface model is developed. The model is based on a free energy functional, partly adapted from the sharp interface model of [Diamant and Andelman 34(8):575–580, (1996)]. The model is implemented as a 2D Lattice Boltzmann scheme, similar to existing microemulsion models, which are coupled to hydrodynamics. Contrary to these microemulsion models, we can describe realistic adsorption isotherms, such as the Langmuir isotherm. From the free energy, functional analytical expressions of equilibrium properties are derived, which compare reasonably with numerical results. Interfacial tension lowering scales with the logarithm of the area fraction of the interface unloaded with a surfactant: . Furthermore, we show that adsorption kinetics are close to the classical relations of Ward and Tordai. Prelimary simulations of droplets in shear flow show promising results, with surfactants migrating to interfacial regions with highest curvature. We conclude that our diffuse interface model is very promising for apprehending the above-mentioned applications as membrane emulsification. Paper presented at the Annual European Rheology Conference (AERC) 2005, April 21–23, Grenoble, France.     

The EHD-driven fluid flow and deformation of a liquid jet by a transverse electric field

The aim of this study is to investigate the effect of a uniform transverse electric field on the steady-state behavior of a liquid cylinder surrounded by another liquid of infinite extent. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electrical and hydrodynamic stresses acting on the interface separating the two fluids is presented, and an expression is found for the interface deformation for small distortions from a circular shape. The electrical stresses acting on the interface of two leaky-dielectric liquids are compared with those acting on an interface separating a perfect dielectric or infinitely conducting core fluid cylinder from a surrounding perfect dielectric fluid. A comparison is made between the results of this study and those of a similar study for fluids with permeable interfaces and the classical results for liquid drops.     

Bag breakup of nonturbulent liquid jets in crossflow

An experimental investigation of the bag breakup of round nonturbulent liquid jets in gaseous crossflow at room temperature and pressure is described. Pulsed photography, pulsed shadowgraphy, and high-speed imaging were used to observe the column and surface waves along the liquid jet and the formation and breakup of bags. Measurements included: wavelengths of column and surface waves, jet velocities, the number of bags along the liquid jet, the number of nodes per bag, droplets sizes and velocities, and trajectories of droplets. Present results show that the column waves of a nonturbulent liquid jet in crossflow within bag breakup regime can be explained based on Rayleigh–Taylor instability. The number of nodes per bag affected the breakup mechanism of the bags. Three distinctive sizes of droplets were produced due the breakup of the bag membrane, the ring strings and the ring nodes. The size of the droplets resulting from the breakup of the bag membrane was constant independent of the crossflow Weber number. Finally different trajectories were observed for the three groups of droplets.     

Large eddy simulation and experiment of shear breakup in liquid-liquid jet: Formation of ligaments and droplets

Understanding the shear breakup in jet flows and the formation of droplets from ligaments is important to determine the final droplet size distribution (DSD). The initial droplet size, which affects the final DSD, is considered to be generated by the shear breakup. Large eddy simulation (LES) was performed to investigate the shear breakup in liquid-liquid jet flows. The explicit Volume of Fluid (VOF) model with the geometric reconstruction scheme was used to capture the oil-water interface. The estimated oil distribution including wave peaks, ligaments, droplets and water streaks were compared to the experiments with a good agreement. The estimated DSD matched with the measurements favorably well. In the simulation, the formation of droplets with a smooth and curved surface from ligaments or sheet-like structures was obtained. Different mechanisms were observed along with the shear layer including the formation of droplets from ligament through the capillary forces, breakage of a droplet into smaller ones and attachment of a droplet to a ligament. The destructive shear forces and resisting surface tension forces were quantified on stretching and retracting ligaments. The influence of internal viscous force was found to be negligible due to low oil viscosity. The critical capillary number was found to be larger than 5.0 for ligaments breaking with the shear breakup. The capillary number was below unity for retracting ligaments. The coalescence of two equal-sized droplets was obtained in the shear breakup region. The shear stress magnitude at the contact region increased more than two folds. The total surface area decreased nearly 20% after the coalescence.