Newtonian Drop Research Articles

Stability of the shape of a translating viscoelastic drop at low Reynolds number

We examine the stability of the shape of a viscoelastic drop translating in an immiscible, otherwise quiescent Newtonian fluid at low Reynolds number. Non-Newtonian stresses in the drop phase are characterized by the finitely extensible nonlinear elastic-Chilcott-Rallison model. A boundary integral method is used to numerically examine the time-evolution of initially perturbed drop shapes over a range of dimensionless parameters. For sufficiently small capillary numbers, the drop achieves an oblate spheroidal steady shape. For large initial deformations or capillary numbers, however, the drop deforms continuously and eventually breaks up through either the formation of an elongated tail or the development of a re-entrant cavity at its trailing end. These mechanisms of drop breakup are qualitatively similar to those reported earlier for Newtonian drops [C. J. Koh and L. G. Leal, Phys. Fluids A 1, 1309 (1989)10.1063/1.857359; C. Pozrikidis, J. Fluid Mech. 210, 1 (1990)10.1017/S0022112090001203; H. A. Stone, Annu. Rev. Fluid Mech. 26, 65 (1994)10.1146/annurev.fl.26.010194.000433]. Compared to the case of a Newtonian drop, drop phase elasticity is found to have a stabilizing (destabilizing) effect for initially oblate (prolate) shape perturbations, due to the development of polymeric stresses caused by deformation of polymer chains in alignment with streamlines of the flow. Polymer viscosity has a strong influence on the stability of the shape of drops, whereas polymer relaxation time and extensibility have relatively weak influences.

On the motion of Newtonian and non-Newtonian liquid drops

In the present study, the motion of Newtonian and non-Newtonian liquid drops has been investigated experimentally. In order to investigate the effect of bulk fluid on drops, we have used water and air, as two fluids with different properties, and various industrial and biological applications. Image processing is utilized to analyze the images obtained by a high speed camera. The research has been separated into two parts. The first part has been devoted to the experiments in which air is the bulk fluid, and the second is related to the experiment carried out in water. The range of Reynolds number is, approximately, 50<Re<500. The major concern of the present study is the size variation of drops and its effect on the drag coefficient. It is proved that the period of size variation of a drop does not vary with properties. Rheological aspects of the problem have also been considered. In air with small density and viscosity, addition of non-Newtonian characteristics to the fluid causes the behavior of the drop to undergo dramatic changes. However, in water, a denser and more viscous bulk fluid, the behavior of Newtonian and non-Newtonian drops (at least for shear thinning fluids) looks the same.

Maximum spreading of a shear-thinning liquid drop impacting on dry solid surfaces

In the present study, the maximum spreading of Newtonian and shear-thinning drops impacting on dry surfaces was investigated experimentally. Water, glycerin drops (Newtonian liquids) and xanthan drops (shear-thinning liquids) were impinged upon the flat surfaces of glass, stainless steel and parafilm-M substrates, which have hydrophilic, moderate and hydrophobic properties, respectively, at the impact velocities ranging from 0.8 to 4.0m/s. The impact velocity and the liquid viscosity with the shear-thinning characteristic had dominant effects on the maximum spreading; on the other hand, the surface wettability effect turned out to be minor. Several maximum spreading models for a Newtonian liquid were reviewed briefly and compared with the experimental measurements. The Roisman model showed the best agreement to the experimental results for Newtonian drops. A new empirical correlation for the maximum spreading of a shear-thinning drop was proposed by modifying the Scheller and Bousfield model with the concept of the effective shear viscosity, which is correlated with the shear rate varying with the impact velocity. The correlation agreed to within ±5% for 91% of the shear-thinning drop experiments.

Effect of interfacial modifier on single drop deformation and breakup in step increasing shear flow

This work deals with in situ visualisation of deformation and breakup of a copolymer modified single Newtonian drop immersed in a Newtonian homogenous matrix. The experiments were carried out on a model system made of poly-isobutylene as the suspending fluid and two poly-dimethylsiloxanes with different molecular weights as the drop phase with viscosity ratios 0.036 and 1.13, below and above but close to unity. Three weight concentrations 0.5%, 2% and 10% of the block copolymer laying below, close to and above the critical concentration of the total drop surface coverage were examined. Single drop deformation experiments were carried out in a home-designed Couette quartz cell connected to a home-modified Paar Physica Rheometer. The variation in the length-to-diameter ratio (L/D) versus shear rate and capillary number was measured both in steady and in transient regimes till breakup. The results indicated a weaker resistance of copolymer modified drops against hydrodynamic stresses at both viscosity ratios as compared to the clean drop. However, the drop deformation was found to be complex and depends on the copolymer concentration and the viscosity ratio.

Observation of the spreading and receding behavior of a shear-thinning liquid drop impacting on dry solid surfaces

In the present work, the impact dynamics of shear-thinning drops on dry solid surfaces was investigated and compared with that of Newtonian drops. The effects of liquid viscosity with shear-thinning characteristics, surface wettability and impact velocity on the spreading and receding behavior of the drops were investigated experimentally. Water, glycerin drops (Newtonian liquids), and xanthan drops (shear-thinning liquids) were impinged upon glass, stainless steel and parafilm-M substrates, which have hydrophilic, moderate and hydrophobic properties, respectively, at the impact velocities ranging from 0.8 to 4.0 m/s. The xanthan drops spread out more widely and receded more rapidly than the glycerin drops because the spreading and receding motion drastically reduced the viscosity and consequently produced a lower level of viscous dissipation. The impact velocity and the liquid viscosity with shear-thinning characteristics had a dominant effect on the spreading phase. In contrast, the surface wettability had only a minor effect on the spreading phase but a very significant effect on the receding phase. The effect of the impact velocity on the receding phase was limited to the low-viscosity drops (with a small degree of shear-thinning) on the parafilm-M (hydrophobic) substrate. That is, when the low-viscosity drops impacted on a hydrophobic substrate, the receding velocity increased greatly with higher impact velocity, resulting in partial or complete rebound behavior. On the other hand, drop receding was significantly suppressed on the glass (hydrophilic) substrate, and the receding velocity was almost insensitive to the impact velocity. Suppression of drop receding was more prominent with the liquid having a larger degree of shear-thinning characteristic (i.e., showing a larger reduction of viscosity with the increase of the shear rate).

Interfacial tension determination of liquid systems in which one of the phases is non-Newtonian using a rheo-optical method

The interfacial tension between two low-density-difference liquids, one of them being non-Newtonian, was determined using a rheo-optical method with a computer-controlled parallel bands apparatus. This determination is featured measuring the steady-state deformation suffered by a Newtonian drop sheared by a non-Newtonian liquid matrix as a function of the applied shear rate. The application of Taylor's theory to the linear region of deformations, taking into account a mean viscosity of the matrix, enables us to determine the interfacial tension between both liquids as a fitting parameter. Such calculated σ values were checked for several oils with different viscosities, immersed in the same continuous phase, finding excellent agreement with the conventional drop detachment measurement.

Interaction of viscous drops in a yield stress material

The interaction of Newtonian drops of various volumes moving in yield stress material is investigated experimentally. Tetrachloroethylene drops move in a rectangular reservoir filled with neutralized 0.07% w/w Carbopol gel under the action of gravity. For initially vertically aligned drop pairs, we present time evolution of separation distance, velocities during the interaction and conditions for coalescence of the drops, which depend on the volumes of the drops and the initial separation between them. For the asymmetric interaction of the pairs, we present interaction patterns, which have been used for estimation of the size of the yielded region and its shape around the leading drop.

Motion and shape of an axisymmetric viscoplastic drop slowly falling through a viscous fluid

Slow sedimentation of a deformable drop of Bingham fluid in an unbounded Newtonian medium is studied using a variation of the integral equation method (Toose et al., J Eng Math 30:131–150, 1996, Int J Numer Methods Fluids 30:653–674, 1999). The Green function for the Stokes equation is used, and the non-Newtonian stress is treated as a source term. The computations are performed for a range of physical parameters of the system. It is demonstrated that initially deformed drop similar to Newtonian ones breaks up for high capillary number, Ca, and stabilizes to steady shapes at low Ca. Estimations of critical capillary number for specific initial deformations demonstrated its growth (increase in the stability of the drop) with the yield stress magnitude both for prolate and oblate initial shapes. Prolate initial shapes become more stable with the increase of the plastic viscosity. In contrast to this, for low yield stress, oblate shapes are destabilized with the growth of the plastic viscosity. This effect is similar to the effect of the viscosity of a Newtonian drop on its stability. However, at higher yield stress, the effect of plastic viscosity is reversed.

The free-fall of viscoplastic drops

The free-fall of liquid drops under the action of gravity is studied experimentally by high-speed imaging, with focus on drops of viscoplastic fluids (i.e., those materials which exhibit a fluid behaviour only when the applied stress exceeds a certain threshold value, called the yield stress), and compared to the behaviour observed in Newtonian drops. The yield stress of the fluid is shown to alter significantly the drop shape during free-fall. The results can be interpreted in terms of a dimensionless number comparing the yield stress magnitude and the capillary pressure.

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.

Formation of viscoplastic drops by capillary breakup

The process of growth and detachment of drops from a capillary nozzle is studied experimentally by high-speed imaging. Newtonian drops are compared to shear-thinning and viscoplastic drops. Both Newtonian and shear-thinning fluid drops grow on the end of the capillary until a maximum supportable tensile stress is reached in the drop neck, after which they become unstable and detach. The critical stress is not influenced by variations in viscosity or in the degree of shear thinning. Viscoplastic fluids show a different behavior: at low values of the yield stress, the critical stability behavior is similar to that of Newtonian and shear-thinning drops. Above a threshold value, characterized in terms of the drop size, surface tension and tensile yield-stress magnitude, yield-stress forces are larger than surface forces, and the maximum tensile stress achievable in the drop neck at the point of critical stability is governed by the von Mises criterion.

Effects of viscoelasticity on the retraction of a sheared drop

Effects of drop and matrix viscoelasticity on the retraction of a sheared drop are numerically investigated. Retraction of an Oldroyd-B drop in a Newtonian matrix is initially faster and later slower with increasing drop Deborah number. The observed behavior is explained using an ordinary differential equation model representing the dominant balance between various forces during retraction. The initial faster relaxation of viscoelastic drops is due to viscoelastic stresses pulling the drop interface at the tips inward. The later slower retraction is due to the slowly-relaxing viscoelastic forces at the equator, where they act against the capillary force. The drop inclination decreases substantially during retraction unlike in a Newtonian case. Matrix viscoelasticity slows the relaxation of a Newtonian drop because of the increasingly slow relaxation of highly stretched polymers near the drop tip with increasing Deborah number. Increasing the ratio of polymeric to total viscosity further accentuates the viscoelastic effects in both cases. For an Oldroyd-B drop in an Oldroyd-B matrix, a competition between the dispersed and the continuous phase elasticities, represented by their ratio, determines the dynamics; larger values of the ratio leads again to initial faster and later slower retraction.

Impact of shear-thinning and yield-stress drops on solid substrates

The behaviour of shear-thinning and viscoplastic fluid drops impacting on solid substratesas compared with that of Newtonian drops is studied experimentally by means ofhigh-speed imaging. In particular, the investigation focuses on the morphological aspects ofdrops after inertial spreading. While the impact morphology of drops of shear-thinningfluids turns out to be qualitatively similar to that of Newtonian fluids, viscoplastic dropscan exhibit central drop peaks at the end of inertial spreading. The influence ofyield-stress magnitude on drop impact behaviour is qualitatively established bymeasuring the size of these central drop peaks. The peaks indicate that dropdeformation during impact is localized: within a threshold radius, shear-stress effects willnot be large enough in magnitude to overcome yield-stress effects, and thereforeviscoplastic fluids within this region will not deform from the drop shape prior toimpact.

A comparison of viscoelastic stress wakes for two-dimensional and three-dimensional Newtonian drop deformations in a viscoelastic matrix under shear

A recent experimental study of a Newtonian drop suspended in a viscoelastic matrix undergoing simple shear displays a transient overshoot in drop deformation which is qualitatively similar to two-dimensional (2D) numerical simulation results. Despite the similarity, an interpretation in light of the 2D result is misleading because the overshoot is absent in the fully three-dimensional (3D) simulation. This motivates a study of regimes where qualitatively different and interesting features such as overshoots in deformation occur for a 2D drop but not for a 3D drop. The influence of viscoelastic “wakes” that emanate from the drop tips is reported. The viscoelastic wakes are larger and of higher magnitude in 3D than in 2D, and lead to more deformation in 3D. During drop evolution, the less deformed drop is found to be aligned more with the flow direction. As the drop-to-matrix viscosity ratio increases from 1 to past 3, drop rotation is promoted, with accompanying retraction when the capillary number is sufficiently high. Thus, a 3D overshoot in deformation is promoted with increasing viscosity ratio.

Motion of a Single Newtonian Liquid Drop Through Quiescent Immiscible Visco-Elastic Liquid: Shape and Eccentricity

The shape of a single Newtonian liquid drop moving freely under gravity in quiescent non-Newtonian liquid (visco-inelastic and visco-elastic) has been studied. Strong effect of fluid elasticity on the shape of Newtonian liquid drop has been observed. Based on experimental observations, a shape regime graph has been suitably modified for elastic fluids. Correlations have also been proposed for the prediction of eccentricity of the Newtonian liquid drop moving through a non-Newtonian visco-elastic liquid.

Confined drop motion in viscoelastic two-phase systems

In this study, we numerically examine the buoyancy-driven, axisymmetric motion of drops through vertical cylindrical capillaries. Combinations of Newtonian and viscoelastic drop and suspending fluid phases are considered. The effects of confinement, material properties, and rheological properties of the two phases on drop mobility and deformation are examined. Four dimensionless parameters (Reynolds number, capillary number, Deborah number, and the drop-to-tube size ratio) play critical roles in determining the drop motion. In general, a Newtonian drop immersed in a viscoelastic fluid experiences an extending trailing edge, while a viscoelastic drop in a Newtonian fluid develops an indentation around the rear stagnation point. Under certain conditions, a cusped drop appears due to fluid viscoelasticity that triggers shape instability.

Influence of viscoelasticity on drop deformation and orientation in shear flow. Part 2: Dynamics

An experimental study of drop dynamics under shear is conducted for five fluid pairs: a reference Newtonian system, two systems with a viscoelastic drop in a Newtonian matrix, one with a Newtonian drop in a viscoelastic matrix, all at drop to matrix viscosity ratio λ = 1.5 , and a separate case at λ = 0.75 . The viscoelastic liquids are either a Boger fluid or a shear-thinning viscoelastic fluid satisfying an Ellis model. Deborah numbers in the range 1–2 and a range of capillary numbers from low to above breakup conditions are addressed. The results focus on three aspects: relaxation after cessation of shear, a new viscoelastic drop breakup scenario, and the effect of shear flow history on drop breakup. Numerical simulations with the 3D volume-of-fluid PROST method complement the experimental results.

Mobility of shear thinning viscous drops in a shear Newtonian carrying flow using DR‐BEM

AbstractThe study of drop behaviour has attracted great interest in the last years due to its importance in different industrial and biological systems. Most available works focus on Newtonian drops, excluding some very important applications such as polymer mixing. Simulations of non‐Newtonian drops have had only limited study, mostly in time‐dependent rheologies or simple flow cases. This work presents a boundary‐only formulation based on the dual reciprocity method to model the motion and deformation of non‐Newtonian shear thinning drops due to a shear Newtonian unbounded carrying flow. Pair‐wise interactions at low Reynolds number between two viscous shear thinning non‐Newtonian drops are numerically simulated in order to obtain mobility magnitudes under linear shear flow of different strengths. Separation of the drops in the direction perpendicular to the imposed flow field at high capillary number (small surface tension) and low viscosity ratio was favoured by shear thinning, increasing in magnitude as the capillary number increases and the viscous ratio decreases. Higher values of this separation occur at higher values of the viscosity ratio when compared with the case of Newtonian drops. In order to obtain a good physical description of the non‐Newtonian drop behaviour, while maintaining good computational performance, the non‐Newtonian viscosity is made to obey the truncated power law model. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Numerical study on the effect of viscoelasticity on drop deformation in simple shear and 5:1:5 planar contraction/expansion microchannel

We implemented a finite element-front tracking method (FE-FTM) to understand the drop dynamics in microfluidic applications. We investigated the effect of viscoelasticity of both drop and medium. The Oldroyd-B model was used for viscoelastic fluid, and DEVSS-G/SUPG/matrix logarithm algorithms were applied to improve numerical stability. We first verified the reliability of the algorithm by comparing with previous results under simple shear flow. The results were in good agreement with previous reports in a wide range of parameters such as capillary number ( Ca) and Deborah number ( De). Then we applied the algorithm to a 5:1:5 planar contraction – narrow channel – expansion flow which is typical microfluidic flows. One of the goal of this study is to explore the effect of viscoelasticity of drop and medium on drop deformation, and to propose the strategy to control the drop shape. When a Newtonian drop was suspended in a viscoelastic medium, in the narrow channel region, we observed an ‘ellipsoid-like drop’ in contrast to a ‘bullet-like drop’ which is the typical shape for the case of a Newtonian drop in Newtonian medium. When a viscoelastic drop is suspended in Newtonian medium, the extent of ‘drop swell’ to the cross-stream direction was enhanced at the exit of the narrow channel. We explained these phenomena in terms of the normal stress difference developed in the viscoelastic fluid. The present study shows that the viscoelasticity plays a significant role on drop dynamics and drop shape in particular. We expect this study will be helpful to understand the drop dynamics in microchannel flow, and provide useful information in manipulating drops in complicated geometries such as microfluidic channels.

Numerical simulation of the motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow using DR-BEM

The motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow is studied using a boundary element numerical simulation. Non-linear effects from the dependency of the viscosity on the velocity field are treated in an implicit manner and the resultant domain integral is transformed into an equivalent series of boundary integrals using the Dual Reciprocity Method. The non-homogeneous (non-linear) system of algebraic equations resulting from the discretization of the boundary element formulation is solved using a modified Newton–Raphson method for drops with values of the power law index of n = 0.8 and 0.6 and compared to the corresponding Newtonian cases ( n = 1 ). The viscosity of the fluid inside the drop follows the truncated power law model. By using this model, the shear-thinning behaviour of the viscosity is correctly represented while avoiding the shear thickening which can be observed using the standard power law in small gradient flows. The simulations showed that the non-Newtonian drops had larger deformations than the corresponding Newtonian drops due to a general decrease in the viscosity. The value of the local viscosities was found to be dependant not only on the velocity field created by the motion of the internal cylinder, but strongly dependant on the surface tension forces. The rate of deformation of the drops was greater in the beginning of the simulation and decreased toward the end showing the drops found a more or less stable shape.