A numerical study on drop formation of viscoelastic liquids using a nonlinear constitutive equation

In this paper, a numerical solution for viscoelastic drop formation from a nozzle into an ambient gas is presented. A volume of fluid (VOF) method is used to predict the formation and break-up process of viscoelastic drop. Here, Giesekus model is used as the constitutive equation. The major features of the phenomenon, such as instantaneous drop length, limiting length of a drop at breakup, minimum drop radius and the volume of the primary drop is determined for a range of the parameter space spanned by the appropriate dimensionless groups. The results reveal that enhancing the mobility factor, Wiessenberg number, and viscosity ratio causes a noticeable decrease in limiting drop length and a small decrease on the primary drop volume. Also, the increasing of gravitational bond number and capillary number causes the limiting drop length increases while the primary drop volume is reduced.     

Collision of a liquid drop on the edge region of a plate heated above the Leidenfrost temperature

A liquid drop colliding on the edge region of a heated plate at a temperature above the Leidenfrost temperature were investigated experimentally. To determine the edge region of a plate, it is not a matter as simple as comparing the size of the drop and the plate. The width of the edge region is largely determined by the tangential velocity component of the colliding drop. The collision processes near the edge of a plate were found to be drastically different from those far from the edge. Generally, a collision process far from the edge comprises simply the flattening, retracting, and reflecting of the colliding drop. A collision near the edge, however, involves more complicated processes. When the tangential velocity component of the drop is small, disintegration of the drop through reflection is weakened while drop splitting or deflection by the edge can be observed. When the tangential velocity component of the drop is large, drop splitting by the edge is lessened while the stretching-out separation of the flattened drop outside the edge greatly enhances the disintegration of the drop. Simple physical models are constructed to describe some of the collision processes.     

Effect of interface deformability on thermocapillary motion of a drop in a tube

The effect of an externally imposed axial temperature gradient on the mobility and deformation of a drop in an otherwise stagnant liquid within an insulated cylindrical tube is investigated. In the absence of bulk transport of momentum and energy, the boundary integral technique is used to obtain the flow and temperature fields inside and outside the deformable drop. The steady drop shapes and the corresponding migration velocities are examined over a wide range of the dimensionless parameters. The steady drop shape is nearly spherical for dimensionless drop sizes <0.5, but becomes slightly elongated in the axial direction for drop sizes comparable to tube diameter. The adverse effect of drop deformation on the effective temperature gradient driving the motion is slightly more pronounced than its favorable effect of reducing drag, thereby leading to a slight reduction in drop mobility with increasing drop deformation. Increasing the viscosity ratio reduces drop deformation and leads to a slight enhancement in the relative mobility (with respect to free thermocapillary motion) of confined drops. When the drop fluid has a lower thermal conductivity than the exterior phase, the presence of the thermally-insulating wall increases the thermal driving force for drop motion (compared to that for the same drop in unbounded domain) by causing more pronounced bending of the isotherms toward the drop. However, the favorable thermal effect of the confining wall is overwhelmed by its retarding hydrodynamic effect, causing the confined drop to always move slower than its unbounded counterpart regardless of the value of the thermal conductivity ratio.     

Nonlinear Capillary Oscillations of a Volumetrically Charged Dielectric Drop

For the axisymmetric capillary oscillations of a charged dielectric fluid drop an expression describing the shape of the generating surface of the drop as a function of time is obtained in the quadratic approximation in the amplitude of an arbitrary initial deformation of its spherical equilibrium shape. It is shown that in contrast to a perfectly conducting charged drop there is no displacement of the drop charge center during oscillation and, hence, such a drop cannot be a source of dipole electromagnetic radiation like a conducting drop in the quadratic approximation.     

Experimental investigation of viscoelastic drop deformation in Newtonian matrix at high capillary number under simple shear flow

The steady state deformation of a viscoelastic drop (Boger fluid) in a Newtonian liquid at high capillary number under simple shear flow is investigated by direct visualization using a specially designed Couette apparatus which enables visualization from two perpendicular directions. Two drop deformation modes are found: (1) Mode I – drop deformation in the flow direction and (2) Mode II – drop deformation in the vorticity direction. The drop deformation mode depends on the relative strength of the elastic contribution to viscous contribution. If the elastic contribution is weak compared to the viscous contribution, the drop elongates in the flow direction via Mode I. If the elastic contribution is strong, the drop elongates in the vorticity direction via Mode II. The drop size also affects the drop deformation. At the same capillary number, bigger drops have larger deformations than smaller drops.     

Drop formation in liquid-liquid systems

Conclusion An experimental method for determining the momentum flux on a drop for a nozzle with an arbitrary fluid velocity profile at the exit has been described. I t was observed that the variation of drop volume with flow rate can differ some-what from that conceived in previous drop formation models, in that drop volume does not necessarily vary smoothly with flow rate. High-speed motion pictures of drop formation and detachment show that the drop oscillates as it grows. I t is believed that this oscillation is related to the observed unsteady variation of drop volume with flow rate. Presently with General Dynamics Space Systems Division, San Diego/CA, USA     

Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear

Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a front tracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of the matrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—the drop shows reduced deformation with increasing Deborah number due to the increased inhibiting viscoelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response is non-monotonic—the steady drop deformation first decreases with increasing Deborah number but above a critical Deborah number, it increases with further increase in Deborah number, reaching higher than the viscous case value for some viscosity ratios. We explain the increase in deformation with Deborah number by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducing the inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclination angle increases with increasing Deborah number. A drop aligned more with the maximum stretching axis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratio of polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angle increases. Our simulation results compare favorably with a number of experimental and computational results from other researchers.     

Estimation of the diameter–charge distribution in polydisperse electrically charged sprays of electrically insulating liquids

The majority of scientific and industrial electrical spray applications make use of sprays that contain a range of drop diameters. Indirect evidence suggests the mean drop diameter and the mean drop charge level are usually correlated. In addition, within each drop diameter class there is every reason to suspect a distribution of charge levels exist for a particular drop diameter class. This paper presents an experimental method that uses the joint PDF of drop velocity and diameter, obtained from phase Doppler anemometry measurements, and directly obtained spatially resolved distributions of the mass and charge flux to obtain a drop diameter and charge frequency distribution. The method is demonstrated using several data-sets obtained from experimental measurements of steady poly-disperse sprays of an electrically insulating liquid produced with the charge injection technique. The space charge repulsion in the spray plume produces a hollow cone spray structure. In addition an approximate self-similarity is observed, with the maximum radial mass and charge flow occurring at r/d ~ 200. The charge flux profile is slightly offset from the mass flux profile, and this gives direct evidence that the spray specific charge increases from approximately 20% of the bulk mean spray specific charge on the spray axis to approximately 200% of the bulk mean specific charge in the periphery of the spray. The results from the drop charge estimation model suggest a complex picture of the correlation between drop charge and drop diameter, with spray specific charge, injection velocity and orifice diameter all contributing to the shape of the drop diameter–charge distribution. Mean drop charge as a function of the Rayleigh limit is approximately 0.2, and is invariant with drop diameter and also across the spray cases tested.     

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.     

Interaction between two consecutive axisymmetric Taylor drops flowing in a heavier liquid in a vertical tube

A study regarding the interaction between two consecutive Taylor drops flowing in a heavier liquid in a vertical tube is reported. Under certain conditions, due to the wake of the leading drop, the trailing drop accelerates, leading to coalescence of the two drops. This study was developed using a numerical model based in the Volume of Fluid method in an axisymmetric geometry. The simulations reported in the present work had to fulfill two conditions: axisymmetry (due to the numerical model) and a high enough drop Reynolds number (which is related to the disturbances in the wake of an isolated drop, and thus to the tendency to drop interaction). Relevant dimensionless numbers are used to assess the effect of the acting forces. Detailed flow patterns and drop shapes are provided. Furthermore, the approaching velocity acquired by the trailing drop is analyzed and velocity profiles between the leading and the trailing drop are also reported. In general, the trailing drop shows an accelerating region, followed by a deceleration near the leading drop. The increase of Eotvos number promotes higher accelerations, while the increase in Morton number and viscosity ratio has the opposite effect. By comparison to literature gas-liquid studies, it was also found that interfacial forces promote the shape stability of the drops.     

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     

MOTION AND DEFORMATION OF VISCOUS DROP IN STOKES FLOW NEAR RIGID WALL

A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerical experiments by the method were performed for different initial stand-off distances of the drop to the wall, viscosity ratios, combined surface tension and buoyancy parameters and ambient flow parameters. Numerical results show that due to the action of ambient flow and buoyancy the drop is compressed and stretched respectively in axial and radial directions when time goes. When the ambient flow action is weaker than that of the buoyancy the drop raises and bends upward and the stress on the wall induced by drop motion decreases when time advances. When the ambient flow action is stronger than that of the buoyancy the drop descends and becomes flatter and flatter as time goes. In this case when the initial stand-off distance is large the stress on the wall increases as the drop evolutes but when the stand-off distance is small the stress on the wall decreases as a result of combined effects of ambient flow, buoyancy and the stronger wall action to the flow. The action of the stress on the wall induced by drop motion is restricted in an area near the symmetric axis, which increases when the initial stand-off distance increases. When the initial stand-off distance increases the stress induced by drop motion decreases substantially. The surface tension effects resist the deformation and smooth the profile of the drop surfaces. The drop viscosity will reduce the deformation and migration of the drop.     

Deformation and breakup of a non-Newtonian slender drop in an extensional flow

The deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid in a simple extensional and creeping flow has been theoretically studied. The power-law was chosen for the fluid inside the drop, and the deformation of the drop is described by a single ordinary differential equation, which was numerically solved. Asymptotic analytical expressions for the local radius were derived near the center and close to the end of the drop. The results for the shape of the drop and the breakup criterion are presented as a function of the capillary number, the viscosity ratio and type of non-Newtonian fluid inside the drop. An approximate analytical solution is also suggested which is in good agreement with the numerical results.     

相近韦伯数条件下激波波后液滴初期变形的影响机制

以实验结合数值模拟与理论分析的方法,研究韦伯数在2 100~2 700区间内,不同组合流动参数对液滴破碎初期变形的影响与作用机制。实验中通过高速摄影捕捉到一系列具有明显差异的液滴变形模态,表明在相近韦伯数下液滴的初期变形仍受到气流速度、密度等具体流动参数的显著影响。以刚性球体替代液滴进行外流数值模拟,利用球体表面气动力分布推算出的液滴表面变形趋势与实际变形形态吻合,表明液滴的初期变形特征与外流流动分离和涡特征具有一致性。对流场和理论变形数据的分析显示,流动分离发展阶段和稳定阶段对液滴作用力以及它们所诱导的液滴变形特征存在很大差异;分离发展与液滴变形过程的特征时间之比可由气液密度比的平方根表示,它决定了液滴早期变形的基本形态。分离发展阶段所占时间比例越高,即实验中气液密度比越高,则液滴更倾向于发展出单个显著的环形突起,反之则趋于形成多个相对均衡的突起。     

Drop and Bubble Deformation in an Electric Field

A steady problem of drop (bubble) shape in a uniform electric field is considered when the drop and the surrounding medium are immiscible. The electric-charge transport includes both the ohmic current across the interphase boundary and convective transport over the interface. If there is no convective transport, the drop (bubble) may be transformed into either an elongated or a flattened spheroid. Under these conditions, the sign of the deformation remains unchanged for arbitrary values of the problem parameters. Convective charge transport along the surface initiates additional motion in both the drop and the surrounding medium. However, with increase in the convective-transport intensity the deformed drops display different behavior. The compression of a flattened drop slows and, under certain conditions, compression is replaced by extension. However, an elongated spheroid cannot be transformed into a flattened spheroid. The calculations were performed under the assumption that the drop is convex. It was found that, for both an elongated and a flattened drop, the maximum ratio of the major and minor spheroid axes is 2:1. In experiments with oils, the possibility of both a decrease in the drop compression rate and deformation sign reversal was demonstrated.     

A boundary‐only approach to the deformation of a shear‐thinning drop in extensional Newtonian flow

The influence of shear thinning on drop deformation is examined through a numerical simulation. A two‐dimensional formulation within the scope of the boundary element method (BEM) is proposed for a drop driven by the ambient flow inside a channel of a general shape, with emphasis on a convergent–divergent channel. The drop is assumed to be shear thinning, obeying the Carreau–Bird model and the suspending fluid is Newtonian. The viscosity of the drop at any time is estimated on the basis of a rate‐of‐strain averaged over the region occupied by the drop. The viscosity thus changes from one time step to the next, and it is strongly influenced by drop deformation. It is found that small drops, flowing on the axis, elongate in the convergent part of the channel, then regain their spherical form in the divergent part; thus confirming experimental observations. Newtonian drops placed off‐axis are found to rotate during the flow with the period related to the initial extension, i.e. to the drop aspect ratio. This rotation is strongly prohibited by shear thinning. The formulation is validated by monitoring the local change of viscosity along the interface between the drop and the suspending fluid. It is found that the viscosity averaged over the drop compares, generally to within a few per cent, with the exact viscosity along the interface.     

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.     

The response of a spherical droplet on a wall executing small sinusoidal vibrations

Meccanica - This study concerns the response of a spherical drop, attached to a sinusoidally vibrating wall. Given that the drop is spherical, the model is more realistic than that of a 2D drop...     

Equilibrium shapes of a drop pendant in an electrostatic field

The stationary shapes of a conducting fluid drop in the gap between the plates of a plane capacitor are studied. The drop is held on the upper plate by the surface tension forces. The self-consistent problem of the determination of the drop shape and the charge distribution over its surface is solved. Estimates are obtained for the maximum volume of the stationary drop at the given fluid parameters and electric field strength.     

Elastocapillary Coiling of an Elastic Rod Inside a Drop

Capillary forces acting at the surface of a liquid drop can be strong enough to deform small objects and recent studies have provided several examples of elastic instabilities induced by surface tension. We present such an example where a liquid drop sits on a straight fiber, and we show that the liquid attracts the fiber which thereby coils inside the drop. We derive the equilibrium equations for the system, compute bifurcation curves, and show the packed fiber may adopt several possible configurations inside the drop. We use the energy of the system to discriminate between the different configurations and find a intermittent regime between two-dimensional and three-dimensional solutions as more and more fiber is driven inside the drop.