The effects of confinement and inertia on the production of droplets

Recent experiments of Sibillo et al., Phys. Rev. Lett. 97:054502, (2006) investigate the effect of walls on flow-induced drop deformation for Stokes flow. The drop and the fluid in which it is suspended have the same viscosities. The capillary numbers vary from 0.4 to 0.46. They find that complex start-up transients are observed with overshoots and undershoots in drop deformation. Drop breakup is inhibited by lowering the gap. The ratio of initial drop radius to wall separation is fixed at 0.34. We show that inertia can enhance elongation to break the drop by examining Reynolds numbers in the range of 1 to 10. The volumes of the daughter drops can be larger than in the unbounded case, and even result in the production of monodisperse droplets.      

Investigation of liquid–liquid drop coalescence using tomographic PIV

High-speed tomographic PIV was used to investigate the coalescence of drops placed on a liquid/liquid interface; the coalescence of a single drop and of a drop in the presence of an adjacent drop (side-by-side drops) was investigated. The viscosity ratio between the drop and surrounding fluids was 0.14, the Ohnesorge number (Oh = μ_d/(ρ_dσD)^1/2) was 0.011, and Bond numbers (Bo = (ρ _d  − ρ _s )gD ²/σ) were 3.1–7.5. Evolving volumetric velocity fields of the full coalescence process allowed for quantification of the velocity scales occurring over different time scales. For both single and side-by-side drops, the coalescence initiates with an off-axis film rupture and film retraction speeds an order of magnitude larger than the collapse speed of the drop fluid. This is followed by the formation and propagation of an outward surface wave along the coalescing interface with wavelength of approximately 2D. For side-by-side drops, the collapse of the first drop is asymmetric due to the presence of the second drop and associated interface deformation. Overall, tomographic PIV provides insight into the flow physics and inherent three-dimensionalities in the coalescence process that would not be achievable with flow visualization or planar PIV only.     

Techniques for generating centimetric drops in microgravity and application to cavitation studies

This paper describes the techniques and physical parameters used to produce stable centimetric water drops in microgravity, and to study single cavitation bubbles inside such drops (Parabolic Flight Campaigns, European Space Agency ESA). While the main scientific results have been presented in a previous paper, we shall herein provide the necessary technical background, with potential applications to other experiments. First, we present an original method to produce and capture large stable drops in microgravity. This technique succeeded in generating quasi-spherical water drops with volumes up to 8 ml, despite the residual g-jitter. We find that the equilibrium of the drops is essentially dictated by the ratio between the drop volume and the contact surface used to capture the drop, and formulate a simple stability criterion. In a second part, we present a setup for creating and studying single cavitation bubbles inside those drops. In addition, we analyze the influence of the bubble size and position on the drop behaviour after collapse, i.e., jets and surface perturbations.     

Investigation of binary drop rebound and coalescence in liquids using dual-field PIV technique

Experiments on binary drop collisions within an index-matched liquid were conducted for Weber numbers (We) in the range of 1–50. Drop pairs of water/glycerin mixture were injected horizontally into silicone oil and, due to gravitational effects, travelled on downward trajectories before colliding. A dual-field high-speed PIV measurement system was employed to quantify drop trajectories and overall collision conditions while simultaneously examining detailed velocity fields at the collision interface. Sequences of velocity and vorticity fields were computed for both larger and smaller fields of view. In the We range examined, both rebounding and coalescing behavior occurred. Coalescence was found to result from a combination of vortical flow within drops and strong drop deformation characteristic of higher We. Flow through the centers of opposing ring vortices, strengthened by drop deformation, enhanced drainage of the thin film in the impact region, leading to film rupture and coalescence. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Diffusion to a chain of drops (bubbles) at large péclet numbers

The problem of diffusion of a substance, dissolved in a flow, to absorbing drops (bubbles) moving one after another in a viscous incompressible fluid is investigated. An approximate analytic expression is obtained for the differential and integral flows of the substance to the surface of each drop with consideration of the changes of the concentration and velocity fields due to the presence of other drops. A chain of spherical drops of equal radius arranged on the axis of a uniform forward flow is examined. It is shown that if the distance between drops, referred to the radius of the drops, satisfies the inequality 1lP^1/2 (P is the Péclet number), then the integral inflow of the substance to the surface of the second drop of the chain is 2.41 times less than the integral inflow to the first (the drops are enumerated along the flow); the total diffusion flow to the surface of an arbitrary drop with number k is determined by the expression I_k=I₁[k^1/2 – (k–1)^1/2], where I_k is the total flow to the first drop of the chain. The case of diffusion interaction of a solid particle and drop is examined. It is shown that for particles moving one after another with the same velocity in a fluid at rest the presence of a drop before the solid particle leads to a marked decrease of the total diffusion flow of the solid particle [by O(P^1/6) times], whereas the presence of a solid particle before a drop does not affect (in the main approximation with respect to the characteristic diffusion parameter) the total flow of the latter.I _k=I _i[k ^1/2–(k–1)^1/2]Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–69, January–February, 1978.     

Heat transfer characteristics of boundary-layer flows induced by continuous surfaces stretched with prescribed skin friction

A continuous surface stretched with velocity u _w=u _w (x) and having the temperature distribution T _w=T _w (x) interacts with the viscous fluid in which it is immersed both mechanically and thermally. The thermal interaction is characterized by the surface heat flux q _w=q _w (x) and the mechanical one by the skin friction τ _w=τ _w (x). In the whole previous theoretical research concerned with such processes, either (u _w and T _w) or (u _w and q _w) have been prescribed as known boundary conditions. The goal of the present paper is to initiate the investigation of the boundary layer flows induced by stretching processes for which either (τ _w and T _w ) or (τ _w and q _w) are the prescribed quantities. The case of an isothermal surface stretched with constant skin friction, (τ _w=const., T _w=const. ≠ T _∞) is worked out in detail. The corresponding flow and heat transfer characteristics are compared to those obtained for the (well known) case of a uniformly moving isothermal surface (u _w=const., T _w=const. ≠ T _∞).     

Collision between immiscible drops with large surface tension difference: diesel oil and water

The collision outcomes of immiscible drops with large surface tension difference, namely, a water drop and a diesel oil drop, were observed experimentally. In a near head-on collision between immiscible drops with large surface tension difference, an “overlaying” action for the drop of the smaller surface tension, i.e., the diesel oil drop, to go around the surface of the drop of the larger surface tension, i.e., the water drop, occurs during the collision. This overlaying action reduces the reflex energy for head-on collisions, making reflex separation more difficult to occur. At the same time, due to the immiscibility, the liquid bridge during stretching separation becomes narrower, which makes stretching separation easier to happen. No coalescence could be observed for a collision of Weber number greater than 60. In addition, compound drops are produced frequently.     

Gravity-driven motion of a deformable drop or bubble near an inclined plane at low Reynolds number

The creeping motion of a three-dimensional deformable drop or bubble in the vicinity of an inclined wall is investigated by dynamical simulations using a boundary-integral method. We examine the transient and steady velocities, shapes, and positions of a freely-suspended, non-wetting drop moving due to gravity as a function of the drop-to-medium viscosity ratio, λ, the wall inclination angle from horizontal, θ, and Bond number, B, the latter which gives the relative magnitude of the buoyancy to capillary forces. For fixed λ and θ, drops and bubbles show increasingly pronounced deformation in steady motion with increasing Bond number, and a continued elongation and the possible onset of breakup are observed for sufficiently large Bond numbers. Unexpectedly, viscous drops maintain smaller separations and deform more than bubbles in steady motion at fixed Bond number over a large range of inclination angles. The steady velocities of drops (made dimensionless by the settling velocity of an isolated spherical drop) increase with increasing Bond number for intermediate-to-large inclination angles (i.e. 45° ? θ ? 75°). However, the steady drop velocity is not always an increasing function of Bond number for viscous drops at smaller inclination angles.     

Visualization study of evaporation of single n-pentane drops in water

A laser shadowgraph system was constructed to enable successive filming of a drop or a bubble rising or falling in an immiscible liquid confined within a vertical column. The assembly was applied to a study of the evaporation of n-pentane drops in a stagnant medium of water. The liquid/vapor two-phase bubble evolving from each pentane drop was observed together with its wake, the morphology and the dynamics of which are our primary concern in considering the mechanism of the medium-to-bubble heat transfer.List of symbols a minor axis of ellipsoidal two-phase bubble - b major axis of ellipsoidal two-phase bubble - D ₀ diameter of saturated-liquid drop set to vaporize - Re Reynolds number based on instantaneous, volume-equivalent spherical diameter and rise velocity of two-phase bubble and kinematic viscosity of the continuous phase - t time lapse after the start of evaporation - T ^* excess of undisturbed continuous-phase temperature above the temperature at which the sum of the saturated vapor pressures of the dispersed- and the continuous-phase fluids is equal to the pressure at the position where evaporation starts - opening angle of wake-covered region on bubble surface - _w zenith angle at flow-separation ring on bubble surface - mass fraction of vapor in two-phase bubble     

Application of liquid-crystal thermometry to drop temperature measurements

A technique has been developed that enables remote sensing of the temperatures of liquid drops in a medium of an immiscible, transparent liquid with the aid of dispersing microcapsules of thermochromic liquid crystal in each drop under illumination by either a planar floodlight or a light sheet which cuts the drop at a meridian. Based on appropriate hue-temperature calibrations made with an isothermal, stationary drop/medium system, one can analyze spatial and time variations of temperature within drops in motion under transient convective heating (or cooling) from the medium.List of Symbols B tristimulus blue component - D drop diameter - G tristimulus green component - H hue angle - h ₀ constant term in the expression for H - H _m mean value of H over drop surface - h vertical coordinate fixed onto test column - R tristimulus red component - Re drop Reynolds number, UD/v _c - r, g, b chromaticity coordinates - r drop radius - s penetration depth of light into drop - T temperature - T _d instantaneous drop temperature - T _d0 initial drop temperature - T _c continuous-phase temperature - U velocity of rise of drop - z vertical coordinate laid on drop - v _c kinematic viscosity of continuous phase - angle of lighting measured from camera axis - _s local view angle (azimuth) - polar angle     

Calculation of motion of a spherical drop in the Bingham medium

Mathematical modeling of the experimentally observed process of approaching of two identical oil drops located in an alcohol-water solution (matrix) with an identical density is performed. It is found that the drop moves in cycles consisting of the state at rest, acceleration, and deceleration; the cycle time is about 10⁻² s. Violation of the balance of forces on the drop boundary in the state at rest is caused by the fact that the shear stresses on this boundary cannot exceed the yield stress of the matrix, and the normal stresses are determined by solving the problem of the elasticity theory, because intermolecular bonds in the quiescent matrix make it similar to a solid. The results of drop motion calculations and experimental data agree well during the entire process of drop approaching, except for the final stage, which can be attributed to the neglect of hydrodynamic interaction of the drops.     

Experimental investigation of the interaction between an electrically charged liquid-drop dispersed phase and a turbulent air-steam jet

The interaction of an electrically charged liquid-drop dispersed phase with a turbulent air-steam jet in a "water drop generator (capillary)-jettarget" system is investigated. The experimental conditions were as follows: volume flow of water through the capillary 0.01 cm³/s. capillary inside diameter 0.8 mm, electric potential applied to the capillary 20 kV. When the electric field is absent, condensation does not develop in the jet despite the existence of oversaturation zones. Two regimes of interaction between the charged dispersed phase and the jet are detected. The first regime (for ϕ<8 kV) is characterized by the regular launching of fairly large charged drops (0.5<r<2 mm) from the capillary and the absence of condensation in thejet. As the potential ϕ increases, the drop size decreases, whereas the drop charge increases. This regime made it possible to model the motion of individual charged clusters of different charge and size in aircraft engine jets. The second regime (ϕ>8 kV) is characterized by the irregular launching of drops from the capillary, drop dispersion with respect to size, a sharp increase in the target current, and the sudden appearance of condensation in the air-steam jet. The possible electrohydrodynamic and heterophase processes are qualitatively analyzed. Moscow, e-mail: vatazhin@ciam.ru.likhter@ciam.ru. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–114, July–August, 2000. The work received financial support from the Russian Foundation for Basic Research (project No. 99-01-00983).     

Mass transfer from a pulsating drop

The primary difficulty in solving the problem of mass transport through an isolated drop (or bubble) moving in a fluid medium at high Reynolds numbers lies in the extreme complexity of the hydrodynamic pattern of the phenomenon. For sufficiently high velocities a separation of the external flow will occur in the rear portion of the drops and bubbles, which leads to the appearance of a turbulent wake and a sharp increase of the hydrodynamic resistance. Beginning with those dimensions for which the resistance force acting per unit surface of the drop or bubble from the external medium becomes greater than the capillary pressure, the surface of the drops and bubbles begins to deform and pulsate. The local variations of the surface tension, resulting either from the process of convective diffusion or from adsorption of surface-active substances, have a large effect on the hydrodynamics of drops and bubbles (particularly on the deformation of their surface) [1, 2], The presence of vortical, and possibly even turbulent, motion within the drops and bubbles may, under certain conditions [1], lead to their fractionation.Naturally, at the present time such complex hydrodynamics cannot be described by exact quantitative relations. Several authors have attempted to solve this problem approximately within the framework of certain assumptions. In particular [3–6], a theory was developed for the boundary layer on the surface of spherical and ellipsoidal gaseous bubbles moving in a liquid, studies were made [7, 8] of the hydrodynamics of drops located in a gas flow and the conditions were found for which fractionation of such drops takes place. Of considerable practical interest is the development of the theory of mass transfer in pulsating drops and bubbles and finding in explicit form the dependence of the mass transfer coefficients on the hydrodynamic characteristics of these systems. Until this relationship is established, every theory which ignores the effect of hydrodynamics on the mass transfer rate from an individual drop or bubble cannot be considered in any way well-founded. This relates particularly to the theories [9, 10] which consider mass transfer in systems with concentrated streams of drops and bubbles. The present paper is devoted to the study of mass transport through the surface of an isolated drop in an irrotational gas or liquid stream for large Peclet numbers P.In conclusion the authors wish to thank V. G. Levich for his helpful discussions.     

Drop in an oscillating liquid

At the present time there is a complete lack of studies devoted to the forced motion of a drop located in an oscillating liquid. However, this problem is of considerable interest. For example, it represents simulation of hydrodynamic processes occurring during the irradiation of drops of one liquid located in another liquid by longwave sound. The stationary flows occurring in this case may have a significant influence on the heat- and mass-transfer processes. In the present article we investigate the velocity field in the interior and exterior of a drop executing forced oscillatory motion as a result of its interaction with the ambient liquid. At a sufficiently large distance from the drop the ambient liquid oscillates in a specified way, where s/R ? 1 (s is the amplitude of displacement of the liquid particles, and R is the radius of the drop).     

Dependence of micro-drop generation performance on dispenser geometry

In this paper, the drop generation performance, represented by the speed of generation and the attainable size range of drops, of λ-junction type micro (∼100 μm) dispensers was examined for various heights, widths and fluid injection angles quantitatively. Target range of drops was about the same size of the channel hydraulic diameter (0.8–1.2 D_h,c) that is known to be most efficient for internal mixing of different components within micro-drops. Viscosities of the disperse and continuous phases were 2.7 and 2.3 mPa s, respectively. Also, the superficial velocity range of the disperse phase was 0.002–0.128 m/s and that of the continuous phase was 0.02–0.15 m/s. Hence, the corresponding ranges of the capillary and the Reynolds numbers (based on the channel width) of the continuous phase were 0.004–0.034 and 1–32, respectively. Within the present test ranges, the drop generation performance was improved with the smaller width ratio (between the side and the main inlets), and at the aspect ratio of about 0.8 and the injection angle of about 120°. Furthermore, through the detailed observations, the geometrical similarity of the bulged part of the disperse phase was confirmed to exist between the cases with different junction dimensions (widths and height), which is an important clue for prediction of drop sizes.     

Sizing problem for a cross flow heat exchanger with wing-type vortex generator

In the present study, sizing of a single pass cross flow heat exchanger with unmixed fluid streams has been investigated. The heat exchanger is a cross flow heat exchanger. It has overall dimensions of 20 × 20 × 20 cm. Two the most common heat exchanger design problems are the rating and sizing problem. Sizing problems deal with designing an exchanger and determining its physical size to meet the specified heat duty, pressure drops and other considerations. It means the determination of the exchanger construction type, flow arrangement, heat transfer surface geometries and materials, and the physical sizes of an exchanger to meet specified heat transfer and pressure drop. In this study, the physical size (length, width, height, mass flow rates of both fluids and surface areas on each side of the exchanger) are determined. Inputs to the sizing problem are surface geometries, fluid mass flow rates, inlet and outlet fluid temperatures and pressure drop on each side. Dimensions of L _a , L _b , and L _c for the selected surfaces were investigated such that the design meets the heat duty and pressure drops on both sides exactly.     

Production of secondary drops during the single water drop impact onto a plane water surface

The normal impact of single water drops onto a plane water surface was studied experimentally to reveal the amount of secondary drops produced from the rim of crown-like interfacial structure. Within the experimental ranges tested, the ratio of the total mass of secondary drops to the mass of primary drop was approximately within 0–1 and correlated well as the function of dimensionless parameter K that consisted of the impact Weber number We and the Ohnesorge number Oh (K = We Oh ^−0.4). The dependences of the number and the mean diameter of secondary drops on K and dimensionless film thickness were also investigated.     

Statistical mechanics of temporary polymer networks I. The equilibrium theory

In part I of this work (the present article) the equilibrium state of temporary polymer networks is treated in the framework of thermodynamics and statistical mechanics. The network is described as an open system. Thereby we use a modified spring-bead model in which the beads represent junctions that decay and reform thus adding a viscous component to the assumed elastic behaviour of the permanent network. The relevant statistical equation — analogous to Liouville's equation — is solved. The grand-canonical probability density function and two of three equations of state are derived. Explicit formulae are given for several relevant probabilities. For instance the probabilityw (z)dz that a network chain connecting two junctions has a contour length betweenz andz +dz is given by the Wien type formulaw(z) =A z ³ exp {–B z} whereA andB do not depend onz.     

Three-dimensional numerical simulation of sedimenting drops inside a vertical channel

Numerical simulation of sedimenting deformable drops inside a vertical channel has been performed at finite Reynolds numbers. The channel is confined by two vertical walls in x-direction and is periodic in y- and z-directions. Results are obtained using a finite difference/front-tracking method. The main dimensionless parameters are the Reynolds number, the Bond number and ratio of the length of channel to the diameter of drops. The effect of these parameters on lateral migration of drops is investigated. It is found that the wall repulsion is the main mechanism of the lateral migration of the drop, and drop migrates toward the channel axis. When the Reynolds number is relatively low, two different lateral migration regimes are observed: migration with monotonic approach and migration with damped oscillations. These regimes are affected by the dimensionless parameters. When the Bond number increases, the oscillations of drop around the centerline of channel are stronger and drop reaches the channel centerline in a larger period. Results of lateral migration of one drop are consistent with perturbation theory, and two-dimensional numerical simulations performed by Feng et al. (1994). The drag coefficient has also been calculated, and effect of various parameters has been discussed. Two drops interaction is similar to that observed by Feng et al. (1994) for two-dimensional circular cylinders. Results are consistent with experiments performed by Wu and Manasseh (1998). Simulations of four sedimenting drops show that depending on the relative size of drops, they either fall in two rows or they form a single horizontal layer and settle with a unique velocity.     

One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P _w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P _w could be either positive or negative. When P _w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P _w  < 0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k–l formulation rather than from the closure for P _w . An alternative closure for P _w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P _w  > 0 but has also a small region of negative P _w in the buffer layer, was also successfully tested. This second model for P _w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.