Drop Viscosity Research Articles

Near-Contact, Thermocapillary Migration of a Nonconducting, Viscous Drop Normal to a Planar Interface

The near-contact, thermocapillary motion of a nonconducting, spherical drop normal to a planar, isothermal interface has been analyzed using a lubrication solution for conditions of small Reynolds and Marangoni numbers. The interface separates two viscous fluids; a third is contained within the drop. A closed-form expression is obtained for the near-contact drop velocity. The drop motion results from a "contact force" that is opposed by a lubrication resistance; the contact force is determined by considering a force balance on a drop tangent to the interface. For drops of moderate viscosity, the contact force is comparable to the thermocapillary force that acts on an isolated, stationary drop; for highly viscous drops, the contact force dominates as the logarithm of the drop viscosity ratio. If both the drop surface and the planar interface are fully mobile or free, the near-contact drop velocity, normalized by the value for a drop in an unbounded fluid, decreases with drop viscosity; however, if the interface is immobile and/or the drop is highly viscous, the normalized velocity increases with drop viscosity. The cases of a drop moving toward rigid and free surfaces and of a drop moving toward a semi-infinite fluid with the same viscosity are examined explicitly. A comparison with buoyancy-driven motion reveals that thermocapillary migration is the more efficient near-contact migration mechanism.

Droplet Growth Due to Brownian, Gravitational, or Thermocapillary Motion and Coalescence in Dilute Dispersions

The time evolution of drop size distributions in dilute, homogeneous dispersions was analyzed by solving population dynamics equations which account for the effects of hydrodynamic interactions on the collision rates of the drops. The collision mechanisms studied individually include Brownian motion, thermocapillary migration, and gravity sedimentation. It is shown that hydrodynamic interactions greatly reduce the predicted rates of drop growth in dispersions. The growth rates are shown to decrease with increasing drop viscosity and/or thermal conductivity due to increasing hydrodynamic interactions. For Brownian coalescence, the results for long times are in good agreement with the predictions of a similarity solution, whereas no similarity solution exists for gravitational and thermocapillary coalescence. For thermocapillary-induced coalescence of highly conducting drops, there is a region in parameter space in which drop collisions are prevented by thermocapillary repulsion, and this leads to slow growth and the evolution of drop-size distributions with long tails of small drops.

The deformation and breakup of liquid drops in low Reynolds number flow through a capillary

The deformation and breakup of a freely suspended, immiscible liquid drop are examined experimentally for low Reynolds number flow in a straight circular capillary tube. Results are reported for capillary numbers from 0.05 to greater than unity and for values of the viscosity ratio that span more than three orders of magnitude. In every case, the size of the drop is comparable to the tube diameter. The steady-state shapes of the drops are compared with previous numerical studies of the problem. However, for a critical value of the capillary number that depends on drop size and viscosity ratio, drop breakup is observed in the experiment. When the viscosity ratio is small, i.e., the drop fluid is less viscous than the outer phase liquid, an indentation in the trailing edge of the drop grows and entrains outer phase liquid inside the boundaries of the original drop. When the viscosity ratio is O(1), the drop stretches along the axis of the tube until it breaks.

Surface-tension-induced mixing following coalescence of initially stationary drops

The momentumless coalescence of drops of the same liquid, separated by an immiscible host, is studied experimentally. Observations show that for low-viscosity drops of unequal sizes, there is considerable mixing following coalescence, with the smaller drop penetrating the larger drop as a vortex. The extreme case of coalescence of a small drop with the bulk of the same liquid at a flat interface with an immiscible liquid is studied in detail. The penetration depths of small drops (1–5 mm) following coalescence are measured and correlated with theoretical predictions. It is found that in the range of the investigation, the penetration depth is proportional to the 5/4 power of drop diameter and inversely proportional to the square root of the drop viscosity.

The lubrication force between two viscous drops

The hydrodynamic force resisting the relative motion of two unequal drops moving along their line of centers is determined for Stokes flow conditions. The drops are assumed to be in near-contact and to have sufficiently high interfacial tension that they remain spherical. The squeeze flow in the narrow gap between the drops is analyzed using lubrication theory, and the flow within the drops near the axis of symmetry is analyzed using a boundary integral technique. The two flows are coupled through the nonzero tangential stress and velocity at the interface. Depending on the ratio of drop viscosity to that of the continuous phase, and also on the ratio of the distance between the drops to their reduced radius, three possible flow situations arise, corresponding to nearly rigid drops, drops with partially mobile interfaces, and drops with fully mobile interfaces. The results for the resistance functions are in good agreement with an earlier series solution using bispherical coordinates. These results have important implications for droplet collisions and coalescence.

Effect of Covalently Bound Fatty Acids and Associated Lipids on the Viscosity of Gastric Mucus Glycoprotein in Cystic Fibrosis

The contribution of associated and covalently bound lipids to the viscosity of gastric mucus glycoprotein in healthy and cystic fibrosis (CF) individuals was investigated. While both preparations exhibited similar contents of protein and carbohydrate, the CF glycoprotein contained 1.3 times more associated lipids and 6 times more covalently bound fatty acids. The viscosity of CF mucus glycoprotein was about 1.8 times higher than that of normal glycoprotein. Extraction of associated lipids lead to 3-fold drop in the viscosity of CF glycoprotein and 5-fold drop in the case of normal glycoprotein. Removal of covalently bound fatty acids caused further 1.6-fold reduction in the viscosity of normal mucus glycoprotein and 6-fold in CF glycoprotein. The viscosity of the delipidated and deacylated CF mucus glycoprotein was only about 10% higher than that of the similarly treated normal glycoprotein. The results suggest that the elevated level of covalently bound and associated lipids is responsible for the increased viscosity of CF mucin.

Mucus glycoprotein fatty acyltransferase in patients with cystic fibrosis: Effect on the glycoprotein viscosity

The presence of an acyltransferase activity which catalyzes the transfer of palmitic acid from palmitoyl coenzyme A to mucus glycoprotein has been demonstrated in the microsomal fraction of human rectal mucosa. The activity of this enzyme in the mucosa of patients with cystic fibrosis (CF) was found to be 3.5 times higher than that from normal individuals. The CF mucus glycoprotein in comparison to that of normal contained 1.3 times more associated lipids and 6 times more covalently bound fatty acids. The viscosity of the intact CF glycoprotein was 1.8 times higher than that of normal glycoprotein. Extraction of associated lipids led to 3-fold drop in the viscosity of CF glycoprotein and 5-fold drop in the case of normal glycoprotein. Further loss in the viscosity occurred following removal of the covalently bound fatty acids. The viscosity of such modified CF mucus glycoprotein was only about 10% higher than that of similarly treated normal glycoprotein.

Experimental study on the behavior of the drop impinging upon a hot surface (2nd report, Effect of the liquid properties on the behavior of a drop)

The behavior of the drop impinging upon a hot surface has been investigated experimentally in this paper. Putting importance on the dynamic aspects, the effects provided by the drop's viscosity and the surface tension on the variation of disintegration and deformation of the drops have been examined. The conclusions are summarized as follows; For drops with kinetic viscosity of upto approximately 60 mm2/s, the disintegration patterns can be identified by We number, even if the surface tension and/or the impinging velocity differ. Also, the non-dimensional maximum spreading diameter dsmax/di and the non-dimensional maximum apparent contact diameter dcmax/di and the non-dimensional maximum apparent contact diameter dcmax/di can be categorized soley under the We number. But the viscosity increases and becomes more than approximately 80 mm2/s, the effect of viscosity on the variation in disintegration patterns and deformation becomes more marked and then cannot be identified solely by We number. In this study, the experimental equations for dsmax/di have been defined by We number and viscosity.

An experimental study on double emulsion drop breakup in uniform shear flow

In this work, the characteristics of the breakup in simple shear flow of double emulsion drops, suspended in a continuous immiscible, Newtonian phase, are studied. Double emulsions have been proposed as a medium for extraction but the breakup of drops reduces the efficiency of the extraction. Double emulsion drop breakup due to viscous forces was studied in a transparent cone-and-plate viscometer. The deformation or breakup parameter is the ratio of the applied viscous force when the drop ruptures to the surface tension force. This parameter can be correlated with the ratio of the apparent viscosity of the double emulsion drop to the viscosity of the continuous phase to yield a breakup curve. Although mechanistically double emulsion drops break up similarly to homogeneous Newtonian drops, the breakup curves for these two systems are different. It is shown that the elasticity of the double emulsion drops is in part responsible for this behavior.

The Use of Hydrocyclones for Small Particle Separation

Abstract Hydrocyclone operating characteristics have been studied by measuring the capacity for separating haematite, magnetite and silica particles suspended in water. The effects of viscosity (achieved by changing temperature) and throughput have been investigated and are satisfactorily correlated. The measurements demonstrate that under appropriate conditions of high pressure drop and low viscosity, particles below the usually accepted limit of 2 μm can be separated. Measured particle separation is expressed as a hydrocyclone efficiency, and its variation with particle size is shown. Anomalous results with magnetite and haematite suspensions are suggested to occur because these materials exist as agglomerates with effective density less than half that of the bulk material.

External and internal streamlines and deformation of drops in linear two-dimensional flows

The creeping flow of a Newtonian fluid around a neutrally buoyant, immiscible spherical liquid drop has been studied theoretically. The streamlines inside and outside the drop and its deformation have been calculated for the family of linear two-dimensional flows, members of which are determined by a dimensionless parameter −1 ⩽ α ⩽ 1. These flows include pure shear flow (α = 1) as one limit and pure rotation (α = −1) as the other, with simple shear (α = 0) as an intermediate case. In the bulk medium, it is found that both an open and a closed streamline region exists for 0 ⩽ α < 1, with the distance from the drop to the limiting streamline dividing the two regions being determined by the ratio of the drop viscosity to that of the medium, λ. All streamlines are open in pure shear flows (α = 1) regardless of λ and for all flows when 0 ⩽ α ⩽ 1 and λ = 0. For flows having −1 ⩽ α < 0, streamlines are always closed. The circulation established in the drop due to the external flow varies according to α and λ. In particular, as many as four pockets of circulation are found for flows with 0 < α ⩽ 1 while for −1 < α ⩽ 0 only two such “pockets” exist. In the trivial case α = −1 there is a single pocket in the drop. Equations giving the axis ratio and orientation of a deformed drop are also derived.

Stokes flow past bubbles and drops partially coated with thin films. Part 2. Thin films with internal circulation – a perturbation solution

In the present study we examine the steady axisymmetric creeping flow due to the motion of a liquid drop or a bubble which is partially covered by a thin immiscible fluid layer or film. The analysis is based on the assumption that surface-tension forces are large compared with viscous forces which deform the drop, and that the circulation in the film is weak. The latter assumption is satisfied provided that the film-fluid viscosity is not too small. A perturbation scheme based on the thinness of the fluid layer is used to construct the solution.One of the principal results is an expression for the drag force on the complex drop. We also find that the extent to which the drop or bubble is covered the film has a maximum value depending on the magnitude of the driving force on the film. In addition, we find the rather interesting result that when the ratio of the primary drop viscosity and bulk fluid viscosity is greater than ½, the circulation within the film may have a double-cell structure.

An experimental study of small-amplitude drop oscillations in immiscible liquid systems

Measurements of the characteristics of small-amplitude shape oscillations of drops immersed in a host liquid have been carried out by acoustical means. The resonance frequencies of the first few modes have been measured, as well as the damping constant for the fundamental mode, as functions of the drop radius and viscosities of the two liquids. A qualitative photographic study during steady oscillations has revealed a simple internal fluid-particle flow field with no circulation. The theory available at the present time has been found to provide results which are in general agreement with experimental findings for low-viscosity liquids.

Calculation of close interaction between drops, with internal circulation and slip effect taken into account

An axisymmetric problem of motion of two spherical drops in a viscous medium is studied in the Stokes approximation. The drop viscosities are assumed finite but large, compared with the viscosity of the surrounding medium. A small degree of slippage is also allowed at the sphere surfaces. An asymptotic solution of the problem is constructed, applicable to the case when the gap between the sphere surfaces is small. In particular cases when slippage or internal circulation are absent, the solution agrees with the results of /1,2/.

Slow motion of two droplets and a droplet towards a fluid or solid interface

Starting from the general formula of Haber et al. (1973) for the drag force arising when two drops are approaching each other along their line of centers in an unbounded liquid, asymptotic expressions for great and small separations are derived. In the former case a generalization of the well-known formulae of Rybczynski (1911), Hadamard (1911) and Lorentz (1907) is obtained. In the latter, new approximate formulae, valid for great and small drop viscosity are derived. The approach of a liquid drop toward a solid plane boundary is considered as well.

The transverse force on a drop in an unbounded parabolic flow

The steady flow in and around a deformable liquid sphere moving in an unbounded viscous parabolic flow and subject to an external body force is calculated for small values of the ratio of the Weber number to the Reynolds number in the creeping-flow regime. It is found that, in addition to the drag force, the drop experiences a force orthogonal to the undisturbed flow direction. When the body force is absent (neutrally buoyant drop), this lift force tends to drive the drop inwards to the axis, where the undisturbed flow velocity is maximum, i.e., towards a position of lower velocity gradient. In the case for which the parabolic flow profile is a Poiseuille flow profile, the lift force is given by the expression. \[ {\bf F}_1 =-6\pi\mu\epsilon U_0\frac{\alpha +\frac{2}{3}}{\alpha + 1}\bigg(\frac{a}{R_0}\bigg)^4{\bf b}F[1+o(\epsilon)]. \] Here a is the radius of the undeformed sphere, R0 is the radial distance from the position of maximum undisturbed flow U0 at the profile axis to the position of zero flow, ε is the ratio of the Weber number to the Reynolds number, given by ε=μU0T−1, where μ is the external fluid viscosity and T is the surface tension of the drop, α is the ratio of the drop and external fluid viscosities, b is the radial vector from the flow axis to the centre of mass of the drop, and F is a function of α and a dimensionless parameter dependent on the body force that is determined in the analysis. Reasonable agreement is found between the observations by Goldsmith &amp; Mason (1962) of the axial drift of liquid drops in Poiseuille flow and the predictions of the theory herein.

The Bursting of Pointed Drops in Slow Viscous Flow

Viscous drops, confined by the slow axisymmetric straining motion of a viscous fluid, are considered when the surface tension is weak. The shape of the drops is determined using slender-body theory, and it is found that steady solutions only exist for sufficiently small drop viscosities. Nonuniqueness exists, with bifurcation from a simple quadratic solution. At high drop viscosities, when there are no steady solutions, a description of the unsteady elongation of shape-preserving drops is obtained. This is the bursting phenomenon described experimentally by Taylor [1].

The flow of suspensions through tubes VII. Rotation and deformation of particles in pulsatile flow

A study of the translation and rotation of rigid spheres, rods, and discs, as well as the deformation of fluid drops in pulsatile and oscillatory flows, was made. Except when within one particle diameter of the wall, the axial displacements of small spheres, rods, and discs in oscillatory flow were in agreement with the theory of Womersley, and in pulsatile flow could be predicted by the superposition of the oscillatory on the steady flow. The rotations of the cylinders were in accord with theory based on slow steady flow when the fluctuating velocity gradient of the undisturbed field was used. When the particles were very close to the wall, however, both translational and rotational slip were observed, the degree of slip being similar to that found in steady flow. The oscillating deformation and orientation of liquid drops of low viscosity were found to be in agreement with calculations based on the theory for steady flow. When the drop viscosity became appreciable, however, the deformation lagged the velocity gradient owing to the dissipation of energy in extensional viscous flow within the drop.

Particle size distribution in dilute liquid/liquid dispersions by light scattering

Abstract Equal energy per unit mass (or volume) proved to be a valid scaling criterion for particle size distribution as well as mean particle size for geometrically similar vessels with a volume range of 4 to 1. Binodal particle size distributions were observed. Although the coarse fraction varied as the −6/5 power of the agitation rate, the tine size was relatively independent of agitation rate. The system with the greater drop viscosity gave a smaller fine‐fraction and a larger coarse‐fraction than the other system (IB/W). Both systems approached equilibrium size slowly (many minutes). Equilibrium was approached more rapidly at high agitation rates. Differences were small between different tank sizes at equal energy dissipation rates per unit mass. The average size of the fine fraction remains about the same through the run (although the amount increases). The size of the coarse fraction decreases with time. More experiments are necessary to elucidate the mechanism behind this.

Mechanism and Speed of Breakup of Drops

A mathematical analysis has been made of the breakup of liquid drops in an air stream; only the mechanism in which the drops flatten, become bowl-shaped, inflate like a parachute, and finally burst is considered. The analysis provides an understanding of this process of breakup and the conditions for which the viscosity and surface tension become important factors. An estimate of the breakup time for a wide range of conditions is obtained as a function of the drop diameter, surface tension, viscosity, drop density, air density, and velocity difference. The results are compared with the available experimental data.