On the interaction forces between evaporating drops in charged liquid-drop systems

The pairwise hydrodynamic and electrostatic interaction between micrometer-sized water droplets at small distances between them due to their evaporation and the presence of an electric charge on at least one of them is considered. The velocities of the steady-state motion of charged water drops with radii of 1 and 10 μm evaporating in air are calculated. It is shown that at small distances between the drops the joint action of hydrodynamic attraction and polarization interaction, always of attraction type, favor the coalescence of the drops (or drops and solid particles), leading to the displacement of the maximum of the function of drop distribution over size to the region of greater sizes and the gravity sedimentation of large drops. At large distances between the drops, when the short-distance hydrodynamic and polarization attractive forces become smaller than the long-distance Coulomb repulsion forces between likely charged particles, this distance tends to increase. These phenomena give a microphysical explanation to the phenomenon of electrostatic blooming in optically dense smokes and mists.     

Calculation of the rate of coalescence of an emulsion in a turbulent flow

The central moment of the theory describing the merging (coalescence) of the drops of an emulsion is determination of the time of the approach of a drop or a number of drops colliding with a given drop in unit time. In the stage immediately preceding the merging of the drops the forces of the hydrodynamic braking of the approaching drops are found to be considerable. The role of these forces has been analyzed earlier for the case of the capture of small drops by large drops in an oncoming flow in the presence of an external electrical field [1] and for the problem of the Brownian coalescence of drops, taking account of the effect of the electric double layer and of surface forces of interaction [2–4]. The present article considers the approach of drops with turbulent diffusion in an electrical field. Of the greatest interest is the sharp slowing of the approach due to the hydrodynamic interaction of the drops, considerably sharper than in the case of molecular diffusion [2]. As a result, the sharp acceleration of the approach and coalescence of drops with the action of an electrical field on an emulsion in a turbulent flow becomes understandable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 47–55, May–June, 1976.The authors are grateful to G. I. Barenblatt, A. I. Leonov, V. I. Loginov, and L. P. Smirnov for their evaluation and criticism of the work.     

Self-similar thinning of a liquid film separating liquid phases or a liquid and a solid phase

The article discusses the thinning of a thin flat film of liquid between coalescing bubbles or coalescing drops due to motion of the liquid under the action of capillary forces, under drag conditions of the surfaces of the film. A study was also made of the outflow of a liquid from a film separating a bubble (drop) from a solid surface. The problem of the decrease in thickness and subsequent breakaway of thin liquid films is of interest in investigations of the kinetics of coalescence and coagulation and in the theory of the stability of foams and lyophobic colloids in the presence of surfactants or electrolytes, as well as in the theory of heterogeneous boiling. Information on some work and on special characteristics of the statement of the problem may be found in [1].Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp, 39–48, March–April, 1973.     

Bimodal model of slurry viscosity with application to coal-slurries. Part 1. Theory and experiment

The rheological behavior of stable slurries is shown to be characterized by a bimodal model that represents a slurry as made up of a coarse fraction and a colloidal size fine fraction. According to the model, the two fractions behave independently of each other, and the non-Newtonian behavior of the viscosity is solely caused by the colloidal fraction, while the coarse fraction increases the viscosity level through hydrodynamic interactions. Data from experiments run with colloidal coal particles of about 2–3 µm average size dispersed in water show the viscosity of these colloidal suspensions to exhibit a highly shearrate-dependent behavior and, in the high shear limit, to match very closely the viscosity of suspensions of uniform size rigid spheres although the coal volume fraction must be determined semi-empirically. Different amounts of coarse coal particles are added to the colloidal suspension and the viscosity of the truly bimodal slurries measured as a function of shear rate. In agreement with the bimodal model, the measured shear viscosities show the coarse fraction to behave independently of the colloidal fraction and its contribution to the viscosity rise to be independent of the shear rate. It is shown that the shear rate exerted on the colloidal fraction is higher than that applied by the viscometer as a result of hydrodynamic interactions between the coarse particles, and that it is this effective higher shear rate which is necessary to apply in the correlations. For determining the coal volume fraction a relatively simple and quite accurate measurement technique is developed for determining the density and void fraction of coarse porous particles; the technique directly relates volume fraction to mass fraction.     

Interaction forces of particles dispersed in an electrolyte

Expressions are derived for the forces acting in a disperse medium in the presence of interaction of the double layers surrounding particles or drops of the dispersed phase when the potential of the dispersed particles is small. It is found that the force produced by the presence of double layers is proportional to the concentration gradient of the dispersed particles. It is shown that this force is comparable with the force produced by Brownian motion of the particles and may even exceed it. The equations of motion for the dispersed phase are derived with allowance for the convective terms, the pressure gradient, and the forces caused by Brownian motion and the presence of the double layers. A generalized Fick's law is obtained with effective diffusion coefficient. The equilibrium distribution of the particle concentration in a uniformly rotating cylinder is found with allowance for the interaction of the double layers.Translated from Izvestiya Akademii Nauk SSSR, Meklianika Zhidkosti i Gaza, No. 5, pp. 98–102, September–October, 1984.     

Pickering emulsions stabilized by colloidal surfactants: Role of solid particles

Pickering emulsions are emulsions stabilized by colloidal surfactants, i.e. solid particles. Compared with traditional molecular surfactant-stabilized emulsions, Pickering emulsions show many advantages, such as high resistance to coalescence, long-term stability, good biocompatibility and tunable properties. In recent years, Pickering emulsions are widely applied in scientific researches and industrial applications. In this review, we focus on the influences of particle properties on Pickering emulsions, including particle amphiphilicity, concentration, size and shape, and summarize the strategies developed for the preparation of amphiphilic Janus particles. The applications of Pickering emulsions in food industry, cosmetic industry, material science, drug delivery, biomedical research and vaccine adjuvant will also be covered. Pickering emulsions are a unique system for multi-disciplinary studies and will become more and more important in the future.     

Inertia and drag of cylinders oscillating in a fluid

The results of an experimental investigation of the hydrodynamic forces acting on a circular cylinder oscillating horizontally in water at rest parallel to the free surface are presented. The coefficients of the hydrodynamic forces — the inertia force proportional to the acceleration and the viscous drag proportional to the square of the velocity -are determined. The force coefficients are shown to depend significantly on the dimensionless (divided by the cylinder diameter) amplitude of the oscillations on the interval of variation from 0.5 to 10. In the experiments the maximum values of the Reynolds numbers, calculated from the maximum velocity and the cylinder diameter, were 2.10³–8.10⁴.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 111–115, July–August, 1989.     

DENSITY FUNCTIONAL FOR STRUCTURES OF COLLOIDS CONFINED IN A SLIT-LIKE PORE

A density functional theory is applied to calculating the local density profiles of colloids confined in a slit-like pore as well as the radial distribution functions of bulk colloids. The interaction between the colloidal particles is described using a hard-core Yukawa model. The excess Helmholtz energy functional is a combination of the modified fundamental measure theory of Yu and Wu （2002） for the hard-core contribution and a corrected mean-field theory for the attractive contribution. Comparison with the results from the Monte Carlo simulations shows that the corrected theory improves the density profiles of colloids in the vicinity of contact over the original mean-field theory. Both the present corrected theory and simulations suggest that there are depletion and desorption for the colloid with strong attraction between particles at low temperature.     

Coalescence of viscous drops translating through a capillary tube

An experimental study of the interaction and coalescence of viscous drops moving through a cylindrical capillary tube under low Reynolds number conditions is presented. The combined pressure- and buoyancy-driven motion of drops in a Newtonian continuous phase is examined. The interaction between two drops is quantified using image analysis, and measurements of the coalescence time are reported for various drop size ratios, Bond numbers, and viscosity ratios. The time scale for coalescence in the non-axisymmetric configuration is found to be substantially larger than that for coalescence in the axisymmetric configuration. Measurements of the radius of the liquid film formed between the two drops at the instant of apparent contact are used in conjunction with a planar film drainage model to predict the dependence of the coalescence time on drop size ratio for coalescence of low viscosity-ratio drops in the axisymmetric configuration.     

Influence of an acoustic wave on a system of two spheres in a viscous fluid

In this article we formulate and solve the problem of the influence of radiation forces (forces created by the radiation pressure) on two spheres in a viscous fluid during the transmission of an acoustic wave. On the basis of these forces we investigate the nature of the interaction between the spheres as determined by the mutual disturbance of the flow fields around them as a result of interference between the primary and secondary waves reflected from the spheres. A previously proposed [2] approach is used in the investigations. The radiation force acting on one of the spheres is filtered by averaging the convolution of the stress tensor in the fluid with the unit normal to the surface of the sphere over a time interval and over the surface of the sphere. The stresses in the fluid are represented, to within second-order quantities in the parameters of the wave field, in terms of the velocity potentials obtained from the solution of the linear problem of the diffraction of the primary wave by the free spheres. The diffraction problem is formulated and solved within the framework of the theory of linear viscoelastic solids [6]. The case of an ideal fluid has been studied previously [3–5, 7]. Radiation forces are one of the causes of the relative drift of solid particles situated in a fluid in an acoustic field.S. P. Timoshenko Institute of Mechanics, Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 33–40, February, 1994.     

Diffusional interaction of drops in a liquid

The method of combining asymptotic expansions (with respect to a large Peclet number) is used to investigate the three-dimensional problem of steady-state convective diffusion to the surface of drops, around which flows a laminar stream of a viscous incompressible liquid whose velocity field is assumed to be known from the solution of the corresponding hydrodynamic problem. It is shown that for large Peclet numbers the heat and mass transfer between drops is completely determined by the mutual arrangement of special (starting or ending at the surface of a drop) lines of flow; under these circumstances, in the flow there are chains of drops which have no mutual diffusional effect on one another, and the total diffusional flow to a drop is determined by diffusion to particles located upstream in the same chain. For the case where the distance between the drops in the chain is much leas than P^1/2 (P is the Peclet number), formulas for the distribution of the concentration and the total diffusional flow to the surface of each drop are obtained. It is shown that the total diffusional flow to the surface of a drop approaches zero in inverse proportion to its order number in a chain, which generalizes [1], in which the axisymmetric case is considered. A solution of the diffusional case is obtained for the case where there are critical lines at the surface of the drop. The problem is solved to the end if the singular flow lines are not closed and depart to infinity. With the presence of a region of closed circulation behind the drops, the problem is reduced to an integral equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika, Zhidkosti i Gaza, No. 2, pp. 44–56, March–April, 1978.The author thanks Yu. P. Gupalo and Yu. S. Ryazantsev for their interest in the work.     

Evaporation of spherical drops in a binary gas mixture at arbitrary Knudsen numbers

The quasisteady evaporation of drops in a binary gas mixture is investigated at arbitrary Knudsen numbers. The analysis is based on the solution of kinetic equations with collision integrals in the Boltzmann form by Lees's method [9]. The obtained solution makes it possible to consider an arbitrary model of intermolecular interactions. Formulas for the evaporation time of the drops are analyzed, the model of rigid elastic spheres being used for the interaction of the molecules.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–118, January–February, 1982.     

Theory of the combustion of small drops of metal

The article discusses the combustion of small drops of metal. It is postulated that the formation of an oxide in the liquid phase starts with the origin of a condensed phase and continues as the result of a reaction between the vapors of the drop and the oxidizer at the surface of the forming particles of the condensed phase. It is shown that the process of the formation of particles of condensed oxide in the gas, for very small drops, has an essentially unsteady-state character. Under these circumstances, a considerable fraction of the vaporization products of a drop does not succeed in condensing after the complete gasification of the drop and remains in the gaseous state.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 47–53, March–April, 1976.     

Rheology of oil-in-water emulsions

The effect of interfacial tension on the steady-flow and dynamic viscoelastic behavior of emulsions are studied experimentally. At very low inter-facial tensions and low volume fractions, the viscosity decreases with increasing shear rate and becomes constant at high shear rates. The high-shear-rate Newtonian viscosity is not affected by interfacial tension, but the transition from pseudoplastic to Newtonian flow shifts to lower shear rates as the interfacial tension decreases. At an interfacial tension of 5 × 10^–3 Nm^–1, the viscosity decreases, passes through a minimum, and then increases as the shear rate is increased. The dilatant behavior may be attributed to elastic responses of interfaces during collision of drops. At high volume fractions, the emulsions show remarkable elasticity resulting from the interfacial energy associated with deformation of liquid films. The modulus and viscosity are proportional to interfacial tension and inversely proportional to drop size.     

Shear thickening in electrically-stabilized colloidal suspensions

A theory is presented for the onset of shear thickening in colloidal suspensions of particles, stabilized by an electrostatic repulsion. Based on an activation model, a critical shear stress can be derived for the onset of shear thickening in dense suspensions for a constant potential and a constant charge approach of the spheres. Unlike previous models, the total interaction potential is taken into account (sum of attraction and repulsion). The critical shear stress is related to the maximum of the total interaction potential scaled by the free volume per particle. A comparison with experimental investigations shows the applicability of the theory.     

On an attempt to simplify the Quartapelle–Napolitano approach to computation of hydrodynamic forces in open flows

In this paper we are interested in the Quartapelle–Napolitano approach to calculation of forces in viscous incompressible flows in exterior domains. We study the possibility of deriving a simpler formulation of this approach which might lead to a more convenient expression for the hydrodynamic force, but conclude that such a simplification is, within the family of approaches considered, impossible. This shows that the original Quartapelle–Napolitano formula is in fact “optimal” within this class of approaches.     

Dissipative particle dynamics simulation of polymer drops in a periodic shear flow

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particle dynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE) bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles, respectively, perpendicular to Z-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions are implemented in the X and Y directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour” method is employed to model immiscible fluids according to Rothman–Keller method; the χ-parameters in Flory–Huggins-type models are also analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shear field, the relationship between the deformation parameter (D_def) and the capillary number (Ca) can be delineated into a linear and nonlinear regime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study, Ca<0.22, in the linear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescence may occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-size distribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease in the first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis shows that the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer drop under extension due to the memory of the FENE chains.     

Analysis of the nonequilibrium two-phase flow with coagulation and crushing of condensate particles for arbitrary mass and velocity distributions of the secondary drops

The problem of the nonequilibrium flow of a two-phase gas-polydisperse condensate drop mixture is considered in a one-dimensional approximation taking account of coagulation and crushing of particles of different size during collisions. A closed system of equations in the flow characteristics is obtained in application to the general case of arbitrary size, velocity, and temperature distributions of the secondary drops formed during crushing. Regularities in the interaction of freely moving drops during collisions are investigated experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–79, March–April, 1975.The authors are grateful to L. E. Sternin for attention to the research and to N. S. Kovalgina for assistance in carrying it out.     

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.     

Effect of drop size distribution on the flow behavior of oil-in-water emulsions

The effect of drop size distribution on the viscosity was experimentally examined for oil-in-water emulsions at volume fractions of = 0.5, 0.63 and 0.8. At = 0.5, the hydrodynamic forces during drop collisions govern the viscosity behavior. The viscosity versus shear rate curve is scaled on the root-mean-cube diameter which is related to the number of drops per unit volume. At = 0.8, the resistance to flow arises from the deformation and rearrangement of thin liquid films between drops. The viscosity at a given shear rate is inversely proportional to the volume-surface mean diameter which is related to the total interfacial area per unit volume. However, since the drops come into contact and the liquid film separating adjacent drops is generated without drop deformation at = 0.63, the viscosity curve is not scaled on the mean diameter. The flow behavior near the critical volume fraction strongly depends not only on the mean drop size, but also on the width of the distribution.