Capture of small drops by large drops in an electric field

In the investigation of the process of the breakdown of emulsions by the coalescence of drops, up to the present time, use has been made of the theory of the coagulation of colloids (for example, [1]). However, there is a considerable difference between colloids and emulsions. Forces of attraction, bringing about the coalescence of two colloidal particles, become predominant at distances much greater than the particle size, so that, in a hydrodynamic sense, it can be assumed that colloidal particles do not interact. On the contrary, the disperse phase in emulsions consists of molecularly smooth spherical drops with a size from a few tenths of a micron or more and, with approach of the drops, forces of hydrodynamic interaction, inhibiting coalescence, become substantial. As a rule, the drops can be regarded as rigid undeformed spheres since their surface is stabilized by surface-active substances. With the approach of such spheres, the layer of liquid between them generates a braking force proportional to the rate of approach of the drops and inversely proportional to the distance between their surfaces. As a result, the approach of drops under the action of a finite force takes place over an infinite time. It follows from this that the process of the coalescence of drops requires the presence of a force of attraction, rising to infinity with approach of the drops, and any theory of coalescence must take simultaneous account of the forces of attraction of the drops and of the hydrodynamic forces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–68, September–October, 1973.     

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.     

The flow of a dielectric liquid

The nature of the forces acting on a dielectric liquid in an electric field in the presence of a temperature gradient is investigated. It is shown that the electrical forces produced by nonuniformity in the electrical conductivity of the medium can give rise to a hydrodynamic flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 136–137, November–December, 1973.     

Nonmonodisperse breakup of capillary jets in a variable electric field

The electrical charging of capillary jets has a strong influence on their stability [1–10]. Well-known theoretical studies have been devoted to the linear [1–6], weakly linear [7], or finite-amplitude [10] stability of such jets in a constant electric field. In the present paper, an investigation is made in the framework of the full nonlinear equations. The main attention is devoted to effects associated with allowance for a time-variable electric field. It is shown that a sharp decrease of the surface charge may lead to an appreciable decrease in the size of the satellite droplets; allowance for the long-wavelength background also leads to a decrease in the size of the satellite droplets. In contrast, a sharp increase of the surface charge increases the relative contribution of the satellite droplets. At the same time, introduction of small-scale background perturbations can lead to a decrease in the contribution of the fine satellite droplets and to a weakening of their reaction to a rapidly increasing electric field. It is shown that the degree of monodisperseness can be increased by a relatively slowly varying electric field. An averaged effect of an electric field that varies rapidly in time is found. Appreciable increase of the initial perturbation amplitude in the case of a periodically varying electric field can lead to an appreciable increase in the degree of monodisperseness. The introduction of short-wavelength perturbations in a periodic electric field with large amplitude of the pulsations can lead to disappearance of the satellite droplets.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 55–62, March–April, 1991.     

Convective stability of a weakly conducting liquid in an electric field

In an inhomogeneously heated weakly conductive liquid (electrical conductivity 10^–12–1 cm^–1) located in a constant electric field a volume charge is induced because of thermal inhomogeneity of electrical conductivity and dielectric permittivity. The ponderomotive forces which develop set the liquid into intense motion [1–6]. However, under certain conditions equilibrium proves possible, and in that case the question of its stability may be considered. A theoretical analysis of liquid equilibrium stability in a planar horizontal condenser was performed in [2, 4]. Critical problem parameters were found for the case where Archimedean forces are absent [2]. Charge perturbation relaxation was considered instantaneous. It was shown that instability is of an oscillatory character. In [4] only heating from above was considered. Basic results were obtained in the limiting case of disappearingly small thermal diffusivity in the liquid (infinitely high Prandtl numbers). In the present study a more general formulation will be used to examine convective stability of equilibrium of a vertical liquid layer heated from above or below and located in an electric field. For the case of a layer with free thermally insulated boundaries, an exact solution is obtained. Values of critical Rayleigh number and neutral oscillation frequency for heating from above and below are found Neutral curves are constructed. It is demonstrated that with heating from below instability of both the oscillatory and monotonic types is possible, while with heating from above the instability has an oscillatory character. Values are found for the dimensionless field parameter at which the form of instability changes for heating from below and at which instability becomes possible for heating from above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 16–23, September–October, 1976.In conclusion, the author thanks E. M. Zhukhovitskii for this interest in the study and valuable advice.     

Experimental determination of the hydrodynamic moment associated with the nonsymmetric penetration of a disk into a compressible fluid

The nonsymmetric penetration of a disk (circular cylinder) into a compressible fluid is investigated. The results are obtained by physical modeling. A fluid with a low speed of sound (finely dispersed medium with gas bubbles, whose dimensionless equation of state coincides with the dimensionless equation of state of water [5]) was used as the working medium. The experiments were carried out at entry angles on the interval 54 < < 88, angles of attack on the interval –15 < < +15 and Mach numbers on the interval 0.002 M 0.2.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 88–94, September–October, 1990.     

Transmission coefficient of a grid electrode

The behavior of conducting emulsion drops (carrying a constant charge) in the region between electrodes connected to dc sources is studied. Assuming that the concentration of the dispersed phase is reasonably low, the problem may be reduced to one of determining the motion of an isolated drop close to the electrode. The trajectories of the drops in a flow passing around the electrode are then calculated, allowing for charge exchange between the drops and the electrode, and the electrode transmission coefficient is determined in relation to the parameters of the problem. An analogous situation was envisaged in earlier papers [1, 2] for a single cylindrical electrode but without allowing for the recharging of the particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 157–158, May–June, 1974.     

Crystallization dynamics of a liquid metal drop impinging onto a multilayered substrate

Thermal and hydrodynamic processes that occur during impingement of a liquid metal drop onto a multilayered substrate are numerically studied. The mathematical model is based on the Navier–Stokes equations for an incompressible liquid and on substrate and drop heattransfer equations that take into account the surfacetension forces and metal solidification. The effect of the impact velocity, initial drop diameter, metal overheating, and temperature and thermophysical characteristics of the substrate on the morphology of the solid drop, its height, contactspot diameter, and total solidification time was examined numerically. The simulation results are found to be in satisfactory agreement with experimental data.     

Magnetic hydrodynamics of two-dimensional flows in a plasma with an anisotropic pressure

The hydrodynamic equations of Chew, Goldberger, and Low [1] are used to analyze certain types of two-dimensional flows of a plasma with an anisotropic pressure (the pressure along the magnetic field p differs from the pressure across it p). In Sec. 1 the relationships derived in [2] for the transition of plasma state across surfaces of strong discontinuity are invoked to investigate the variation of the hydrodynamic parameters in weak shock waves in the linear approximation. The flow around bodies which only slightly perturb the main flow is investigated in Sec. 2 in the linear approximation. Similar problems for the case of an isotropic pressure are studied in detail in [3–5], for example.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–10, November–December, 1970.     

Evolution of energy spectrum of an eddy system

The turbulent flow of an ideal fluid is represented by a discrete model, in which the turbulence is concentrated at individual points and the cloud of these point eddies determines the turbulent flow. In the given mathematical model of turbulence (in contrast to other models of finite dimensions), the accurate hydrodynamic equation is not replaced by a finite system of coupled ordinary differential equations, and singular initial data are used. Advantages of such an approach are that the interactions between the point eddies are obvious, and their dynamics can be described by comparatively simple equations. The present work is a numerical investigation of the interactions between 100 identical point eddies which are initially distributed uniformly over a circle. This model is used to trace the tendency to directional energy transfer from small scale to large scale in two-dimensional turbulence. This phenomenon, which has been shown to be possible in several theoretical works and numerical investigations [1–3], may be attributed to the static irreversibility of turbulence [4]. A similar effect can be observed directly in the earth's atmosphere, in that the general circulation of the atmosphere is fed by cyclonic energy; this is referred to as negative viscosity [5]. A spectral analysis is carried out for such a system of point eddies, using a new procedure based on the formula obtained in [6]. In the evolution of the spectrum, sections close to the –5/3 law and the –3 law are obtained. Secondary instability of the eddy system is noted. At the end of the numerical experiment, a quasisteady state is established in the system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 43–48, November–December, 1976.In conclusion, we thank E. A. Novikov for formulating the problem and for his constant interest in the work.     

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.     

Small perturbation spectrum for plane-parallel flows of viscoelastic fluids

The investigation of the occurrence of a transition from the laminar to the turbulent flow regime in weak polymer solutions is of great practical interest. Experimental data indicate both an increase in flow stability and an occurrence of early turbulence [1]. Paper [2] explains the discrepancy in the experimental data for the numerical investigation of the first-mode symmetric perturbations, which are unstable for a Newtonian fluid. Paper [3] shows that other modes also become unstable in the case of the flow of a viscoelastic Maxwellian fluid in a channel. These features of the hydrodynamic stability of viscoelastic fluids indicate a significant rearrangement of the small perturbation spectrum. In the present paper, the perturbation spectrum for plane-parallel flows of viscoelastic Oldroyd and Maxwellian fluids is investigated at small Reynolds numbers, and at large and small wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 164–167, November–December, 1984.     

Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

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.     

Numerical Study of Hydrodynamics and Heat and Mass Transfer of a Ducted Gas–Vapor‐Droplet Flow

A computational model has been developed to predict heat and mass transfer and hydrodynamic characteristics of a turbulent gas–vapor–droplet flow. Turbulent characteristics of the gas phase are computed using the k– model of turbulence. It is shown that, with increasing inlet droplet diameter, the rate of heat transfer between the duct surface and the vapor–gas mixture decreases appreciably, whereas the wall friction increases only insignificantly. The predicted values agree fairly well with available experimental and numerical data     

Development of the region of a turbulized liquid in a stratified medium

The article discusses the plane unsteady-state problem of the development of a region of turbulent pulsations in an incompressible stratified liquid. At the initial moment of time, the energy of the turbulence is given inside a region of finite dimensions. A semiempirical system of equations describing this process is proposed. The article gives the data from numerical calculations, illustrating the original expansion of the region as a result of turbulent diffusion, its subsequent compression along a vertical (collapse) under the action of the forces of buoyancy, and the internal waves generated by the collapse.The work was reported at the International Symposium on Stratified Flows (Novosibirsk, August 29–31, 1972).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 45–52, May–June, 1974.     

Calculation of turbulent viscosity

Within the widely popularized statistical model of turbulence, in which the source of the turbulent energy is an external random force such as the Gaussian white noise, a computation is made of the effective viscosity characterizing the response of the turbulent fluid to the external perturbation. The method used in order to calculate the effective viscosity is that of the renormalization group, which makes it possible, by starting from the lower approximation in the theory of perturbations, to find the sum of a certain infinite subsequence of the series in the theory of perturbations. The expression obtained for the effective viscosity coincides in the inertial interval with the Kolmogorov power law, and gives a dependence different from the power dependence in the region of incomplete self-similarity. The main result of the study is to illustrate the possibility of describing the multimode cascade processes characteristic of developed hydrodynamic turbulence within the theory of perturbations, improved by means of the method of the renormalization group.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 29–36, July–August, 1987.The author wishes to express his gratitude to S. S. Moiseev, V. I. Tatarskii, and A. M. Yaglom for discussions and critical observations.     

Electrohydrodynamic flow in a two-dimensional channel with an axially disposed electrode-emitter

A study is made of the flow of an incompressible nonviscous unipolar charged liquid in a two-dimensional channel ¦ x ¦ <, ¦ y ¦ h with conducting walls and an axially disposed electrode-emitter (along y = 0). The charged particles have an arbitrary constant mobility. The charge distribution on the emitter is approximated as a unit step function. The problem is solved by linearizing the equations with respect to the electrohydrodynamic interaction. The behavior of the electrical parameters is determined, and the deformed profiles of velocity and pressure downstream of the zone in which the electrostatic forces are rotational in character are calculated. These profiles can be determined without having to solve the linearized partial differential hydrodynamic equations in the entire region occupied by the flow, although the profiles then depend on the distribution of the electrical parameters along the entire length of the channel.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 35–40, November–December, 1970.In conclusion the author wishes to thank A. B. Vatazhin for suggesting the problem and for his constant attention throughout the work.     

Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, (where λ is the characteristic relaxation time of polymer, and is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.     

Flows in the initial sections of channels with permeable walls

Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h<30 and L/a< 50 (2h is the distance between permeable walls of a flat channel anda is the tube radius) the velocity distributions in the laminar and turbulent flow regimes differ little and are nearly self-similar for solutions obtained in [4]. For sufficiently large Reynolds numbers, Re₀>100 (Re₀=v₀h/ or Re₀=v₀ a/, where v₀ is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(/2)x× cos (y/2) in a flat channel and u=x cos (y²/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u₀ a/<3.10³ fluid injection with v₀/u₀>0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v₀/u₀>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 43–48, September–October, 1976.