Motion of drops

A. N. Frumkin and V. G. Levich, Zh. Fiz. Khim. 21, 1183 (1947) (in Russian). This work, as well as related research on the motion of drops and bubbles in fluids, is summarized in the textbook (translated from the Russian) V G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1962). 

Some additional effects were considered for the thermocapillary motion of drops and bubbles in an external temperature gradient interaction of a drop with a plane wall [285], and interaction of drops with bubbles or of bubbles with each other [12, 146], In particular, it was shown in [12] that the interaction of drops of radius a decreases with increasing distance l between them as (a/i f for thermocapillary drift, compared with a/l for the motion in the gravitational field. 

M. Hemmat and A. Borhan, Buoyancy-Driven Motion of Drops and Bubbles in a Periodically Constricted Capillary, Chem. Eng. Commun., 150 (1996). 

The basic classical theories, such as the film, boundary layer,transient film, and penetration hypotheses are obviously outside the scope of this chapter, but the reader is assumed to be familiar with their basic concepts. Harriott s (H8) recent review on mass transfer to interfaces is recommended in this connection. An excellent treatise on the motion of drops and bubbles in fluid media is found in Levich s Physicochemical Hydrodynamics (L8, Ch. 8). 

Slow motion of drops subject to deformations presents rather difficult mathematical problems because it involves solving the Stokes equation for external and internal liquids taking into account kinematic and dynamic conditions at originally unknown mobile interface of the drop. The method of solution of such problems is based on integral representation [11]... 

In this coordinate system, the flow of the ambient liquid moves relative to the larger drop. On a large distance from the drop, the velocity can be assumed constant and equal to the sedimentation velocity of the drop. Another drop of smaller size moves relative to the larger drop together with the flow, goes around it, and either touches it or passes by. Due to their small sizes, the motion of drops can be... 

If drop sizes are small enough, and the difference in densities of the internal and external liquid is sufficiently small, the flow of the liquid can be considered as slow, and the motion of drops - as inertialess. This is the case when we separate emulsions of the w/o (water-oil) type. Then equations (11.32) are reduced to the equations of inertialess motion of drops in the quasi-stationary approximation... 

If the liquid flow and the relative motion of drops are axially symmetric, then 6 does not depend on , the shape of the collision cross-section is circular, and... 

In order to determine the collision frequency of drops, it is necessary to consider the relative motion of drops subject to interaction forces - mutual interaction of drops and interaction of drops with the ambient liquid. 

Because of the linearity of Stokes equations, the required expressions for Fg. and Ta, can be found by superposition and approximation of the known particular solutions. We can take the particular solutions for the motion of drop S2 relative to drop Si in a quiescent liquid, and in a flow with the velocity U at the infinity that goes around the two motionless drops S2 and SI separated by a fixed distance. Based on the above, the force Fu and the torque Tj. can be presented as... 

A decrease of the clearance d between drops Si and S2 results in the velocity figuring in the dependences (11.82). To take a proper account of the influence of distortion of the velocity field at small 3, one has to include the resistance factors into (11.82). It should be kept in mind that resistance factors have different forms for the motion of drop S2 parallel and perpendicular to the surface Si, therefore... 

Consider now the hydrodynamic force and torque caused by the proper motion of drop S2. This problem has been studied extensively. The available solutions en-... 

A detailed analysis of relative motion of drops along the line of centers for small clearances between drops is performed in [49]. The approximate expression for the resistance factor of drop S2 looks like... 

Consider first the case when the external electric field Eo is parallel to the direction of the gravity force, that is, to the vector g. Then the motion of drop S2 relative to drop Si can be considered as planar motion in a meridian plane spherical system of coordinates (r, 6, angle between vector Eo and the line of centers of drops coincides with the angle 0 of spherical system of coordinates. The expressions for electric forces acting on two conducting charged drops have been obtained received in Section 12.3. The components of these forces along axes r and 9 for each drop are... 

Taking various values Ao and Oq, it is possible to determine trajectories of motion of drop S2 relative to Si. During the motion, the drop S2 either collides with Si or passes by. Trajectories corresponding to these cases, form two families trajectories terminating at r = 1 2 + l i from the center of sphere Si (collision), and trajectories passing by Si and extended to infinity. These two families are separated by the limiting trajectory. 

The situation is more complicated when 0 < Oq < n/2, that is, when the electric field 0 makes an acute angle with vector U or g (Fig. 13.5). In this case the motion of drop 2 is not planar, its trajectories do not lie in a meridian plane, and it is necessary to consider 3-dimensional motion. To simplify calculations, we neglect the viscous resistance, taking S > 1. In this case the components of electric force are... 

The assumptions made allow us to consider the coalescence of drops with a mobile surface in the same manner as that of drops with a fully retarded surface. The main difference from the case considered in Section 13.6 is in the form of the hydrodynamic resistance factor. If the drops are placed far apart, the factor of hydrodynamic resistance for the relative motion of drops is determined by the formula (11.71), where each of the factors hi and hi is determined according to Hadamar-Rubczynskis formula... 

The coefficient of turbulent diffusion without consideration of the hindered motion of drops is given by [2] ... 

We shall restrict ourselves to considering the sedimentation of drops in a horizontal gravitational separator. If the size of drops does not change, and they are settling with Stokes velocity, then the transfer function is determined by the formula (18.15). Let the size of drops grow according to the law (18.53). Then the trajectories of averaged motion of drops are determined by the equation... 

Thus, at the angular velocity of the swirling flow ra < 50 c the equations of motion of drops of radius R <2 10 m (it is precisely such drops that reach the entrance of centrifugal branch pipes), look like... 

The actual value of capture eflEciency lies between these two numbers, and can be determined by considering the motion of drops as the flow goes around the cylinder. 

The data shown in Fig. 6.4-1 are valid for the motion of drops in a continuous hq-uid phase, i.e., in a spray colunm. Most extractors have internals (trays, packings)... 

A special case of Marangoni instability is its effect on the drag coefficient, that is the motion of drops. The presence of stationary instability immobilises the interface and thus increases the drag coefficient by elimination of internal circulation (Fig. 

A. Borhan, C.F. Mao, Effect of surfactants on the motion of drops through circular tubes. Physics of Fluids A, 1992, 4, 2628-2640. 

We shall now examine more quantitatively the motion of drops and puddles on horizontal surfaces bearing the imprint of a wettability gradient in the. r-direction. Both 7so( ) and depend on the position. x, and... 

FIGURE 10.7. Experimental setup for studying the motion of drops under the influence of a thermal gradient (courtesy F. Rondelez). 

Motion of Drops at Boundary of Hydrophilic-Hydrophobic Surface... 

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