Hydrodynamic interaction drops

Mixing and dispersion of viscous fluids—blending in the polymer processing literature—is the result of complex interaction between flow and events occurring at drop length-scales breakup, coalescence, and hydrodynamic interactions. Similarly, mixing and dispersion of powdered solids in viscous liquids is the result of complex interaction between flow and... 

A further option is to forget about simulating the flow and the processes in the whole vessel and to zoom into local processes by carrying out a DNS for a small box. The idea is to focus on the flow and transport phenomena within such a small box, such as mass transport and chemical reactions in or around a few eddies or bubbles, or the hydrodynamic interaction of a limited number of bubbles, drops, and particles including their readiness to collisions and coalescence. Examples of such detailed studies by means of DNS are due to Ten Cate et al. (2004) and Derksen (2006b). 

When two emulsion drops or foam bubbles approach each other, they hydrodynamically interact which generally results in the formation of a dimple [10,11]. After the dimple moves out, a thick lamella with parallel interfaces forms. If the continuous phase (i.e., the film phase) contains only surface active components at relatively low concentrations (not more than a few times their critical micellar concentration), the thick lamella thins on continually (see Fig. 6, left side). During continuous thinning, the film generally reaches a critical thickness where it either ruptures or black spots appear in it and then, by the expansion of these black spots, it transforms into a very thin film, which is either a common black (10-30 nm) or a Newton black film (5-10 nm). The thickness of the common black film depends on the capillary pressure and salt concentration [8]. This film drainage mechanism has been studied by several researchers [8,10-12] and it has been found that the classical DLVO theory of dispersion stability [13,14] can be qualitatively applied to it by taking into account the electrostatic, van der Waals and steric interactions between the film interfaces [8]. 

E. P. Ascoli, D. S. Dandy, and L. G. Leal, Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall, Int. J. Numer. Methods Fluids 9, 651-88 (1989) E. P. Ascoli, D. S. Dandy, and L. G. Leal, Buoyancy-driven motion of a deformable drop toward a planar wall at low Reynolds number, J. Fluid Mech. 213, 287-311 (1990). 

In the gravitational field, the steady-state velocities of particles and drops of various shapes (or mass) are different [179, 417]. Therefore, the distance between the centers of particles is not constant, and hence, the entire problem about the hydrodynamic interaction is, strictly speaking, nonsteady. It was shown in [417] that for Re l/a this problem can be treated as quasisteady. 

The results of numerous papers dealing with the hydrodynamic interaction of two drops were analyzed in [154, 183,517]. Some results of calculations for the drag force are given there (for various drop radii, values of drop viscosity, and various distances between drops). 

D.A. Sessoms, M. Belloul, W. Engl, M. Roche, L. Courbin, and P. Paitizza, Droplet motion in microfluidic networks Hydrodynamic interactions and pressure-drop measurements, Physical Review E, 80, (2009). 

Zinchenko A. Z., To calculation of hydrodynamical interaction of drops at small Reynolds numbers, Appl. Math. Mech., 1978, Vol. 42. No. 5. p. 955-959 (in Russian). 

The effect of hydrodynamic interaction of particles on the factor of mutual diffusion of particles has been studied in paper [22]. It formulated a theoretical basis for the determination of collision frequency of particles in a turbulent flow. The effect of internal viscosity of drops on their collision frequency was studied in [23-26]. It was shown that the correct accounting for hydrodynamic interaction ensures a good agreement between the theory and the experiment. 

A review of early works on hydrodynamic interaction between two solid spherical particles is contained in [13]. For the most part, these works focus on derivation of forces and torques acting on particles placed relatively far apart, forces and torques on a particle moving perpendicular or parallel to a flat surface, and forces and torques on a particle moving relative to another particle along, or perpendicular to, the line connecting the particle centers. The more general case of translation and rotation of two rigid particles was considered in [40, 41]. Hydrodynamic interaction of two drops with a proper account of the mobility of their surfaces and the internal circulation is analyzed in [39, 42-50]. 

Figure 13.2 shows the characteristic trajectories of motion of a small drop relative to the big one (k = 0.1) for two values of parameter of electro-hydrodynamic interaction Sj = 5 (full lines) and S = 0.1 (dashed lines). The trajectories close to critical are denoted as 1 and 2. The corresponding phase trajectories for S = 5 are shown in Fig. 13.3. 

Since in the case considered the direction Eq is parallel to g Oo = 0), the cross section of collision represents a circle of radius d. Shown in Fig. 13.4 is the dependence of dimensionless radius of collision cross section d/Ri on parameter of electro-hydrodynamic interaction S2 and the ratio of drop radiuses k = R2/R1. The collision cross section grows with k and S2. 

Consider the coalescence of drops with fiilly retarded (delayed) surfaces (which means they behave as rigid particles) in a developed turbulent flow of a lowconcentrated emulsion. We make the assumption that the size of drops is much smaller than the inner scale of turbulence R Ao), and that drops are non-deformed, and thus incapable of breakage. Under these conditions, and taking into account the hydrodynamic interaction of drops, the factor of mutual diffusion of drops is given by the expression (11.70). To determine the collision frequency of drops with radii Ri and Ri (Ri < Ri), it is necessary to solve the diffusion equation (11.36) with boundary conditions (11.39). Place the origin of a spherical system of coordinates (r, 0,0) into the center of the larger particle of radius i i. If interaction forces between drops are spherically symmetrical, Eq. (11.36) with boundary conditions (11.39) assumes the form... 

This expression was obtained by combining the far and near asymptotics of the force of hydrodynamic interaction between approaching drops that move along the line of centers (see Section 11.4). 

The principal shortcoming of the turbulent coagulation model offered by Levich [19] and rejected by many researchers is that it seriously overestimates the collision frequency of drops. Therefore the shear coagulation model [110] of particle coagulation in a turbulent flow has emerged as by far the most popular one. Since Levidfs model does not take into account the hydrodynamic interaction of particles, let us estimate the effect of this interaction on the collision frequency. 

Finally, consider the case of interacting drops having viscosities and //2 that are different from the viscosity of the ambient liquid. It should be treated in the same way as the previous case where the drops have equal internal viscosities, except that we must change the expression for the factor of hydrodynamic resistance h. To this end, consider two drops of types 1 and 2 which move with abso-... 

Let us find the collision frequency of conducting uncharged spherical drops in a turbulent fiow of a dielectric liquid in the presence of a uniform external electric field. Just as before, we assume a developed fiow, with drop sizes smaller than the inner scale of turbulence. We assume the drops to be undeformed, which is possible if the external electric field strength Eo does not exceed the critical value and the size of drops is sufficiently small. Under these conditions, the factor of mutual diffusion of drops of two types 1 and 2 with regard to hydrodynamic interaction is given by (13.86), while h and are given by the expressions (13.85) that apply to drops with a completely retarded surface. We must also take into account molecular and electric interaction forces acting on the drops. 

However, despite such effect, even sufficiently strong electric fields are not able to fully counterbalance the influence of hydrodynamic interaction between the approaching drops on their coalescence frequency in a turbulent flow. To estimate the influence of hydrodynamic interaction, consider the ratio of fluxes without and with such interaction, as we did in Section 13.7. 

To estimate colloidal forces acting between the droplets during the collision, x, z coordinates of the initial (x, Zj) and final (xf, Zf) positions of the mobile droplet before and after the collision are needed. When several pairs of x, z coordinates are plotted on a graph a speeifie seattering pattern will appear. This pattern can be analyzed by eompar-ing the experimental final positions with the ones ealeulated from a theory (2,3), which covers hydrodynamic interactions between the droplets and between the mobile droplet and the wall as well as all external forees aeting on the mobile drop, e.g., those described by DLVO theory. Thus, in the calculations, we assume the existenee of a certain force described by a certain function of the droplet-droplet separation. The final position of the droplet is then calculated and compared with the experimental results. The best match between experimental and theoretical final droplet positions yields the optimum set of parameters or the optimum force-distance profile. 

The Gibbs elasticity characterizes the lateral fluidity of the surfactant adsorption monolayer. For high values of the Gibbs elasticity the adsorption monolayer at a fluid interface behaves as tangentially immobile. Then, if two oil drops approach each other, the hydro-dynamic flow pattern, and the hydrodynamic interaction as well, is the same as if the drops were solid particles, with the only differenee that under some conditions they could deform in the zone of contact. For lower values of the Gibbs elastieity the... 

Marangoni effect appears, see Eq. (1), which can considerably affect the approach of the two drops. These aspects of the hydrodynamic interactions between emulsion drops are considered in Sec. IV. 

Note, however, that if the two drops are not quiescent, but instead approach each other, the critical distance is influenced by the hydrodynamic interactions — see the next section. 

First, we consider the hydrodynamic interactions between two emulsion drops, which remain spherical when the distance between them decreases (Sec. IV.A) this is the transition A—in Fig. 2. Second, we consider the thinning of the film formed between two emulsion drops (Sec. IV.B) this is stage D in Fig. 2. In both cases the effect of surfactant is taken into account and the critical distance (thickness) for drop coalescence is quantified. 

The solution to the problem of hydrodynamie interaction between two rigid spherieal partieles, approaehing each other aeross a viseous fluid, was obtained by Taylor (122). Two spherieal emulsion drops of tangentially immobile sur-faees (due to the presenee of dense surfactant adsorption monolayers) are hydrodynamically equivalent to the two rigid partieles eonsidered by Taylor. The hydrodynamic interaction is due to the dissipation of kinetic energy when the liquid is expelled from the gap between the two spheres. The resulting friction force decreases the velocity of the two spherical drops proportionally to the decrease in the surface-to-surface distance h in accordance with the Taylor (122) equation ... 

To illustrate the effeets of various factors on the velocity of approach of two deforming emulsion drops (Fig. 15a) we used the general expression from Ref 137 (the infinite series expansion) to calculate the mobility factor the results are shown in Figs 16 and 17. First of all, in Fig. 16 we illustrate the effects of bulk and surface diffusion. For that reason Qy = V-ylV- Q is plotted versus the parameter b, related to the bulk diffusivity, for various values of hjk, is related to the siuface diffusivity, see Eq. (54). If the hydrodynamic interaction were operative only in the film, then one would obtain V- IV- q > 1. However, all calculated values of are less than 0.51 (Fig. 16) this fact is ev-... 

For imaginary drops, which experience neither longrange stuface forces (Wfj = 0) nor hydrodynamic interactions (P = 1), Eq. (105) yields a collision efficiency Ej. = 1, and Eq. (98) reduces to the Smoluchowski (173, 174) expression for the rate constant offast irreversible coagulation. In this particular case, Eq. (96) represents an infinite set of nonlinear differential equations. If all flocculation rate constants are the same and equal to ay, the problem has an exact analytical solution (173, 174) ... 

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