Drops shape/deformation

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

This latter case is the same result as Einstein calculated for the situation where slip occurred at the rigid particle-liquid interface. Cox15 has extended the analysis of drop shape and orientation to a wider range of conditions, but for typical colloidal systems the deformation remains small at shear rates normally accessible in the rheometer. The data shown in Figure 3.11 was calculated from Cox s analysis. His results have been confirmed by Torza et al.16 with optical measurements. The ratio of the viscous to interfacial tension forces, Rf, was given as ... 

Drops accelerated by an air stream may split, as described in Chapter 12. For drops which do not split, measured drag coefficients are larger than for rigid spheres under steady-state conditions (R2). The difference is probably associated more with shape deformations than with the history and added mass effects discussed above. For micron-size drops where there is no significant deformation, trajectories may be calculated using steady-state drag coefficients (SI). 

Figure 2-15. Photographs of the relaxation of a pair of initially deformed viscous drops back to a sphere under the action of surface tension. The characteristic time scale for this surface-tension-driven flow is tc = fiRi 1 + X)/y. The properties of the drop on the left-hand side are X = 0.19, /id = 5.5 Pa s, ji = 29.3 Pa s, y = 4.4 mN/rn, R = 187 /an, and this gives tc = 1.48 s. For the drop on the right-hand side, X = 6.8, lid = 199 Pa s, //. = 29.3 Pa s, y = 4.96 mN/m, R = 217 /an, and tc = 9.99 s. The photos were taken at the times shown in the figure. When compared with the characteristic time scales these correspond to exactly equal dimensionless times (/ = t/tc) (a) t = 0.0, (b) t = 0.36, (c) t = 0.9, (d) t = 1.85, (e) t = 6.5. It will be noted that the drop shapes are virtually identical when compared at the same characteristic times. This is a first illustration of the principle of dynamic similarity, which will be discussed at length in subsequent chapters.

Now, the drop shape will be non-spherical if this is necessary to satisfy the boundary conditions at its surface, and, specifically, to satisfy the normal-stress balance, (2 135). To determine the condition that leads to small deformations, we can nondimensionalize the boundary conditions using the same characteristic scales that were used for the governing equations, (7 198) and (7-199). The result for the normal-stress balance is... 

In general, the problem just defined is nonlinear, in spite of the fact that the governing, creeping-flow equations are linear. This is because the drop shape is unknown and dependent on the pressure and stresses, which in turn, depend on the flow. Thus n and F are also unknown functions of the flow field, and the boundary conditions (2-112), (2-122), (2-141), and (8-58) are therefore nonlinear. Thus, for arbitrary Ca, for which the deformation may be quite significant, the problem can be solved only numerically. Later in this chapter, we briefly discuss a method, known as the boundary Integral method, that may be used to carry out such numerical calculations. Here, however, we consider the limiting case... 

The value of the boundary-integral method is particularly evident if we consider problems in which one or more of the boundaries is a fluid interface. Here, for simplicity, we consider the generic problem of a drop in an unbounded fluid that is undergoing some mean motion that causes the drop to deform in shape. This type of problem is particularly difficult because the shape of the interface is unknown and is often changing with time. We shall see that the boundary-integral formulation provides a powerful basis to attack this class of problems, and in fact, is largely responsible for much of the considerable theoretical progress that has... 

Calculations for larger drops are complicated by phenomena such as shape deformation, wake oscillations, and eddy shedding, making theoretical estimates of E difficult. The overall process of rain formation is further complicated by the fact that drops on collision trajectories may not coalesce but bounce off each other. The principal barrier to coalescence is the cushion of air between the two drops that must be drained before they can come into contact. An empirical coalescence efficiency Ec suggested by Whelpdale and List (1971) to address droplet bounce-off is... 

Consider the motion of a drop in an electrolyte solution under the action of applied electric field [52]. Suppose that the thickness of the electric double layer is small in comparison with the drop radius Xp a), and that the drop is ideally polarizable, i.e. no discharge or formation of ions occurs at the drop surface and no current flow through the drop. It is also assumed that the drop has spherical shape. It win be shown in Part V that the drop may deform under the action of externa] electric field, extending parallel to the field strength vector and assuming the shape of on ellipsoid. The spherical shape assumption is valid if the strength of external electric field does not surpass some critical value. Electric potential is described by the Poisson equation (9.24)... 

Suppose that the drop is deformed slighdy. Then the shape of the drop can be approximated by a sphere. Laplace s equation in a spherical coordinate system under the spherical symmetry condition is written as... 

Even further increases in the Weber number can lead to a more pronounced deformation in the drop. This is shown in Fig. 4.7 for We = 100. For this case, there is no returning point and the drop continuously deforms and spreads out into a sheet-like shape. A drop with this much deformation ultimately breaks up into smaller pieces by either particles getting pinched off its tip or the whole flat drop breaking into several pieces. Panel (a) in Fig. 4.7 clearly shows how the enhanced vortex in the leeward side of the jet helps stretching the drop into a thin shape which in turn expands the vortex itself. As expected, the continuous deformation leads to a smooth increase in the drag coefficient. 

Kelbaliyev, G. and Ceylan, K., Development of new empirical equations for estimation of drag coefficient, shape deformation, and rising velocity of gas bubbles ot liquid drops. Chem. Eng. Commun. 194, 1623-1637, 2007. 

If the droplet is rotating while oscillating, its oscillation frequency is modified. Busse [2] considered this problem and extended Rayleigh solution to rotating flows. He assumed that rotation-induced shape deformation remained small and axisym-metric. His results for the shift in frequency Aco of axisymmetric oscillation of a liquid drop with angular frequency of rotation of Q are ... 

Ibrahim et al. [12] proposed the Droplet Deformation Breakup (DDB) model, which is based on the drop s dynamics in terms of the motion of the center-of-mass of the half-droplet. It is assumed that the liquid drop is deformed due to a pure extensional flow from an initial spherical shape of radius r into an oblate spheroid having an ellipsoidal cross-section with major semi-axis a and minor semi-axis b. The internal energy of the half-drop comes from the sum of its kinetic and potential energies, E, expressed as follows ... 

The electrical potential causes the deformation of the fluid drop, and when the applied voltage develops enough force and balances with the fluid surface tension of the polymer solution, the drop is deformed under a cone shape with a semivertical angle of 30°. Beyond this critical value (Rayleigh limit], the electrostatic forces generated by the charge carriers overcome the surface tension and the deformed droplet undergoes a transition zone just before the fiber jet is initiated to the collector screen. By this way, fluid is... 

The surface properties of polymers are important in technology of plastics, coatings, textiles, films, and adhesives through their role in processes of wetting, adsorption, and adhesion. We will discuss only surface tensions of polymer melts that can be measured directly by reversible deformation or can be inferred from drop shapes. Those inferred from contact angles of liquids on solid polymers ( critical surface tension of wetting ) are excluded, as their relations to surface tensions are uncertain. 

The film stability was measured either directly in the capillaries, by observing the time needed for the breakdown of (oil or water) film, or in separate measurements carried out on glass plates of the same material. The latter method which is more accurate, consists of determining the time that elapses from placing water (or solution) drops of 100 mm volume on the surface until a sudden deformation of the drop shape takes place. This time is indicative of the rupture of the oil film due to gravity. 

An example of these simulations is illustrated in Figure 4.26a, which gives two different views of an oil drop trapped in a Plateau border for which the ratio of oil-water to air-water surface tensions is a realistic 0.1 and the ratio of the spherical equivalent drop radius to the reciprocal Plateau border curvature is unity. The equivalent radius of the drop is therefore almost an order of magnitude greater than the minimum radius We see from the figure that the drop is indeed markedly deformed as a result of the high capillary pressure due in turn to the relatively high air-water surface tension. The drop shape in cross-section appears to closely conform to the... 

In SSF or Couette flow, it is not possible to break a drop if the viscosity ratio is greater than about 3. The deformed drop shape is stabilized by internal circulation. It has been concluded that extension is more effective than shear at breaking drops. With respect to practical flows, it is important not to interpret this statement too literally since in practical applications, a single steady shear gradient rarely exists. Bear in mind that the physical definitions of shear and extension depend on the environment seen by the drop along its trajectory, while the mathematical definitions are related to the choice of coordinate system. 

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