Deformed drop

A. Z. Zinchenko, R. H. Davis 2002, (Shear flow of highly concentrated emulsions of deformable drops by numerical simulations),/. Fluid Mech. 455, 21. 

Illustration Drop size distributions produced by chaotic flows. Affinely deformed drops generate long filaments with a stretching distribution based on the log-normal distribution. The amount of stretching (A) determines the radius of the filament locally as... 

As noted in Chapter 2, bubbles and drops remain nearly spherical at moderate Reynolds numbers (e.g., at Re = 500) if surface tension forces are sufficiently strong. For drops and bubbles rising or falling freely in systems of practical importance, significant deformations from the spherical occur for all Re > 600 (see Fig. 2.5). Hence the range of Re covered in this section, roughly 1 < Re < 600, is more restricted than that considered in Section II for solid spheres. Steady motion of deformed drops and bubbles at all Re is treated in Chapters 7 and 8. 

A good approximation to the shape of deformed drops in air may therefore be obtained from knowledge of the system properties and drop size. The ratios (bi + b2)/2a and fii/(fii + 2) e calculated from Eo using Eqs. (7-25) and (7-27). From geometric considerations... 

Figure 10.30 Deforming drop inside a rhomboidal mixing section.

As a conclusion, if the viscosity ratio p between the internal and external phases lies between 0.01 and 2, the shear applied on a polydisperse emulsion made of large drops leads to a monodisperse one with a mean diameter governed by the stress. This fragmentation occurs through elongation of the drops and the development of a Rayleigh instability with a characteristic time of the order of one second. The obtained monodispersity probably results from the fact that the Rayleigh instability develops under shear for a critical diameter of the deformed drops. 

Summarizing, it can be concluded that a relatively sharp (within 2-4% of deformation) drop in H is observed for copolymers of PBT but in comparison with homo-PBT this transition occurs at much higher deformations (between 25 and 30%). This difference as well as the following increase and decrease of H are related to the structural peculiarities of thermoplastic elastomers - the presence of a soft amorphous phase which first deforms and the existence of a physical network. The very low H values obtained for PEE are related to the fact that the PBT crystallites are floating in an amorphous matrix characterized by a low viscosity. 

Combining Eqs. (67)-(69) and (71), Ross and Curl obtained the coalescence efficiency for two deformable drops... 

In the case of deformed drops (R 0), the drainage time, x, is determined by Equation 5.273, and in such a case the fluid particles approach each other in the Reynolds regime." " The dependence of x on R in Equation 5.273 is very complex, because the driving force, F, and the film radius, R, depend on R. The film radius can be estimated from the balance of the driving and capillary force " " ... 

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). 

L. G. Leal, The slow motion of slender rod-like particles in a second-order fluid, J. Fluid Mech. 69, 305-37 (1975) B. P. Ho and L. G. Leal, Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid, J. Fluid Mech. 76, 783-99 (1976) P. C. H. Chan and L. G. Leal, The motion of a deformable drop in a second-order fluid, J. Fluid Mech. 92, 131-70 (1979) L. G. Leal, The motion of small particles in non-Newtonian fluids, J. Non-Newtonian Fluid Mech. 5, 33-78 (1979) R. J. Phillips, Dynamic simulation of hydro-dynamically interacting spheres in a quiescent second-order fluid, J. Fluid Mech. 315, 345-65 (1996). 

Problem 8-13. Lateral Migration of a Deformable Drop in 2D Poiseuille Flow. A viscous drop of viscosity fi and density p is carried along in the unidirectional motion of an incompressible, Newtonian fluid of viscosity p and density/) = p between two infinite plane walls. The radius of the undeformed drop is denoted as a, and the distance between the walls is d. We assume that the capillary number, Ca = apXl/rr, is small so that the drop deformation is also small. Here, a is the interfacial tension, and G is the mean shear rate of the undisturbed flow. 

Derive dimensionless equations and boundary conditions whose solution would be sufficient to determine the drop velocity (and shape) to 0(8). Use the method of domain perturbations to express all boundary conditions at the deformed drop interface in terms of equivalent conditions at the spherical surface of the undeformed drop. Show that 5 = Ca. 

The static mixer creates a hydrodynamic environment where critical viscous shear stresses deform drops, causing them to break. The dispersion process decreases the drop sizes and increases the... 

The spherical form of a drop or a bubble in Stokes flow follows from the fact that the flow is inertia-free. However, even for the case in which the inertia forces dominate viscous forces and the Reynolds number cannot be considered small, the drop remains undeformed if the inertia forces are small compared with the capillary forces. The ratio of inertial to capillary forces is measured by the Weber number We = p U a/a, where cr is the surface tension at the drop boundary. For small We, a deformable drop will conserve the spherical form. 

Many empirical relations for steady-state velocity of deformed drops and bubbles of various shapes, including shapes more complicated than the ellipsoidal shape, are presented in [94]. Laminar flow past nonspherical drops was studied numerically in [98, 517]. 

Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number was studied numerically in [55],... 

Under strong deformations, drops split into smaller ones, that is, are destroyed. The destruction process for drops is very complicated and is determined by surface tension, viscosity, inertia forces and some other factors. For various characteristic velocities of the relative phase motion, the character of destruction may be essentially different. A comparative analysis of many experimental and theoretical studies of drop destruction was given in [154, 312]. It was pointed out that there are six basic mechanisms of drop destruction, which correspond to different ranges of the Weber number. 

The problem on finite deformations arising in the motion of a solid sphere toward a free interface and in the motion of a deformed drop to the solid plane wall, which is vital for chemical industry, was studied numerically in [17, 517],... 

Bozzi, L. A., Feng, J. Q., Scott, T. C., and Pearlstein A. J., Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number, J. Fluid Mech., Vol. 336, pp. 1-32, 1997. 

Petrov, A. G., Circulation inside viscous deformed drops moving in gas with constant velocity, J. Appl. Mech. Techn. Phys., No. 6, 1989. 

To date, the code is capable of simulating a single, deforming drop, as shown in Figs. 5.4a and 5.46. The code will be expanded to provide the required capabilities for this project. The work will proceed through the following phases ... 

Figure 4.9. Schematic representation of the inteifacial tension measurements by the spinning drop method ffi is the angular velocity, and d the deformed drop diameter.

---