Drop breakup deformation

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

Bubble and drop breakup is mainly due to shearing in turbulent eddies or in velocity gradients close to the walls. Figure 15.11 shows the breakup of a bubble, and Figure 15.12 shows the breakup of a drop in turbulent flow. The mechanism for breakup in these small surface-tension-dominated fluid particles is initially very similar. They are deformed until the aspect ratio is about 3. The turbulent fluctuations in the flow affect the particles, and at some point one end becomes... 

Fig. 16. Drop breakup in the journal bearing flow. The drop initially in the chaotic region of the flow deforms into a thin filament that breaks to produce a fine dispersion of droplets. The drop initially in the regular region of the flow (island) remains undeformed (Tjahjadi and Ottino, 1991).

Breakup by end-pinching occurs when a drop is deformed at Ca close to Cacrit, and the flow is stopped abruptly. 

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

Although the dominant mixing mechanism of an immiscible liquid polymeric system appears to be stretching the dispersed phase into filament and then form droplets by filament breakup, individual small droplet may also break up at Ca 3> Ca. A detailed review of this mechanism is given by Janssen (34). The deformation of a spherical liquid droplet in a homogeneous flow held of another liquid was studied in the classic work of G. I. Taylor (35), who showed that for simple shear flow, a case in which interfacial tension dominates, the drop would deform into a spheroid with its major axis at an angle of 45° to the how, whereas for the viscosity-dominated case, it would deform into a spheroid with its major axis approaching the direction of how (36). Taylor expressed the deformation D as follows... 

The most efficient mechanism of drop breakup involves its deformation into a fiber followed by the thread disintegration under the influence of capillary forces. Fibrillation occurs in both steady state shear and uniaxial extension. In shear (= rotation + extension) the process is less efficient and limited to low-X region, e.g. X < 2. In irrotatlonal uniaxial extension (in absence of the interphase slip) the phases codeform into threadlike structures. 

Figure 11 Phenomena connected with drop breakup (a) deformation (splitting) of droplets, (b) deformation into lenticular shape, (c) development and separation of the boundary layer, (d) velocity distribution near the separation point of the boundary layer, (e) Kelvin Helmholtz instability, and (f) Rayleigh Taylor instability, pi < P2-...

Also if t]c breakup is difficult. At t]D/t]c = 10 4, which is about the magnitude in most foams, Wecr would be as large as 30. At such a small viscosity ratio, the bubble or drop is deformed into a long thread before breaking. However, for some protein surfactants, the surface layer of the drop can be stagnant (Section 10.8.3) and then the drop can presumably break at a smaller Weber number. 

A recent analysis of 2- to 3-millimeter diameter, viscous and viscoelastic drop breakup under high accelerations at Mach 2 to 3 offered in [6] reveals the sequences of the breakup events including the deformation, bag and bag-stamen formation, and, finally, mist formation. Breakup at these large accelerations is attributed to the onset of the Rayleigh-Taylor instability. 

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 ... 

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

The remainder of the chapter focuses on the actual spray modeling. The exposition is primarily done for the RANS method, but with the indicated modifications, the methodology also applies to LES. The liquid phase is described by means of a probability density function (PDF). The various submodels needed to determine this PDF are derived from drop-drop and drop-gas interactions. These submodels include drop collisions, drop deformation, and drop breakup, as well as drop drag, drop evaporation, and chemical reactions. Also, the interaction between gas phase, liquid phase, turbulence, and chemistry is examined in some detail. Further, a discussion of the boundary conditions is given, in particular, a description of the wall functions used for the simulations of the boundary layers and the heat transfer between the gas and its confining walls. 

For K > 2 the drops deform into stable filaments, which only upon reduction of k disintegrate by the capillarity forces into mini-droplets. The deformation and breakup processes require time - in shear flows the reduced time to break is tb > 100- When values of the capillarity number and the reduced time are within the region of drop breakup, the mechanism of breakup depends on the viscosity ratio, A, - in shear flow, when X > 3.8, the drops may deform, but they cannot break. Dispersing in extensional flow field is not subjected to this limitation. Furthermore, for this deformation mode Kcr (being proportional to drop diameter) is significantly smaller than that in shear (Grace 1982). 

If the drops are such that the capillary number is smaller than its critical value, deformation will not lead to drop breakup and after the deformation has ended, the drop retracts (Fig. 5.4). 

In practice, in a mixture much larger drops can be found than predicted by the critical capillary number because Grace s observations were based on single drops. In actual systems, where many drops exist, coalescence will occur. Because material elements also undergo varying levels of shear forces in time, the mixing process in polymer systems can be considered as a complex interaction between deformation, drop breakup, coalescence, and retraction. 

At high snrfactant concentration, the resistance to deformation may be due solely to drop viscosity, and/or the ultimate size may be dictated by thermodynamic considerations. Koshy et al. (1988a) developed a model for drop breakup in the presence of snrfactants. Unfortunately, there are few experimental data to support its implementation. 

Marks, C. R. (1998). Drop breakup and deformation in sudden onset strong flows, Ph.D. dissertation. University of Maryland, College Park, MD. 

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