Droplet breakup viscous forces

Breakup Criteria. Generally, droplet breakup in a flowing stream is governed by its surface tension and viscous forces, and dynamic pressure. For liquids of low viscosities, droplet breakup is primarily controlled by the aerodynamic force and surface tension force, and may begin when a critical condition, i.e., an equilibrium between these two forces, is attained ... 

This formula applies to planar extensional flow as well as to shear, if the shear rate y in Eq. (9-11) is replaced by 2e, where is the extension rate. Taylor predicted that droplet breakup should occur when the viscous stresses that deform the droplet overwhelm the surface tension forces that resist deformation this occurs when D reaches a value Db given approximately by... 

Effect of Flow Velocity. The flow velocity determines the shear rate and the pressure gradient. Therefore, the magnitude of a viscous force acting on a water droplet is directly related to flow velocity. This viscous force determines whether droplets can pass through pore throats smaller than themselves. It is also a factor in breakup of droplets into smaller droplets. 

The mechanisms governing deformation and breakup of drops in Newtonian liquid systems are relatively well understood. However, within the range of compounding and processing conditions the molten polymers are viscoelastic liquids. In these systems the shape of a droplet is determined not only by the dissipative (viscous) forces, but also by the pressure distribution around the droplet that originates from the elastic part of the stress tensor. Therefore, the characteristics of drop deformation and breakup in viscoelastic systems may be quite different from those in Newtonian ones. Some of the pertinent papers on the topic are listed in Table 9.3. 

In the spray process the particle size and morphology depends obviously on temperature and pressure and also on several operating parameto-s like jet breakup, mass transfer rates between droplets and antisolvent phase. The droplet sizes and the jet breakup depend on the balance between the different forces present in the system (inertial and external forces are against the existence of the droplet and viscous and int focial tension are fovorable to the droplet formation). These forces can be varied... 

The basic principle of droplet deformation is that if the viscous forces exceed the cohesive forces of surface tension, the droplet will break up. Prior to the breakup, the droplet deforms to a cylindrical elongated form. The deformation and breakup conditions for Newtonian liquids can be estimated by the Weber number (We) obtained from... 

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number... 

The reactor vessel is usually a stirred tank. The monomer phase is subjected either to turbulent pressure fluctuations or to viscous shear forces, which break it into small droplets that assume a spherical shape under the influence of interfacial tension. These droplets undergo constant collisions (collision rate >1 s ), with some of the collisions resulting in coalescence. Eventually, a dynamic equilibrium is established, leading to a stationary mean particle size. Individual drops do not retain their unique identity, but undergo continuous breakup and coalescence instead. In some cases, an appropriate dispersant can be used to induce the formation of a protective Aim on the droplet surface. As a result, pairs of clusters of drops that tend to coalesce are broken up by the action of the stirrer before the critical coalescence period elapses. A stable state is ultimately reached in which individual drops maintain their identities over prolonged periods of time [247]. 

Based on an analogy between the oscillations of a two-dimensional (2D) droplet and a mass spring system (similar to the Taylor analogy breakup (TAB) model), we assume that the deformation of our 2D liquid droplet is dependent on the viscous (Fv), surface tension (Fj), and inertial (Fa) forces. So, performing a force balance in the X2-direction for the half element (shaded) in Fig. 29.2c, we can write... 

A review of the past literature on the available correlations on the mean droplet size produced by splash plate nozzles shows that there are large discrepancies between the results. The prediction of the droplet sizes generated by splash plate nozzles is based on the Kelvin-Helmholtz (K-H) instability theory for a liquid sheet. Dombrowski and Johns [14], Dombrowski and Hooper [18] and Fraser et al. [13] developed such a theoretical model to predict droplet sizes from the breakup of a liquid sheet. They considered effects of liquid inertia, shear viscosity, surface tension and aerodynamic forces on the sheet breakup and ligament formation. Dombrowski and Johns [14] obtained the following equation for droplets produced by a viscous liquid sheet ... 

When immiscible fluid streams are contacted at the inlet section of a microchannel network, the ultimate flow regime depends on the geometry of the microchannel, the flow rates and instabilities that occur at the fluid-fluid interface. In microfluidic systems, flow instabilities provide a passive means for co-flowing fluid streams to increase the interfacial area between them and form, e.g. by an unstable fluid interface that disintegrates into droplets or bubbles. Because of the low Reynolds numbers involved, viscous instabilities are very important At very high flow rates, however, inertial forces become influential as well. In the following, we discuss different instabilities that either lead to drop/bubble breakup or at least deform an initially flat fluid-fluid interface. Many important phenomena relate to classical work on the stability of unbounded viscous flows (see e.g. the textbooks by Drazin and Reid[56]and Chandrasekhar [57]). We will see, however, that flow confinement provides a number of new effects that are not yet fully understood and remain active research topics. 

Surface tension-driven breakup into droplets is rarely important in melt spinning, where the large viscous and elastic forces overwhelm the surface tension forces, ft is an important mechanism in the formation of the dispersed phase in polymer blends, and it is important in solution processing. The surface tension-driven breakup of a viscoelastic filament has been analyzed using both thin filament equations and a transient finite element analysis, but we will not pursue the topic here because it is not relevant to our present discussion. 

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