Breakup capillary instability

Breakup due to capillary instabilities dominates when the length of the filament is more than 15 times the initial radius of the drop. 

The number of satellite drops produced upon breakup by capillary instabilities decreases as p increases (minimum of 3 to maximum of 16). 

The interaction between the dispersed-phase elements at high volume fractions has an impact on breakup and aggregation, which is not well understood. For example, Elemans et al. (1997) found that when closely spaced stationary threads break by the growth of capillary instabilities, the disturbances on adjacent threads are half a wavelength out of phase (Fig. 43), and the rate of growth of the instability is smaller. Such interaction effects may have practical applications, for example, in the formation of monodisperse emulsions (Mason and Bibette, 1996). 

Linear stability theories have also been applied to analyses of liquid sheet breakup processes. The capillary instability of thin liquid sheets was first studied by Squire[258] who showed that instability and breakup of a liquid sheet are caused by the growth of sinuous waves, i.e., sideways deflections of the sheet centerline. For a low viscosity liquid sheet, Fraser et al.[73] derived an expression for the wavelength of the dominant unstable wave. A similar formulation was derived by Li[539] who considered both sinuous and varicose instabilities. Clark and DombrowskF540 and Reitz and Diwakar13161 formulated equations for liquid sheet breakup length. 

Frenkel, A. L. Babchin, A. J. Levich, B. Shlang, T. Sivashinsky, G. I. Annular Flows Can Keep Unstable Films from Breakup Nonlinear Saturation of Capillary Instability. J. Colloid Interface Sci. 1987.115,225. 

Abstract This chapter deals with capillary instability of straight free liquid jets moving in air. It begins with linear stability theory for small perturbations of Newtonian liquid jets and discusses the unstable modes, characteristic growth rates, temporal and spatial instabilities and their underlying physical mechanisms. The linear theory also provides an estimate of the main droplet size emerging from capillary breakup. Formation of satellite modes is treated in the framework of either asymptotic methods or direct numerical simulations. Then, such additional effects like thermocapiUarity, or swirl are taken into account. In addition, quasi-one-dimensional approach for description of capillary breakup is introduced and illustrated in detail for Newtonian and rheologically complex liquid jets (pseudoplastic, dilatant, and viscoelastic polymeric liquids). 

Figure 1.5 shows the growth rate of the capillary instability for different liquid viscosities. Viscosity dampens the instability with a damping coefficient of hpl lp and shifts the fastest growing perturbations toward longer waves. For p = 0, Rayleigh solution is obtained, whereas for very viscous jets with (3pk /2p) al2pa , a> = a/ pd) — k a ). The breakup length for a viscous jet is found as ...

Sheets produced in zone A (Re < 800) have relatively stable rim. The Reynolds numbers are low, therefore, viscous effects are dominate. Open rim sheets are typically broken at their open edges. As the Reynolds number is increased, the rims become unstable rapidly. Therefore, in zone B (800 < Re < 3,000), the sheet breakup process includes both a laminar capillary instability at the rims and Taylor instability on open edges of the sheet. Similar breakup process is observed in zone C, except that the whole flow is turbulent. Therefore, the breakup process in this zone is identified as turbulent rim instability combined with turbulent sheet instability. The data points in this zone are within 7,000 < Re < 18,000. For Re > 18,000 the rim cannot be distinguished, and the breakup process is mainly turbulent sheet breakup [26]. 

Twin-screw extruder In the melting section of CORI extruders, virtually all the degradation mechanisms that can essentially be distinguished, such as quasi-steady drop breakup, folding, end pinching, and decomposition through capillary instabilities, take place in parallel Polente et al. 2001... 

If the interfacial tension between the polymers is high, the lamellar structures may not form at all. In this case, the domains deform directly in the form of threads, which in turn undergo breakup by capillary instability [154—162]. 

The application of chaotic flows leads to much smaller droplets than allowed by the equilibrium between the shear and interfacial forces [204-206] and than those obtained in some commercial mixers. The smaller size of droplets can be directly traced back to their precursor thinner flbrils as the droplets originate from the fibrils by capillary instability and breakup. The exponential stretching encountered in chaotic mixers subdues the growth of interfacial instabilities in both lamellas and fibrils and consequently smaller diameter fibrils are produced. In some cases, fibrils with an aspect ratio as high as 1000 are produced [204]. In addition, the chaotic mixing conditions have been shown to slow down the rate of coalescence of droplets [207]. Another study utilized the rapid intermaterial area generation in chaotic flows to promote chemical reactions in the synthesis of thermoplastic polyurethanes [140]. 

Considerations and measurements of interfacial tension between polymer melts dates to the 1960s and 1970s [54 to 57]. Several different methods have been used to measure interfacial tension. Extensive use has been made of 1) the shape of drops emerging from a capillary into a second phase (falling drop) [56 to 60] and 2) thread breakage, the breakup of stationary hlaments in a second liquid phase by a capillary instability [59 to 61 ]. The latter analysis is based on the work of Tomotika [62]. Other methods have been used. 

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