On the evolution and breakup of slender drops in an extensional flow

Abstract

The evolution of the shape of an elongated drop embedded in an extensional flow is studied in the framework of slender body theory. The external flow has a weak but not neglected inertia. The problem is governed by three dimensionless parameters: the capillary number, the external Reynolds number, and the viscosity ratio between the drop and the external fluid, and exhibits a multiplicity of stationary shapes with only one being stable. Evolution of the drop surface from initial shapes was studied when the flow intensity was either kept constant or subjected to a sudden change. It was shown that the dynamics of the shape evolution can lead to a breakup of the drop or to a stable stationary shape. Two modes of breakup are revealed: an indefinite elongation and a center pinching mode. The former appears when the viscous forces dominate the inertia effects, with the typical case being that of a creeping flow. The latter breakup mode takes over in the presence of inertia when the drop viscosity diminishes with the extreme example being that of an inviscid drop.

Mass transfer around slender drops in an extensional flow: Inertial effects at large peclet numbers

“…Again, the shape of the slender drop is a parabola with pointed ends (Taylor, [4] Buckmaster, [6] and Acrivos and Lo [7] ): Figure 2 shows the deformation curve, which is composed of a lower branch where 2 n 2:4 and an upper branch in which 2:4 n < 1. Recently, Favelukis et al [10] studied the evolution of such a drop and found that the breakup mechanism of the viscous slender drop in creeping flow is by indefinite elongation. Acrivos and Lo [7] performed a stability analysis on small disturbances of the steady shapes and showed that the deformation of the drop along the lower branch is stable, while the deformation in the upper branch is unstable.…”

“…The solution of the problem contains several lobes depending on the values of parameter n. The first solution is obtained at 2 < n < 4, the next solution is 6 < n < 8, and so on. The recent evolution studies of Favelukis et al [10] predict that the breakup mechanism of an inviscid drop under a small amount of external inertia is by centre pinching. The lobe contains two branches separated by the bifurcation turning point in which f 1 is maximum.…”

“…Thus, the bifurcation turning point is considered as the breakup point. Recently, Favelukis et al [10] studied the evolution of such a drop and found that the breakup mechanism of the viscous slender drop in creeping flow is by indefinite elongation.…”

“…The numerical results at the breakup point are: n ¼ 2:54, L Ã ¼ 30:0Ca 2 , and f 1 ¼ 0:207. The recent evolution studies of Favelukis et al [10] predict that the breakup mechanism of an inviscid drop under a small amount of external inertia is by centre pinching. A simple approximated analytical solution to this problem is also described in Favelukis et al [9] By considering the first two terms of Equation (16), that is, a parabolic shape, we get: The deformation curve of this approximated solution, also plotted in Figure 3 (dashed line), shows very close agreement with the exact solution, up to the breakup point.…”

“…Their results predict that external inertia facilitates the breakup of the slender drop. An extension to Acrivos and Lo, [7] external inertial effects together with a stability analysis was performed by Favelukis et al [9] Recently, the evolution and breakup of a slender drop in axisymmetric extensional flow was treated by Favelukis et al [10] Two modes of breakup were found: indefinite elongation and a centre pinching mode, depending on the relative amount of external inertia. Effects of both internal and external inertia were studied by Brady and Acrivos.…”