Deformation and breakup of a non-Newtonian slender drop in an extensional flow: inertial effects and stability

Abstract

We consider the deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid, subject to an axisymmetric extensional flow, and the influence of inertia in the continuous phase. The non-Newtonian fluid inside the drop is described by the simple power-law model and the unsteady deformation of the drop is represented by a single partial differential equation. The steady-state problem is governed by four parameters: the capillary number; the viscosity ratio; the external Reynolds number; and the exponent characterizing the power-law model for the non-Newtonian drop. For Newtonian drops, as inertia increases, drop breakup is facilitated. However, for shear thinning drops, the influence of increasing inertia results first in preventing and then in facilitating drop breakup. Multiple stationary solutions were also found and a stability analysis has been performed in order to distinguish between stable and unstable stationary states.

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

“…An interesting result from the theory is the existence of multiple (stable and unstable) stationary solutions. Recently, Favelukis et al 15 reported a stability analysis to small disturbances that confirmed and expanded Acrivos and Lo 6 inertia effect findings. This linear stability analysis indicates that the drop shape may depart from the steady, yet unstable, state but it cannot predict the evolution process as time progresses.…”

“…Favelukis et al 15 suggested the use of following rescaled variables: y = RCa and x = z/Ca 2 , having the order of magnitude of 1, that reduce the governing equations to the form . Recall that, in order to obtain a slender drop (R/L 1) the following conditions must be met: Ca 3 1, λ 1/2 1, λCa 3 1, and ReCa 1.…”

“…Finally, following Favelukis et al, 15 we define a dimensionless parameter h, characterizing the relative strength of inertia,…”

“…The structure and stability of the various stationary states is shortly reviewed and the dynamics of shape evolution emerging from these states is followed. In all the runs and figures the initial shapes of all evolution patterns are calculated using the previous theory (Favelukis et al 13,15 ).…”

“…(8) have been obtained by Acrivos and Lo. 6 The solution also admits a closed form (Favelukis et al 15 ). The deformation curve of x L vs. f 1 consists of multiple lobes and that each lobe contains two branches separated by a bifurcation turning point and argued that, for the first lobe, along the lower branch, the deformation is stable, while the upper branch is unstable.…”