|Scaling in pinch-off of generalized Newtonian fluids
Doshi P (1), Suryo R (1), Yildirim OE (1), McKinley GH (2), Basaran OA (1)
(1) Purdue Univ, Sch Chem Engn, 1283 Chem Engn Bldg, W Lafayette, IN 47907 USA
(2) MIT, Dept Mech Engn, Cambridge, MA 02139 USA
Pinch-off dynamics of slender liquid bridges of generalized Newtonian fluids without and with inertia are studied using asymptotic analysis and numerical computation. The deformation-rate-dependent rheology is described by power law and Carreau models. Because the bridges are slender, their dynamics are governed by a pair of spatially one-dimensional (1D), non-linear evolution equations for the bridge shape and axial velocity. A bridge of a power law fluid under creeping flow conditions exhibits self-similar dynamics in the vicinity of the axial location where the bridge radius is a minimum. The scaling exponents that determine the variation with time remaining to breakup of the bridge radius or radial length scale, axial length scale, and axial velocity are evaluated by a combined analytical and numerical approach. Similarity solutions are obtained by collapsing numerically computed profiles of both the bridge shape and the axial velocity in the vicinity of the axial location where the bridge radius is minimum by rescaling of the transient profiles with radial and axial scalings deduced from theory. This scaling behavior is transitory and inertial effects become significant as pinch-off is approached. Thereafter, a new balance is established between viscous, capillary, and inertial forces that leads to a new self-similar regime which persists until pinch-off. The scaling exponents appropriate to this regime are also determined. Moreover, it is shown theoretically that interface shapes in the vicinity of the singularity are non-slender for values of the power law exponent below 2/3. Similarity solutions are once again obtained in the same manner as that used in the creeping flow limit. Low-viscosity bridges of Carreau fluids are known to exhibit a transition from potential flow (PF) scaling to Newtonian scaling. Here it is demonstrated that high-viscosity bridges of Carrean fluids exhibit a transition from power law scaling to Newtonian scaling. The point of transition between the latter two regimes is predicted in terms of parameters of the Carreau model.