Self-similar pinch-off of power law fluids

Abstract

Pinch-off dynamics of slender liquid threads of power law fluids without inertia are studied by asymptotic analysis. Because the threads are slender, their dynamics are governed by a pair of spatially one-dimensional, nonlinear evolution equations for the thread shape and axial velocity that results from a long-wave asymptotic expansion of the creeping flow equations. By means of an approach that differs from those used previously in analyses of capillary pinching of threads of Newtonian fluids, a similarity transformation is derived that reduces the evolution equations to two coupled similarity equations. As in the Newtonian case, it is shown that for each value of the power law exponent n where 0⩽n⩽1, there is a family of similarity solutions for capillary pinching of threads of power law fluids. For a given family of solutions, the radial and axial scales vary with time τ to pinch-off as τn and τδ, respectively, where δ is the axial scaling exponent. It is shown that for a given family of solutions characterized by a fixed value of n, each member of the family has a different scaling exponent δ. Since the viscosity of a power law fluid varies as |γ̇|(n−1), where γ̇ is the deformation rate, for each value of n a numerical method based on domain splicing is used to compute the values of the axial scaling exponent δ and the similarity solutions.