Consistent formulation of the power-law rheology and its application to the spreading of non-Newtonian droplets

Abstract

In this work, we introduce a general form of the Navier-Stokes equations for Generalized Newtonian fluids with an Ostwald power-law. The derivation, based on the covariant formalism, is frame-independent and gives rise to a source term in the Navier-Stokes equations referred to as the Ostwald vector which is characterized by the power-law exponent. The governing equations are then simplified in the long-wave approximation framework and applied to the spreading of an axisymmetric gravity current in the creeping flow regime. Well-known spreading laws are recovered through similarity solutions and a new derivation based on scaling arguments is proposed. Experimental results related to the spreading of gravity current are then presented and the potential to infer unknown rheological parameters from spreading rates is critically discussed in the context of a thorough error analysis.