Stress singularities in non-Newtonian stick-slip and edge flows

Abstract

The J-integral technique was originally developed as a method of looking at energy release rates in cracked solids. It seems adaptable to non-Newtonian fluids, and here the use of J-integral methods for inelastic stick-slip and edge flows is demonstrated. This enables one to predict the strength and form of the singularity from simple integrals over the main flow; this method has been used to correct the numerical estimate of singularity strength in Richardson's Newtonian analysis. The intensification of stress and/or velocity gradients due to viscosity variations is shown using a biviscosity model. Intensity factors connected to the easily computed J-integrals are found for power-law fluid singularities both for the stick-slip and edge flow cases. Consideration of various viscoelastic models in the stick-slip flow shows that (i) the second-order model leads to non-integrable wall forces; (ii) the singularity in the Oldroyd-B case cannot generally be Newtonian in form; (iii) the Phan-Thien-Tanner (PTT) model does not lead to a separable singularity of simple form; (iv) the Maxwell case is unresolved, but no flow may exist. Acceptable solutions for edge flows, by contrast, exist for all cases except the second-order model.