New Transient Algorithms for Non-Newtonian Flows

Abstract : A fully dynamic method for shear flows is presented that treats the short time-scales associated with Newtonian viscosity (or short relaxation processes) and shear-wave propagation implicitly, while treating the long relaxation processes explicitly. The method is generalized to flows with non- constant strain-rate histories in the context of the well-known fiber-drawing problem. The linearized stability of the methods is analyzed, and extension of these methods to planar flows is given. The approach taken in the case of non- trivial deformation histories is that of an Oldroyd difference quotient (ODQ) that approximates the convected derivatives of the differential constitutive equation in Lagrangian fashion along the portion of the streamline upstream of the stress evaluation point. Techniques based on earlier ideas of drift-function tracking are used to develop a weighting scheme for the ODQ that permits the use of low-order, Co stress elements. The numerical methods are discussed and analyzed in the context of a Johnson-Segalman fluid model with added Newtonian viscosity. The resulting initial-boundary-value problem is globally well-posed and possesses the key feature: the steady shear stress is a non-monotone function of the strain rate. Such models will be seen to display the spurt phenomenon in plane Poiseuille flow and apparently related phenomena in step strain experiments. (KR)