Inelastic and flow-type parameter models for non-Newtonian fluids

Generalised Newtonian Fluid (GNF) models which assume that the viscosity of a non-Newtonian fluid can be modelled inelastically as a time-independent, i.e. instantaneous, scalar function of the rate of strain, are used frequently. Classical examples of such models include the power-law, Carreau and Cross models (plus many others). In general flows these models are simply a function of the magnitude of the strain rate tensor (the magnitude being related to the second principal invariant of this tensor). Here we review the various attempts that have been proposed in the literature that retain the central idea of rather simple "inelastic" non-Newtonian fluid models whilst attempting to move beyond the constraints of making such models a function of just the second invariant of the rate of strain tensor and potentially suitable for more general flows. One approach has been to make the model a function of both the second and third invariants such that different viscosity behaviour can be predicted in shear and extension. These models have been shown to be somewhat successful in predicting some well-known features of polymeric flows normally associated with elasticity/strong extensional viscosity effects. However, such models suffer from the fact that the third invariant of the strain rate tensor vanishes in two-dimensional planar flows (and thus planar extensional effects cannot be incorporated). The second approach is to make the model a function of both the second invariant of the rate of strain tensor and a kinematic measure of the local flow "type" such that extensional effects can be incorporated in all stretching flows (i.e. even those in 2D). However, the issue then arises as to how to accurately determine the flow type in a general flow in an objective and straightforward manner. We show that a number of proposed models in the literature are not objective, whilst also highlighting the difficulties associated with this approach and the suc-cesses once fully objective and general measures are used. Although these approaches have not (yet) found wide-spread adoption within the literature, the benefits and utility of such models in general engineering applications remains obvious. This short review thus concludes with an outlook of how such models could be taken further.