K Constitutive Equations for Non-Newtonian Fluids

In the next section, we briefly discuss some specific constitutive models and their derivation. 

In the preceding section, we have seen that there are many fluids that cannot be approximated as Newtonian under normal flow conditions. An obvious question is whether successful generalizations of the Newtonian fluid model exist that can be used to solve flow and transport problems for this class of materials  

If we interpret this question as asking whether models exist for the general class of complex/non-Newtonian fluids that are known to provide accurate descriptions of material behavior under general flow conditions, the current answer is that such models do not exist. Currently successful theories are either restricted to very specific, simple flows, especially generalizations of simple shear flow, for which rheological data can be used to develop empirical models, or to very dilute solutions or suspensions for which the microscale dynamics is dominated by the motion deformation of single, isolated macromolecules or particles/drops.24 

It should be emphasized that many constitutive models have been proposed especially for polymeric solutions and melts, and there is a great deal of current research that is aimed at both new models25 and numerical analysis of fluid motions by use of the existing models 26 The problem is that few have been carefully compared with the behavior of real fluids outside the highly simplistic flows of conventional rheometers, and then mainly under flow conditions in which the perturbations in material structure are weak. Thus there is currently no model that is known to provide quantitatively accurate or even qualitatively reliable descriptions of real complex fluids for a wide spectrum of flows. 

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is 

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