Goddard-Miller model

Genovese, D. B. and Rao, M. A. 2003. Apparent viscosity and first normal stress of starch dispersions role of continuous and dispersed phases, and prediction with the Goddard-Miller model. Appl. Rheol. 13(4) 183-190. 

In addition to relationships between apparent viscosity and dynamic or complex viscosity, those between first normal stress coefficient versus dynamic viscosity or apparent viscosity are also of interest to predict one from another for food processing or product development applications. Such relationships were derived for the quasilinear co-rotational Goddard-Miller model (Abdel-Khalik et al., 1974 Bird et al., 1974, 1977). It should be noted that a first normal stress coefficient in a flow field, V i(y), and another in an oscillatory field, fri(ct>), can be determined. Further, as discussed below, (y) can be estimated from steady shear and dynamic rheological data. 

The quasilinear co-rotational Goddard-Miller model can be written as (Abdel-Khalik et al., 1974, Bird et al., 1974,1977) ... 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

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