Various math models were empirically developed by different researchers, in order to fit trends they saw in their own data sets. The Math Model is a "best fit" line that can be used to characterize data from a specific test and has the advantage for easy data storage and comparison with similar tests by looking at slope and y-intercept.
There are some simple models that provide a reasonable fit to many data sets, and have parameters that have some meaning to various practitioners - whether those people are researchers, QA/QC people, or process engineers. The modified Casson model, for example, works well for testing chocolates, among other things. The Herschel-Bulkley model is useful for materials that have a yield point and then "shear-thin" after yielding. This may be good for "gel-like" materials, for example.
Brookfield's "Analysis" module in our applications software (Rheocalc and Wingather) lists the curve-fit parameter results, along with a "Coefficient of Fit". One could try a few of our models, and select the one with the best "CoF", for example - the closer to "1.00", the better the fit. It may be tough to accurately predict product behavior. Conservatively used, rheology models may be good for interpolating apparent viscosities at various shear rates, for example. Nonetheless, some models may be used to extrapolate yield stresses, for example, at shear rate values of zero. The more data points taken, the more reliable the fit.
Please also bear in mind that a certain model may have been used just because it's "simple" and "good enough for a reasonable estimate". That does not mean that it would be the best model to use. One example is in the Petroleum industry: "drilling mud" rheology has been examined with the Bingham model for years, despite the fact that these materials are highly non-Newtonian! The Bingham model assumes Newtonian behavior after yield. Nonetheless, field personnel felt it is was "good enough" to give them an idea of how the material was handling. More practitioners have started using the "H-B" model during the past few years.