Nonlinear Regression Applied to Non-Newtonian Flow

Abstract

Modifications of the Gauss-Newton method are among the most widely used methods for nonlinear regression analysis. One such modification, which the authors found applicable to a wide variety of pharmaceutical systems, is described. Its application in describing the flow behavior of non-Newtonian systems and the FORTRAN IV program utilized are presented. One of the more useful, accurate, and physically significant equations for describing non-Newtonian flow is the structure equation: F = f + η∞ S − bνe−aS, in which F is shear stress, S is shear rate, and the other terms are constants. The equation was originally evaluated through the use of multiple regression, with the constant a assumed to be equal to 0.001, which gave good fit to a variety of flow systems. Since the original equation was developed, nonlinear regression techniques have appeared which make it possible to examine the structure equation in greater detail. It was found, for example, that for dispersions of a wax, consisting of a mixture of polyethoxylated higher fatty alcohols, in water the constant a varied from 0.013 to 0.049 rather than remaining fixed at 0.001. Some of the original data, upon which the structure equation was based, were reevaluated using nonlinear regression analysis. These data were for suspensions of salicylamide (varying concentration) in methylcellulose solutions of varying concentration. The constant a was found to vary from 0.00109 to 0.00172 as the concentration of methylcellulose increased and was independent of salicylamide concentration. In all instances, allowing a to vary as an adjustable parameter gave a better fit to the data than assuming it to be constant at 0.001. The use of nonlinear regression analysis served to emphasize the usefulness of the structure equation.