Constitutive models that exceed the fitting capabilities of the Herschel–Bulkley model: A case study for shear magnetorheology

Abstract

A significant proportion of non-Newtonian fluids exhibits the phenomena of shear-thinning and yield stress. The Herschel-Bulkley model is probably the most widespread equation to describe the rheological behaviour of such systems, although greater attention is being paid to discerning more accurate mathematical models for visco-plastic systems. In this regard, the work presented herein investigates existing, unconventional viscoplastic models and evaluates their fitting and predictive capabilities in relation to collected magnetorheological data. The four-parameter models of modified Carreau, modified Cross (and its variant extended with the Quemada equation), Shulman and Nasiri-Ashrafizadeh were applied to appraise the shear data of the magnetorheological fluids. Their suitability was also statistically evaluated and compared with that attained for the conventional Herschel-Bulkley model and recently employed analogues, such as the Robertson-Stiff and Mizrahi-Berk models. It was found that the performance of all models for describing magnetorheological flow curves was adequate, with relatively high correlation coefficients. Applying the four-parameter models, however, provided higher accuracy, demonstrated by notably reduced statistical coefficients of regression analysis. Both of the modified Carreau and Cross models exhibited superior fitting performance (up to 4-times lower root mean square error), while the Shulman and Nasiri-Ashrafizadeh models proved questionable in application. Although interpreting four-parameter equations is a rather complex matter, they provided more reliable prediction of shear stress data across the entire range of shear rate than the Herschel-Bulkley approximation, especially the modified Carreau and modified Cross models. It is believed that these findings can be extended to describe the rheology of other systems with viscoplastic characteristics.