Viscosity functions of shear flows for fractional maxwell fluids

Abstract

Fractional calculus is applied increasingly to fluid dynamics. We derive exact analytical solutions for shear stress growth rheological responses of the fractional Maxwell fluid (FMF) in extra-stress tensor form. We do so by using the Laplace transform and its inverse. By shear stress growth, we mean the sudden inception of steady shear flow. We first determine the shear stress growth viscosity, then extend this to the steady shear viscosity using empirical Gleissle mirror relations. We choose to explore the FMF because of its four-parameter versatility and because it describes fluid elasticity measurements accurately. We compare with the ordinary non-fractional Maxwell fluid (OMF) in shear stress growth. Our FMF exact solution agrees well with available measurements on aqueous xanthan gum solutions, so long as the initial residual stresses in the sample are accounted for. We discover that, in practice, positive initial residual shear stress shifts the viscosity in shear stress growth downward. Finally, we construct concentration master curves for our xanthan gum solution shear stress growth rheological functions. We do so to generalize predictions of viscosity under varying conditions. Our worked examples illustrate how to use this dimensionless master curve.