In our previous work (Promjariyakoon 2025; Promjariyakoon et al. 2026), we derive exact analytical solutions for (1) steady shear viscosity, and (2) shear stress growth rheological responses of the fractional Maxwell fluid (FMF) generalized to tensor form. By shear stress growth, we mean the sudden inception of steady shear flow. We found good agreement with experimental observation, though this left some room for improvement. In this work, following our previous method, we improve upon the FMF by adding one more fractional derivative for retardation to get the fractional Jeffreys model (FJM), a fractional Kelvin-Voigt model arranged in series with a spring-pot. We determine (1) complex viscosity, (2) the shear stress growth viscosity function, (3) then extend these to the steady shear viscosity between the stress and the local properties of theusing Gleissle mirror relations, and (4) shear stress relaxation following cessation of steady shear flow. Our FJM exact solution improves significantly upon the FMF, agreeing well with available measurements on aqueous xanthan gum solutions, so long as the initial residual stresses in the sample are accounted for. We show that its material functions are experimentally measurable and physically interpretable quantities. Finally, we construct temperature master curves for a low-density polyethylene melt. We do so to generalize predictions of viscosity under varying temperatures. Our worked example illustrates how to use our main results.
