Quantifying the oscillatory behavior in start-up shear by analytically solving the Johnson-Segalman/Gordon-Schowalter model

Abstract

Since its introduction, back in the late 1970s, by Gordon and Schowalter (GS) and later by Johnson and Segalman (JS), the non-affine or slip parameter, ξ, has been routinely employed by numerous constitutive models. Its use results in spurious oscillations in the transient shear viscosity in start-up shear flow. Recent experimental evidence has shown that such an oscillatory behavior does occurs in polymer solutions under start-up shear, at least for the transient shear viscosity. In the present work, we aim to quantify this oscillatory behavior by providing the analytical solution of the GS-JS model, which we derive here in its entirety using the matrix exponential method. The oscillatory behavior of the model solutions stems directly from the fact that the matrix of which we take the exponential bears complex eigenvalues. Although this simple model is unable to compare favorably against the full spectrum of experimental rheological data, the analytical solution provided herein can improve our understanding of the transient behavior of concentrated polymer solutions.