Spatiotemporal Linear Stability Analyses

This chapter describes the temporal and spatiotemporal linear stability analyses of viscoelastic, sub-diffusive, plane Poiseuille flowPoiseuille flows obeying the Fractional Upper Convected Maxwell (FUCM) equation in the limit of low to moderate Reynolds number (Re) and low Weissenberg number (We). In particular, we demonstrate how the exponent in the sub-diffusive power-law timescale ( \(t^\alpha \) , with \(0 < \alpha \le 1\) ) at the microscale is related to the fractional order of the time derivative, \(\alpha \) , of the corresponding nonlinear stress constitutive equation in the continuum. A description of the various kinds of temporal and spatiotemporal instability occurring in viscoelastic flows, namely the absolute instabilities, the convective instabilities and the evanescent modes, is provided. Results are limited to two exponents for the fractional order of time-derivative: monomer diffusion in Rouse chain melts, \(\alpha =\nicefrac {1}{2}\) , and in Zimm chain solutions, \(\alpha =\nicefrac {2}{3}\) . The temporal stability analysis indicates that with decreasing order of the fractional derivative: (a) the most unstable mode decreases, (b) the peak of the most unstable mode shifts to lower values of Re, and (c) the peak of the most unstable mode, for the Rouse model precipitates towards the limit \(Re \rightarrow 0\) . The Briggs idea of analytic continuation is deployed to demarcate the boundaries of the various types of instabilities. The spatiotemporal phase diagram indicates an abnormal region of temporal stability at high fluid inertia, revealing the presence of a non-homogeneous environment with hindered flow, thus highlighting the potential of the model to effectively capture certain experimentally observed flow-instability transition in sub-diffusive flows.