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The temporal and spatiotemporal linear stability analyses of viscoelastic,subdiffusive, plane Poiseuille and Couette flows obeying the Fractional UpperConvected Maxwell (FUCM) equation in the limit of low to moderate Reynoldsnumber ($Re$) and Weissenberg number ($We$), is reported to identify theregions of topological transition of the advancing flow interface. Inparticular, we demonstrate how the exponent in the subdiffusivepower{\color{black}-}law scaling ($t^\alpha$, with $0 {\color{black} < } \alpha\le 1$) of the mean square displacement of the tracer particle, in themicroscale [Mason and Weitz, Phys. Rev. Lett. {\bf 74}, 1250-1253 (1995)] isrelated to the fractional order of the derivative, $\alpha$, of thecorresponding non-linear stress constitutive equation in the continuum. Thestability studies are limited to two exponents: monomer diffusion in Rousechain melts, $\alpha=\nicefrac{1}{2}$, and in Zimm chain solutions,$\alpha=\nicefrac{2}{3}$. The temporal stability analysis indicates that withdecreasing order of the fractional derivative: (a) the most unstable modedecreases, (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 modelprecipitates towards the limit $Re \rightarrow 0$. The Briggs idea of analyticcontinuation is deployed to classify regions of temporal stability, absoluteand convective instabilities and evanescent modes. The spatiotemporal phasediagram indicates an abnormal region of temporal stability at high fluidinertia, revealing the presence of a non-homogeneous environment with hinderedflow, thus highlighting the potential of the model to effectively capturecertain experimentally observed, flow-instability transition in subdiffusiveflows.

Spatiotemporal linear stability of viscoelastic subdiffusive channel flows: a fractional calculus framework