  The temporal and spatiotemporal linear stability analyses of viscoelastic,
subdiffusive, plane Poiseuille and Couette flows obeying the Fractional Upper
Convected Maxwell (FUCM) equation in the limit of low to moderate Reynolds
number ($Re$) and Weissenberg number ($We$), is reported to identify the
regions of topological transition of the advancing flow interface. In
particular, we demonstrate how the exponent in the subdiffusive
power{\color{black}-}law scaling ($t^\alpha$, with $0 {\color{black} < } \alpha
\le 1$) of the mean square displacement of the tracer particle, in the
microscale [Mason and Weitz, Phys. Rev. Lett. {\bf 74}, 1250-1253 (1995)] is
related to the fractional order of the derivative, $\alpha$, of the
corresponding non-linear stress constitutive equation in the continuum. The
stability studies are limited to two exponents: 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 classify regions of temporal stability, absolute
and convective instabilities and evanescent modes. 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 subdiffusive
flows.
