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

TL;DR

This paper investigates the linear stability of viscoelastic subdiffusive flows using a fractional calculus framework, revealing how fractional order affects flow stability and transition regions, with implications for understanding flow instabilities in complex fluids.

Contribution

It introduces a fractional calculus-based stability analysis for viscoelastic subdiffusive flows, linking subdiffusive exponents to flow stability and identifying transition regions.

Findings

Decreasing fractional order reduces the most unstable mode.

Unstable mode peaks shift to lower Reynolds numbers with lower fractional order.

High inertia regions show abnormal stability, matching experimental observations.

Abstract

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.