Rheodynamics of viscoelastic subdiffusive channel flows: Low Weissenberg number regime

Abstract

Direct numerical simulations of viscoelastic, subdiffusive, and plane Poiseuille flow, representing thick polymer solutions including polymer melts, flows of liquid crystals, as well as biological flow such as mucus, are reported. The stress strain equation is fundamentally derived by relating the timescale exponent at the microscale (tα, with 0<α≤1) with the fractional order of the time derivative, α, of the corresponding non-linear equation in the continuum. The resultant stress constitutive equation is the fractional variant of the well-known upper convected Maxwell equation. In order to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia), the space of symmetric positive definite polymer conformation tensors is visualized using a Riemannian metric along with its three scalar invariants. Numerical simulations of the channel flow, in the regime of low to moderate Reynolds number and low Weissenberg number, effectively capture these flow-structures by providing (i) a better resolution of the instantaneous regions of elastic shocks (which are the alternating regions of expanded and compressed polymer volume, in comparison with the volume of the mean conformation tensor), and (ii) a better resolution to detect neighborhoods where the mean conformation tensor tends to be significantly different in comparison to the instantaneous conformation tensor, thereby corroborating the experimentally observed flow-instability transition of subdiffusive flows. Finally, the strength of the subdiffusive flow model and the invariant theory is highlighted through an application of an electrohydrodynamic micropump.