Rheodynamics of Sub-diffusive Channel Flows

Direct numerical simulations of viscoelastic, sub-diffusive, plane Poiseuille flow, representing thick polymer solutions including polymer melts, flows of liquid crystals, as well as biological flow such as mucus, are described. The computations are carried out using a recently developed family of time-asymptotically stable, implicit-explicit, adaptive, time integration methods (denoted with the \(\theta \) -method) applicable for the general solution of the fractional advection-diffusion-reaction (FADR) equations. The spectral analysis of the method (involving the group velocity and the Phase speedphase speed) indicates a region of favourable dispersion for a limited range of Peclet numberPeclet number. The accuracy and the efficacy of the method are benchmarked using the two-dimensional fractional diffusion equation, originally proposed by Brunner (J. Comput. Phys. 229 6613-6622 (2010)). Numerical simulations of the viscoelastic channel flow effectively capture the non-homogeneous regions of high viscosity at moderate to high fluid inertia (or the so-called ‘Spatiotemporal macrostructuresspatiotemporal macrostructures’), experimentally observed in the flow-instability transition of sub-diffusive flows.