Elastic-like instabilites in non-Newtonian parallel flows

Mixing control in fluid environments having very low Reynolds numbers is a crucial need for many practical purposes ranging from biochemistry analysis in microfluidic devices, where mixing has to be rapid and efficient, to lab-on-a-chip applications, where mixing has to be reduced to avoid spurious effects as in microfluidic rheometer applications. The ability to control fluid mixing properties is clearly subjected to a deep understanding of physical mechanisms able to originate such a mixing for very small Reynolds numbers. In this respect, a simple model to capture mesoscopic effects of order-disorder transitions of an underlying non-Newtonian fluid microstructure subjected to shear is proposed and its behaviors investigated (numerically and by means of asymptotic perturbative methods) in relation to the possible emergence of fluid elatic-like instabilities occurring for arbitrarily small flow inertia (i.e. zero Reynolds numbers). A crucial ingredient for instabilities to emerge has been identified in the finite-time response of the network structure to strain where the order-disorder transition corresponds to a change from low-to-high fluid viscosity (and not viceversa). Our results generalizes the concept of ``elastic instabilities'' in viscoelastic fluids to a more general and larger class of non-Newtonian fluids.