Unsteady electroosmotic flow of Carreau–Newtonian fluids through a cylindrical tube

Abstract

The present study aims to investigate the fully developed electroosmotic flow of Carreau-Newtonian fluids within a circular tube, considering two different boundary conditions of zeta potential. The unsteady behaviour of Carreau-Newtonian fluids, is attributed to the pulsatile pressure-driven flow. Studying two distinct formulations, slip and no-slip zeta potentials, illustrates the relative movement of fluid particles and the charged surface. The proposed study’s framework is divided into two regions: a central region containing a non-Newtonian Carreau fluid exhibiting both shear-thinning and thickening behaviour, and a peripheral region containing a Newtonian fluid with the effect of an electrical double layer at the solid wall of the cylindrical tube. The Poisson–Boltzmann equation under the assumption of Debye–Hückel approximation governs the potential distribution due to the electric double layer, facilitating the examination of electroosmotic flow through tube. Obtaining analytical solutions for the momentum equations in both regions becomes challenging due to the inclusion of pulsatile pressure-driven flow and a non-Newtonian Carreau fluid. Asymptotic series expansions are employed, utilizing small parameters such as the Weissenberg number (We≪1) and a pulsatile Reynolds number (α≪1), to derive velocity expressions for both regions. Wall shear stress fluctuations, as indicated by time-averaged wall shear stress and oscillatory shear index are assessed to gauge the temporal variation of wall shear stress from the dominant blood flow direction. By utilizing potential and hydrodynamic variables, an asymptotic formulation for the streaming potential is developed to ascertain the asymptotic expression of electrokinetic energy conversion efficiency and scrutinize the impacts of pertinent physical parameters on it. The proposed study prominently shows that the Debye–Hückel parameter, Weissenberg number, pulsatile Reynolds number, and time-dependent pressure gradient significantly influence the fluid velocity, flow rate, and flow resistance. The notable determination of the present study is that the velocity amplitude and electrokinetic energy conversion efficiency are enlarged in the slip-dependent case relative to the no-slip zeta potential. It is noteworthy that as inertial forces become more prominent compared to viscous forces, both time-average wall shear stress and oscillatory shear index experience a slight decrease, although the reduction in oscillatory shear index is minimal owing to the absence of obstruction within the tube wall. The NDSolve numerical scheme in Mathematica software is employed to calculate numerical solutions for the governing equations, which are subsequently verified against results obtained through asymptotic analysis. The findings of this study are expected to deepen our understanding of unsteady electroosmotic flows of Carreau fluid and contribute to the design and development of advanced microfluidic devices with enhanced performance for various applications.