Numerical analysis of dynamic electro-osmotic flows of non-Newtonian fluids in rectangular microchannels Cunlu Zhao* and Chun Yang School of Mechanical and Aerospace Engineering, Nanyang Technological University 50 Nanyang Avenue, 639798, Republic of Singapore Address correspondence to this author. E-mail: [email protected] Abstract Numerical analyses of transient electro-osmosis of a typical non-Newtonian liquid induced by DC and AC electric fields in a rectangular microchannel are conducted in the framework of continuum fluid mechanics. The famous power-law constitutive model is used to express the fluid dynamic viscosity in terms of the velocity gradient. Transient start-up characteristics of electro-osmotic power-law liquid flow in rectangular microchannels are simulated by using finite element method. Under a DC electric field, it is found out and the fluid is more inert to the external electric field and the steady-state velocity profile becomes more plug-like with decrease of the flow behavior index of the power-law liquids. The numerical calculations also confirm the validity of the generalized Smoluchowski slip velocity which can serve as the counterpart for the classic Smoluchowski slip velocity when dealing with electrokinetic flow of non-Newtonian power-law fluids. Under AC electric fields, the fluid is more obviously accelerated during oscillations and the amplitude of the oscillating velocity is closer to the magnitude of the generalized Smoluchowski velocity as the fluid behavior index increases. These dynamic predictions are of practical significance for the design of microfluidic devices that 1 manipulate non-Newtonian fluids such as biofluids, polymer solutions and colloidal suspensions. Keywords: Electro-osmotic pumping, Non-Newtonian power-law fluids, Transient characteristics, Microfluidics. 1 Introduction Microfluidic technologies eventually promise a conventional biochemical laboratory to be constructed on a single, disposable microchip. The generic microfluidic systems involve buffer fluid and sample manipulations such as pumping, valving, mixing, injection, dispensing, etc (Bousse et al., 2000; Harrison et al., 1993; Squires and Quake, 2005; Whitesides, 2006). Fundamental understanding of the liquid flow characteristics in microchannels is thus essential to optimum design and precise control of microfluidic devices. In general, liquid motion can be generated by either applying a pressure gradient or imposing an electric field, leading to respective pressure-driven flow or electrokinetically-driven flow. Traditionally, in large-sized channels flow is often driven by pressure that is usually generated by mechanical pumps. In microchannels however it becomes increasingly difficult to utilize pressure-driven flow mode as the channel size shrinks, especially down to micro-and submicron range. Moreover, some parts like microvalves and micropumps with moving components are difficult to fabricate, and they are prone to mechanical failure due to fatigue and fabrication defects. Alternatively, electro-osmotic flow enjoys numerous advantages (over pressure-driven flow), including ease of fabrication and control, no need for moving parts, high reliability, no noise etc. Specifically, a plug-like velocity profile in electro-osmotic flow can result in reduced 2 dispersion of sample species, making capillary electrophoresis become one of most successful technologies for chemical and biomedical analyses (Manz et al., 1994). When considering fluid motion on micro-scales, viscous relaxation is typically achieved within milliseconds. The time evolution of microflows is nevertheless highly relevant to a growing variety of applications of electro-kinetics in microfluidics and colloidal dynamics in the sub-millisecond range(Yossifon et al., 2009). These include inter alia biochip operation in high-speed electrophoretic separation processes (Fan and Harrison, 1994; Jacobson et al., 1998; Jacobson et al., 1994), the use of short-duration pulsed electric fields (Söderman and Jönsson, 1996) in order to distinguish between particle velocities and background electro-osmotic flow or to suppress thermal-zone broadening (Dose and Guiochon, 1993) as well as micro-mixing and micro-pumping by means of AC or modulated DC fields (Ajdari, 2000; González et al., 2000; Ramos et al., 1999). Owing to the short time scales involved, it is essential to account for the unsteady dynamics of the electro-kinetic flows in these applications. Most previous analyses of transient electro-kinetic flow have focused on Newtonian fluids. They have theoretically studied dynamic characteristics of electro-osmosis (Campisi et al., 2005; Hanna and Osterle, 1968; Ivory, 1983; Kang et al., 2002; Keh and Tseng, 2001; Mishchuk and González-Caballero, 2006; Söderman and Jönsson, 1996; Yang et al., 2002; Yang and Kwok, 2003; Zhang et al., 2006) in response to various modes of suddenly applied external fields. Obviously, the short sub-millisecond time scales involved get in the way of an experimental characterization of dynamic behavior of electro-osmotic flows. Nevertheless, the recent advances in the state of the art µPIV techniques (Yan et al., 2007b; Yan et al., 2007a) enable experimental characterizations of 3 dynamics of electrokinetic flows. However, microfluidic devices are usually used to analyze biofluids which may not be treated as Newtonian fluids. Thus, the more general Cauchy momentum equation, instead of the Navier-Stokes equation should be used to describe the flow characteristics of non-Newtonian fluids provided that proper constitutive equations are available. The aim of constructing constitutive equations for non-Newtonian fluids is to find correlations between dynamic viscosity and shear rate. The Power-law model (Graham and Jones, 1994), Carreau model (Khellaf and Lauriat, 2000) , Moldflow first-order model (Koh et al., 2004), and Bingham model (Das et al., 2008) have been successfully developed to analyze non-Newtonian fluid flow and heat/mass transfer. As for electro-osmotic flows of non-Newtonian fluids, they also may behave differently from their Newtonian counterparts. Understanding the rheology in such situations is a crucial aspect for the design and operation of microfluidic devices, thus several authors are currently studying the electrokinetic flow of non-Newtonian fluids in microchannel (Afonso et al., 2009; Berli, 2009; Das and Chakraborty, 2006; Park and Lee, 2008; Tang et al., 2009; Zhao and Yang, 2010; Zhao et al., 2008; Zimmerman et al., 2006). Unfortunately, all the studies listed above for non-Newtonian electro-osmosis are conducted for the steady state. The only relevant work (Zhao and Yang, 2009) just gave the analytical solutions for the transient electro-osmosis of viscoelastic fluids of Oldroyd-B type in a rectangular microchannel, and however did not report how the rheological properties of the fluids affects the dynamics of the electro- osmosis. This study reports numerical analyses of the transient electro-osmotic flow of typical power-law fluids in a rectangular microchannel driven by both DC and AC electric fields. The numerical simulations are carried out by using finite element method 4 which is verified through the comparison with the available exact solutions for Newtonian fluids. Parametric studies are performed to show the dynamics of electroosmosis for power-law fluids and examine effects of fluid rheology (fluid behavior index) on the transient velocity distributions of the electro-osmotic flow of power-law fluids. 2 Problem Formulations Figure 1 shows the dimensions of the micro-channel and the coordinate systems considered in this work. The channel is filled with a liquid solution of dielectric constant, r . It is assumed that the channel wall is uniformly charged with a zeta potential, w , and the liquid solution is a typical non-Newtonian fluid whose behavior can be described by the well-known power-law model. When an external dynamic electric field E0 f (t) is imposed along the x-axis direction, the fluid in the micro-channel sets in motion due to electro- osmosis. f (t) is a time-dependent function and characterizes the dynamic behavior of the applied electric fields In this study, we consider a DC driving electric field (f (t)=1) and a AC driving electric field (f (t)=sin(ωt), in which ω is the frequency). Because of symmetry, the analysis is restricted in the first quadrant of the y-z plane. 2. 1 Electric field in double layer When the liquid in the micro-channel contacts the solid wall, an interfacial charge is established which causes the free ions in the liquid to rearrange so as to form a thin region with non zero net charge density. This region is commonly referred to as the 5 electrical double layer (EDL). According to electrostatics theory, the electric potential distribution in the EDL region is governed by the Poisson equation, which is expressed as 22 e 22 (1) yz 0 r -12 where ε0 is the permittivity of vacuum (ε0=8.854 ×10 F/m), ρe is the net charge density in the EDL region and is related to ionic number concentrations and the EDL potential by using the assumption of Boltzmann distribution (assuming an symmetric electrolyte) zev evvn n ze 2sinh zen (2) kTB where n and n is respectively ionic number concentrations of positive and negative ions in the EDL region. Introducing the dimensionless groups: yyD / h , zzD / h ,K=κDh, and =zveψ/(kBT), then substituting equation(2) in to equation(1), one can show that the electrical potential profile in the EDL is governed by the Poisson–Boltzmann