Study of pulsatile pressure-driven electroosmotic flows through an elliptic cylindrical microchannel with the Navier slip condition
© The Author(s) 2017
Received: 31 January 2017
Accepted: 17 May 2017
Published: 7 June 2017
This paper aims to study an unsteady electric field-driven and pulsatile pressure-driven flow of a Newtonian fluid in an elliptic cylindrical microchannel with Navier boundary wall slip. The governing equations of the slip flow and distributions of electric potential and charge densities are the modified Navier-Stokes equations, the Poisson equation and the Nernst-Planck equations, respectively. Analytical and numerical analyses based on the Mathieu and modified Mathieu equations are performed to investigate the interplaying effects of pulsatile pressure gradients and the slip lengths on the electroosmotic flow.
Over the past decades, a fluid flow manipulation under a very small scale known as microfluidics has become an active field of scientific research due to the emergence of their several applications, for example, lab-on-the-chips, computer chips, medical diagnostic devices, and drug delivery systems [1–5]. Taking the advantage of a small scale system, a microfluidic application not only reduces the requirement for the samples, but also increases the efficiency and speed of the reaction. As a consequence, the microflow characteristics have been widely studied in both experimental and analytical ways in order to develop various system controls and device designs [6–10]. One particular technique to precisely manipulate a flow in a microscale is the use of a pressure force combined with the well-known electrokinetic force, namely, electroosmosis. Electroosmosis phenomenon relies on the formation of the electrical double layer (EDL) generated by two parallel layers of charged ions: the first layer, the layer of ions on the inner wall surface due to the chemical reaction between the fluid and the channel wall; the second layer, the layer of counter ions in the fluid attracted to the first layer by the Coulomb force. The movement of the ions in the second layer, induced by the application of an external electric field, will lead to the motion of the entire fluid caused by the drag force .
Fluid flow problems have been carried out traditionally under the no-slip boundary condition [12–14], which dictates that the velocity, relative to the wall channel, of the fluid adjacent to the wall is zero. However, in microfluidics, the appearance of the fluid slip at the wall interface has been widely reported, and its influence has been investigated [15–17]. For this reason, the velocity slip condition turns into an important factor to achieve the realistic microflow behavior.
In the literature, an analytic solution of an electroosmotic flow in microchannels has been studied under a constant pressure gradient. Goswami and Chakraborty  investigated the semi-analytic solution of a steady electroosmotic flow with the interfacial slip condition in microchannels of various complex cross-sectional shapes under the constant pressure gradient assumption. Na et al. , Chinyoka and Makinde , and Reshad et al.  found the analytic solution of transient electroosmotic and pressure-driven flows with a constant pressure gradient through a microannulus, a slip microchannel, and rectangular microchannels, respectively. However, some microfluidic systems are driven by a pulsatile pressure gradient due to the nature of some systems such as blood flow or the integrated micropump of displacement type. Moreover, a report of Bandopadhyay and Chakraborty  on the investigation of electroosmotic flow in a slip microchannel shows that the overestimation result can be obtained under the avoidance of a pulsating pressure gradient. As a result, a microfluidic investigation combined with a pulsating pressure gradient is a key to keeping the problem suitable in many situations. Recently, the solution of combined pulsating pressure gradient and electroosmotic flow was found by Chakraborty et al. , but the pressure gradient was simplified to just a sinusoidal function.
Due to the fact that the geometry in many microfluidic devices and some systems such as a blood vessel is of a circular or elliptic cross-section, the use of elliptic geometry takes an advantage of embodying the solution for circular geometry and making the problem tractable for any eccentricity.
According to the aforementioned arguments, we here derive the solution of combined pulsatile pressure-driven and electroosmotic flow through an elliptic cylindrical microchannel under the Navier slip condition to describe the flow behavior in a more realistic situation than the previous works. The pressure gradient term in Navier-Stokes equations is precisely expanded by the Fourier series. Moreover, an influence of a pulsatile pressure gradient, the number of the Fourier expansion terms for the pressure gradient, and a slip length are investigated on the volumetric flow rate which plays a more important role in the flow control in microfluidic devices compared to the velocity profile.
In this section, we introduce the elliptic cylindrical coordinates, the Mathieu and modified Mathieu functions which are used throughout this paper.
2.1 Elliptic cylindrical coordinate system
2.2 Mathieu functions
3 Mathematical modeling
4 Solution of the boundary value problem
5 Numerical results and discussions
The comparative results of volumetric flow rate on various numbers of the Fourier expansion terms, \(N+1\), for the pressure gradient defined in equation (10) and on various slip lengths l are presented. Moreover, an influence of oscillating term in a pressure gradient expression on electroosmotic flows with various external electric fields are investigated.
The primary objectives of the present work were twofold. One was to find the solution of an unsteady pulsatile pressure-driven electroosmotic flow through an elliptic cylindrical microchannel with the Navier slip condition. The solution was solved with the use of the Mathieu and modified Mathieu functions. The other was to investigate our numerical results to develop a better understanding of the underlying physical processes in microfluidics. In particular, we compared the volumetric flow rate corresponding to the oscillatory and the constant pressure gradient, the volumetric flow rate with the number of the Fourier expansion terms for the pressure gradient, and the volumetric flow rate with the slip length. We found that when the flow was clearly driven by a combination of pressure and electrokinetic forces, the oscillatory behavior of pulsatile pressure became crucial especially when external electric field was low. The volumetric flow rate is more accurate as we use a higher number of terms in the Fourier expansion for pressure gradient. Moreover, an increment in slip lengths gives rise to an increment in volumetric flow rate in a proportional way.
This work is partially supported by Development and Promotion of Science and Technology Talents Project (DPST).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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