# Electroosmotic Flow of Viscoelastic Fluid in a Nanoslit

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

^{+}and Cl

^{−}in a long channel of length L, height H and width W under externally applied potential difference V

_{0}across the channel. We assume that the channel height is much smaller than both the length and the width (i.e., $H\ll L,H\ll W$), then the problem can be simplified to a 2D problem schematically shown in Figure 1. Cartesian coordinates O-xy are adopted with y-axis in the length direction and the origin fixed on one of the channel walls.

**c**for the LPTT model is governed by

**c**) is the trace of the conformation tensor

**c**.

^{+}(Cl

^{−}) ions, respectively.

## 3. Numerical Method and Validation

**c**, significantly impedes its accuracy and stability at high Wi [38]. A huge amount of effort has been made to resolve this problem when calculating the evolution of polymeric elastic stress, e.g., by introducing the artificial diffusion term [39,40], reconstructing better discretization schemes [41], and decomposing or reformulating the conformation tensor

**c**[41,42]. In this study, log conformation reformulation (LCR) method [42] is implemented into the new solver. This method calculates the conformation tensor

**c**by solving its logarithm instead of solving it directly, thereby, guaranteeing its SPD property automatically. Meanwhile, the deviation between polynomial fitting and exponential variation profiles of the conformation tensor

**c**is eliminated.

**c**is a SPD matrix, it can be decomposed as

**R**is an orthogonal matrix composed by the eigenvectors of

**c**, and $\Lambda $ is a diagonal matrix whose diagonal elements are the eigenvalues of

**c**. The matrix logarithm of the conformation tensor

**c**is introduced as

**c**can be reformulated in terms of this new variable $\mathit{\Psi}$ as

**c**can be recovered from matrix-exponential of $\mathit{\Psi}$ as

## 4. Results and Discussion

_{0}= 0.01, 0.1, and 10 mM, corresponding to $kH/2=0.52,1.64,\mathrm{and}16.45$. Other parameters are set as ${\sigma}_{0}=-0.01\mathrm{C}/{\mathrm{m}}^{2},$ ${V}_{0}=0.05\mathrm{V}$, $\epsilon =0.25\mathrm{and}\beta =0.1$ unless they are specifically stated.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Vogel, R.; Anderson, W.; Eldridge, J.; Glossop, B.; Willmott, G. A Variable Pressure Method for Characterizing Nanoparticle Surface Charge Using Pore Sensors. Anal. Chem.
**2012**, 84, 3125–3131. [Google Scholar] [CrossRef] [PubMed] - Hsu, W.L.; Daiguji, H. Manipulation of Protein Translocation through Nanopores by Flow Field Control and Application to Nanopore Sensors. Anal. Chem.
**2016**, 88, 9251–9258. [Google Scholar] [CrossRef] [PubMed] - Mei, L.; Chou, T.H.; Cheng, Y.S.; Huang, M.J.; Yeh, L.H.; Qian, S. Electrophoresis of pH-regulated Nanoparticles: Impact of the Stern Layer. Phys. Chem. Chem. Phys.
**2016**, 18, 9927–9934. [Google Scholar] [CrossRef] [PubMed] - Karnik, R.; Fan, R.; Yue, M.; Li, D.; Yang, P.; Majumdar, A. Electrostatic Control of Ions and Molecules in Nanofluidic Transistors. Nano Lett.
**2005**, 5, 943–948. [Google Scholar] [CrossRef] [PubMed] - Siria, A.; Poncharal, P.; Biance, A.L.; Fulcrand, R.; Blase, X.; Purcell, S.T.; Bocquet, L. Giant Osmotic Energy Conversion Measured in a Single Transmembrane Boron Nitride Nanotube. Nature
**2013**, 494, 455–458. [Google Scholar] [CrossRef] [PubMed] - Yuan, Z.; Garcia, A.L.; Lopez, G.P.; Petsev, D.N. Electrokinetic Transport and Separations in Fluidic Nanochannels. Electrophoresis
**2007**, 28, 595–610. [Google Scholar] [CrossRef] [PubMed] - Sparreboom, W.; van den Berg, A.; Eijkel, J.C.T. Principles and Applications of Nanofluidic Transport. Nat. Nanotechnol.
**2009**, 4, 713–720. [Google Scholar] [CrossRef] [PubMed] - Green, N.G.; Ramos, A.; Morgan, H. Ac Electrokinetics: A Survey of Sub-micrometre Particle Dynamics. J. Phys. D Appl. Phys.
**2000**, 33, 632–641. [Google Scholar] [CrossRef] - Zhao, C.; Yang, C. Advances in Electrokinetics and Their Applications in Micro/Nano Fluidics. Microfluid. Nanofluid.
**2012**, 13, 179–203. [Google Scholar] [CrossRef] - Mei, L.; Yeh, L.H.; Qian, S. Buffer Anions Can Enormously Enhance the Electrokinetic Energy Conversion in Nanofluidics with Highly Overlapped Double Layers. Nano Energy
**2017**, 32, 374–381. [Google Scholar] [CrossRef] - Reuss, F.F. Charge-induced Flow. Proc. Imp. Soc. Nat. Mosc.
**1809**, 3, 327–344. [Google Scholar] - Tang, Z.; Hong, S.; Djukic, D.; Modi, V.; West, A.C.; Yardley, J.; Osgood, R.M. Electrokinetic Flow Control for Composition Modulation in a Microchannel. J. Micromech. Microeng.
**2002**, 12, 870–877. [Google Scholar] [CrossRef] - Haywood, D.G.; Harms, Z.D.; Jacobson, S.C. Electroosmotic Flow in Nanofluidic Channels. Anal. Chem.
**2014**, 86, 11174–11180. [Google Scholar] [CrossRef] [PubMed] - Prabhakaran, R.A.; Zhou, Y.; Zhao, C.; Hu, G.; Song, Y.; Wang, J.; Yang, C.; Xuan, X. Induced Charge Effects on Electrokinetic Entry Flow. Phys. Fluids
**2017**, 29, 062001. [Google Scholar] [CrossRef] - Burgreen, D.; Nakache, F.R. Electrokinetic Flow in Ultrafine Capillary Slits. J. Phys. Chem.
**1964**, 68, 1084–1091. [Google Scholar] [CrossRef] - Petsev, D.N.; Lopez, G.P. Electrostatic Potential and Electroosmotic Flow in a Cylindrical Capillary Filled with Symmetric Electrolyte: Analytic Solutions in Thin Double Layer Approximation. J. Colloid Interface Sci.
**2006**, 294, 492–498. [Google Scholar] [CrossRef] [PubMed] - Wang, C.; Wong, T.N.; Yang, C.; Ooi, K.T. Characterization of Electroosmotic Flow in Rectangular Microchannels. Int. J. Heat Mass Transf.
**2007**, 50, 3115–3121. [Google Scholar] [CrossRef] - Yossifon, G.; Mushenheim, P.; Chang, Y.C.; Chang, H.C. Nonlinear Current-Voltage Characteristics of Nanochannels. Phys. Rev. E
**2009**, 79, 046305. [Google Scholar] [CrossRef] [PubMed] - Atalay, S. Role of Surface Chemistry in Nanoscale Electrokinetic Transport. Ph.D. Thesis, Old Dominion University, Norfolk, VA, USA, 2014. [Google Scholar]
- Huang, M.J.; Mei, L.; Yeh, L.H.; Qian, S. ph-Regulated Nanopore Conductance with Overlapped Electric Double Layers. Electrochem. Commun.
**2015**, 55, 60–63. [Google Scholar] [CrossRef] - Baldessari, F. Electrokinetics in Nanochannels: Part I. Electric Double Layer Overlap and Channel-to-Well Equilibrium. J. Colloid Interface Sci.
**2008**, 325, 526–538. [Google Scholar] [CrossRef] [PubMed] - Das, S.; Chakraborty, S. Implications of Interactions between Steric Effects and Electrical Double Layer Overlapping Phenomena on Electro-Chemical Transport in Narrow Fluidic Confinements. arXiv, 2010; arXiv:1010.5731. [Google Scholar]
- Choi, W.; Joo, S.W.; Lim, G. Electroosmotic Flows of Viscoelastic Fluids with Asymmetric Electrochemical Boundary Conditions. J. Non-Newton. Fluid Mech.
**2012**, 187, 1–7. [Google Scholar] [CrossRef] - Zhao, C.L.; Yang, C. Electro-osmotic Mobility of Non-Newtonian Fluids. Biomicrofluidics
**2011**, 5, 014110. [Google Scholar] [CrossRef] [PubMed] - Das, S.; Chakraborty, S. Analytical Solutions for Velocity, Temperature and Concentration Distribution in Electroosmotic Microchannel Flows of a Non-Newtonian Bio-Fluid. Anal. Chim. Acta
**2006**, 559, 15–24. [Google Scholar] [CrossRef] - Zimmerman, W.B.; Rees, J.M.; Craven, T.J. Rheometry of Non-Newtonian Electrokinetic Flow in a Microchannel T-Junction. Microfluid. Nanofluid.
**2006**, 2, 481–492. [Google Scholar] [CrossRef] - Olivares, M.L.; Vera-Candioti, L.; Berli, C.L.A. The EOF of Polymer Solutions. Electrophoresis
**2009**, 30, 921–929. [Google Scholar] [CrossRef] [PubMed] - Zhao, C.; Zholkovskij, E.; Masliyah, J.H.; Yang, C. Analysis of Electroosmotic Flow of Power-Law Fluids in a Slit Microchannel. J. Colloid Interface Sci.
**2008**, 326, 503–510. [Google Scholar] [CrossRef] [PubMed] - Park, H.M.; Lee, W.M. Helmholtz–Smoluchowski Velocity for Viscoelastic Electroosmotic Flows. J. Colloid Interface Sci.
**2008**, 317, 631–636. [Google Scholar] [CrossRef] [PubMed] - Park, H.M.; Lee, W.M. Effect of Viscoelasticity on the Flow Pattern and the Volumetric Flow Rate in Electroosmotic Flows through a Microchannel. Lab Chip
**2008**, 8, 1163–1170. [Google Scholar] [CrossRef] [PubMed] - Afonso, A.M.; Alves, M.A.; Pinho, F.T. Analytical Solution of Mixed Electro-osmotic/Pressure Driven Flows of Viscoelastic Fluids in Microchannels. J. Non-Newton. Fluid Mech.
**2009**, 159, 50–63. [Google Scholar] [CrossRef] - Ferrás, L.L.; Afonso, A.M.; Alves, M.A.; Nóbrega, J.M.; Pinho, F.T. Analytical and Numerical Study of the Electro-osmotic Annular Flow of Viscoelastic Fluids. J. Colloid Interface Sci.
**2014**, 420, 152–157. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Afonso, A.M.; Alves, M.A.; Pinho, F.T. Electro-osmotic Flow of Viscoelastic Fluids in Microchannels under Asymmetric Zeta Potentials. J. Eng. Math.
**2011**, 71, 15–30. [Google Scholar] [CrossRef] - Sousa, J.J.; Afonso, A.M.; Pinho, F.T.; Alves, M.A. Effect of the Skimming Layer on Electro-osmotic-Poiseuille Flows of Viscoelastic Fluids. Microfluid. Nanofluid.
**2011**, 10, 107–122. [Google Scholar] [CrossRef] - Dhinakaran, S.; Afonso, A.M.; Alves, M.A.; Pinho, F.T. Steady Viscoelastic Fluid Flow between Parallel Plates under Electro-osmotic Forces: Phan-Thien-Tanner Model. J. Colloid Interface Sci.
**2010**, 344, 513–520. [Google Scholar] [CrossRef] [PubMed] - Ma, Y.; Yeh, L.H.; Lin, C.Y.; Mei, L.; Qian, S. ph-Regulated Ionic Conductance in a Nanochannel with Overlapped Electric Double Layers. Anal. Chem.
**2015**, 87, 4508–4514. [Google Scholar] [CrossRef] [PubMed] - Mei, L.; Yeh, L.H.; Qian, S. Gate Modulation of Proton Transport in a Nanopore. Phys. Chem. Chem. Phys.
**2016**, 18, 7449–7458. [Google Scholar] [CrossRef] [PubMed] - Keunings, R. On the High Weissenberg Number Problem. J. Non-Newton. Fluid Mech.
**1986**, 20, 209–226. [Google Scholar] [CrossRef] - El-Kareh, A.W.; Leal, L.G. Existence of Solutions for All Deborah Numbers for a Non-Newtonian Model Modified to Include Diffusion. J. Non-Newton. Fluid Mech.
**1989**, 33, 257–287. [Google Scholar] [CrossRef] - Min, T.; Yoo, J.Y.; Choi, H. Effect of Spatial Discretization Schemes on Numerical Solutions of Viscoelastic Fluid Flows. J. Non-Newton. Fluid Mech.
**2001**, 100, 27–47. [Google Scholar] [CrossRef] - Vaithianathan, T.; Collins, L.R. Numerical Approach to Simulating Turbulent Flow of a Viscoelastic Polymer Solution. J. Comput. Phys.
**2003**, 187, 1–21. [Google Scholar] [CrossRef] - Fattal, R.; Kupferman, R. Constitutive Laws for the Matrix-Logarithm of the Conformation Tensor. J. Non-Newton. Fluid Mech.
**2004**, 123, 281–285. [Google Scholar] [CrossRef] - Zhang, H.N.; Li, D.Y.; Li, X.B.; Cai, W.H.; Li, F.C. Numerical Simulation of Heat Transfer Process of Viscoelastic Fluid Flow at High Weissenberg Number by Log-Conformation Reformulation. J. Fluids Eng. Trans. ASME
**2017**, 139, 091402. [Google Scholar] [CrossRef] - Leonard, B.P. A Stable and Accurate Convective Modelling Procedure Based on Quadratic Upstream Interpolation. Comput. Meth. Appl. Mech. Eng.
**1979**, 19, 59–98. [Google Scholar] [CrossRef] - Harten, A. High Resolution Schemes for Hyperbolic Conservation Laws. J. Comput. Phys.
**1983**, 49, 357–393. [Google Scholar] [CrossRef] - Issa, R.I. Solution of the Implicitly Discretized Fluid Flow Equations by Operator-Splitting. J. Comput. Phys.
**1986**, 62, 40–65. [Google Scholar] [CrossRef] - Issa, R.I.; Gosman, A.D.; Watkins, A.P. The Computation of Compressible and Incompressible Recirculating Flow by a Non-Iterative Implicit Scheme. J. Comput. Phys.
**1986**, 62, 66–82. [Google Scholar] [CrossRef]

**Figure 2.**Dimensionless y-component velocity profile for Newtonian and viscoelastic fluids at different Wi: analytical results of Afonso et al. [31], (solid line) and current numerical results (symbol).

**Figure 3.**The distribution of dimensionless y-component velocity for various Wi at $kH/2=16.45$. Inset: Dependence of dimensionless flow rate on Wi.

**Figure 4.**The distribution of dimensionless y-component velocity for various Wi at $kH/2=1.64$. Inset: Dependence of dimensionless flow rate on Wi.

**Figure 5.**The distribution of dimensionless y-component velocity for various Wi at $kH/2=0.52$. Inset: Dependence of dimensionless flow rate on Wi.

**Figure 8.**The distribution of dimensionless y-component velocity for various $\beta $ at $kH/2=1.64$ and $\mathsf{\epsilon}=0.25$.

**Figure 9.**The distribution of dimensionless y-component velocity for various $\epsilon $ at $kH/2=1.64\mathrm{and}\beta =0.1$.

**Table 1.**The maximum velocity at the centerline and the enhancement of the maximum velocity for different $kH/2$ when Wi = 3.

Variable | $\mathit{k}\mathit{H}/2=16.45$ | $\mathit{k}\mathit{H}/2=1.64$ | $\mathit{k}\mathit{H}/2=0.52$ |
---|---|---|---|

Maximum velocity | 0.15 | 0.34 | 0.22 |

Enhancement of the maximum velocity | 2.50 | 2.05 | 1.75 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mei, L.; Zhang, H.; Meng, H.; Qian, S.
Electroosmotic Flow of Viscoelastic Fluid in a Nanoslit. *Micromachines* **2018**, *9*, 155.
https://doi.org/10.3390/mi9040155

**AMA Style**

Mei L, Zhang H, Meng H, Qian S.
Electroosmotic Flow of Viscoelastic Fluid in a Nanoslit. *Micromachines*. 2018; 9(4):155.
https://doi.org/10.3390/mi9040155

**Chicago/Turabian Style**

Mei, Lanju, Hongna Zhang, Hongxia Meng, and Shizhi Qian.
2018. "Electroosmotic Flow of Viscoelastic Fluid in a Nanoslit" *Micromachines* 9, no. 4: 155.
https://doi.org/10.3390/mi9040155