Extension of the Improved Bounce-Back Scheme for Electrokinetic Flow in the Lattice Boltzmann Method
<p>Schematic illustrations for the mid-point bounce-back scheme.</p> "> Figure 2
<p>Electric potential distributions from our LBM simulation (symbols) and the analytical solution (black lines) with different height between two identically charged plates in an electrolyte solution.</p> "> Figure 3
<p>Electric potential distributions from present treatment, classical Bounce-Back treatment and the analytical solution with different offset (red symbol <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mtext>=</mtext> <mn>0.5</mn> </mrow> </semantics> </math>; blue symbol <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mtext>=</mtext> <mn>0.7</mn> </mrow> </semantics> </math>; green symbol <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mtext>=</mtext> <mn>0.2</mn> </mrow> </semantics> </math>).</p> "> Figure 4
<p>The error for electric potential between two identically charged plates in an electrolyte solution.</p> "> Figure 5
<p>Electric potential distributions between two coaxial circular surfaces.</p> "> Figure 6
<p>The electric potential distribution (<b>a</b>) and (<b>b</b>) and electro-osmotic flow (<b>c</b>) and (<b>d</b>) around the spherical particle in the <math display="inline"> <semantics> <mrow> <mtext>y</mtext> <mo>=</mo> <msub> <mi>y</mi> <mi>c</mi> </msub> </mrow> </semantics> </math> plane.</p> ">
Abstract
:1. Introduction
2. Macroscopic Governing Equations for EOF
3. Numerical Method
3.1. Lattice Boltzmann Model for the NS Equations
3.2. Lattice Boltzmann Model for Poisson–Boltzmann Equation
3.3. Boundary Conditions
4. Validation and Discussions
4.1. Electric Potential with Flat Surface
4.2. Electric Potential with Complex Geometry
4.3. Application in Electro-Osmotic Flows
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- Stone, H.A.; Stroock, A.D.; Ajdari, A. Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid. Mech. 2004, 36, 381–411. [Google Scholar] [CrossRef]
- Darguji, H.; Yang, P.D.; Majumdar, A. Ion transport in nanofluidic channels. Nano Lett. 2004, 4, 137–142. [Google Scholar] [CrossRef]
- Ho, C.M.; Tai, Y.C. Micro-electro-mechanical-systems (MEMS) and fluid flow. Annu. Rev. Fluid. Mech. 1998, 30, 579–612. [Google Scholar] [CrossRef]
- Wang, C.Y. Fundamental models for fuel cell engineering. Chem. Rev. 2004, 104, 4727–4766. [Google Scholar] [CrossRef] [PubMed]
- Sinton, D.; Li, D.Q. Electroosmotic velocity profiles in microchannels. Colloids Surf. A Physicochem. Eng. Asp. 2003, 222, 273–283. [Google Scholar] [CrossRef]
- Li, B.; Kwok, D.Y. Lattice Boltzmann model of microfluidics with high Reynolds numbers in the presence of external forces. Langmuir 2003, 19, 3041–3048. [Google Scholar] [CrossRef]
- Li, B.; Kwok, D.Y. Electrokinetic microfluidic phenomena by a lattice Boltzmann model using a modified Poisson–Boltzmann equation with an excluded volume effect. J. Chem. Phys. 2004, 120, 947–953. [Google Scholar] [CrossRef] [PubMed]
- Tian, F.Z.; Li, B.M.; Kwok, D.Y. Tradeoff between mixing and transport for electroosmotic flow in heterogeneous microchannels with non-uniform surface potentials. Langmuir 2005, 21, 1126–1131. [Google Scholar] [CrossRef] [PubMed]
- Chakraborty, S. Augmentation of peristaltic microflows through electro-osmotic mechanisms. J. Phys. D Appl. Phys. 2006, 39. [Google Scholar] [CrossRef]
- Benzi, R.; Biferale, L.; Sbragaglia, M.; Succi, S.; Toschi, F. Mesoscopic two-phase model for describing apparent slip in micro-channel flows. Europhys. Lett. 2006, 74. [Google Scholar] [CrossRef]
- Sbragaglia, M.; Benzi, R.; Biferale, L.; Succi, S.; Toschi, F. Surface roughness-hydrophobicity coupling in microchannel and nanochannel flows. Phys. Rev. E 2006, 97, 204503. [Google Scholar] [CrossRef]
- Karniadakis, G.; Beskok, A.; Aluru, N. Microflows and nanoflows; Springer: New York, NY, USA, 2005. [Google Scholar]
- Chen, S.; Doolen, G.D. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 1998, 30, 329–364. [Google Scholar] [CrossRef]
- Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond; Oxford University Press: Oxford, UK, 2001. [Google Scholar]
- Zhang, J.F. Lattice Boltzmann method for microfluidics: models and applications. Microfluid Nanofluid 2011, 10. [Google Scholar] [CrossRef]
- Benzi, R.; Succi, S.; Vergassola, M. The Lattice Boltzmann equation-theory and applications. Phys. Rep. 1992, 222, 145–197. [Google Scholar] [CrossRef]
- De Rosis, A.; Ubertini, S.; Ubertini, F. A comparison between the interpolated bounce-back scheme and the immersed boundary method to treat solid boundary conditions for laminar flows in the lattice Boltzmann framework. J. Sci. Comput. 2014, 61, 477–489. [Google Scholar] [CrossRef]
- Warren, P.B. Electroviscous transport problems via lattice-Boltzmann. Int. J. Mod. Phys. C 1997, 8, 889–898. [Google Scholar] [CrossRef]
- He, X.Y.; Li, N. Lattice Boltzmann simulation of electrochemical systems. Comput. Phys. Commun. 2000, 129, 158–166. [Google Scholar] [CrossRef]
- Zhang, J.; Kwok, D.Y. A 2D lattice Boltzmann study on electrohydrodynamic drop deformation with the leaky dielectric theory. J. Comput. Phys. 2005, 206, 150–161. [Google Scholar] [CrossRef]
- Melchionna, S.; Succi, S. Electrorheology in nanopores via lattice Boltzmann simulation. J. Chem. Phys. 2004, 120, 4492–4497. [Google Scholar] [CrossRef] [PubMed]
- Wu, J.; Srinivasan, V.; Xu, J.; Wang, C.Y. Newton–Krylov-Multigrid algorithms for battery simulation. J. Electrochem. Soc. 2002, 149. [Google Scholar] [CrossRef]
- Wang, J.; Wang, M.; Li, Z. Lattice Boltzmann simulations of mixing enhancement by the electro-osmotic flow in microchannels. Mod. Phys. Lett. B 2005, 19, 1515–1518. [Google Scholar] [CrossRef]
- Wang, J.; Wang, M.; Li, Z. Lattice Poisson–Boltzmann simulations of electro-osmotic flows in microchannels. J. Colloid Interface Sci. 2006, 296, 729–736. [Google Scholar] [CrossRef] [PubMed]
- Tang, G.H.; Li, Z.; Wang, J.K.; He, Y.L.; Tao, W.Q. Electroosmotic flow and mixing in microchannels with the lattice Boltzmann method. J. Appl. Phys. 2006, 100, 094908. [Google Scholar] [CrossRef]
- Tang, G.H.; He, Y.L.; Tao, W.Q. Numerical analysis of mixing enhancement for micro-electroosmotic flow. J. Appl. Phys. 2010, 107, 104906. [Google Scholar] [CrossRef]
- Chai, Z.; Guo, Z.; Shi, B. Study of electro-osmotic flows in microchannels packed with variable porosity media via lattice Boltzmann method. J. Appl. Phys. 2007, 101, 014913. [Google Scholar] [CrossRef]
- Wang, M.; Wang, J.; Chen, S.; Pan, N. Electrokinetic pumping effects of charged porous media in microchannels using the lattice Poisson–Boltzmann method. J. Colloid Interface Sci. 2006, 304, 246–253. [Google Scholar] [CrossRef] [PubMed]
- Chai, Z.; Guo, Z.; Zheng, L.; Shi, B. Lattice Boltzmann simulation of surface roughness effect on gaseous flow in a microchannel. J. Appl. Phys. 2008, 104, 014902. [Google Scholar] [CrossRef]
- Wang, M.; Wang, J.; Chen, S. Roughness and cavitations effects on electro-osmotic flows in rough microchannels using the lattice Poisson–Boltzmann methods. J. Comput. Phys. 2007, 226, 836–851. [Google Scholar] [CrossRef]
- Ladd, A.J.C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 1994, 271, 285–309. [Google Scholar] [CrossRef]
- Yin, X.W.; Zhang, J.F. An improved bounce-back scheme for complex boundary conditions in lattice Boltzmann method. J. Comput. Phys. 2012, 231, 4295–4303. [Google Scholar] [CrossRef]
- Chen, Q.; Zhang, X.B.; Zhang, J.F. Improved treatments for general boundary conditions in the lattice Boltzmann method for convection-diffusion and heat transfer processes. Phys. Rev. E 2013, 88, 033304. [Google Scholar] [CrossRef]
- Chen, Q.; Zhang, X.B.; Zhang, J.F. Numerical simulation of Neumann boundary condition in the thermal lattice Boltzmann model. Int. J. Mod. Phys. C 2014, 25. [Google Scholar] [CrossRef]
- Li, L.; Mei, R.W.; Klausner, J.F. Boundary conditions for thermal lattice Boltzmann equation method. J. Comput. Phys. 2013, 237, 366–395. [Google Scholar] [CrossRef]
- Li, L.; Chen, C.; Mei, R.W.; Klausner, J.F. Conjugate heat and mass transfer in the lattice Boltzmann equation method. Phys. Rev. E 2014, 89, 043308. [Google Scholar] [CrossRef]
- Filippova, O.; Hänel, D. Lattice Boltzmann simulation of gas-particle flow in filters. Comput. Fluids 1997, 26, 697–712. [Google Scholar] [CrossRef]
- Bouzidi, M.; Firdaouss, M.; Lallemand, P. Momentum transfer of a Boltzmann-lattice fluid with boundaries. Phys. Fluids 2001, 13, 3452–3459. [Google Scholar] [CrossRef]
- Yu, D.Z.; Mei, R.W.; Luo, L.-S.; Shyy, W. Viscous flow computations with the method of lattice Boltzmann equation. Prog. Aerosp. Sci. 2003, 39, 329–367. [Google Scholar] [CrossRef]
- Oulaid, O.; Chen, Q.; Zhang, J.F. Accurate boundary treatments for lattice Boltzmann simulations of electric fields and electro-kinetic applications. J. Phys. A Math. Theor. 2013, 46. [Google Scholar] [CrossRef]
- Honig, B.; Nicholls, A. classical electrostatics in biology and chemistry. Science 1995, 268, 1144–1149. [Google Scholar] [CrossRef] [PubMed]
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Chen, Q.; Zhou, H.; Jiang, X.; Xu, L.; Li, Q.; Ru, Y. Extension of the Improved Bounce-Back Scheme for Electrokinetic Flow in the Lattice Boltzmann Method. Entropy 2015, 17, 7406-7419. https://doi.org/10.3390/e17117406
Chen Q, Zhou H, Jiang X, Xu L, Li Q, Ru Y. Extension of the Improved Bounce-Back Scheme for Electrokinetic Flow in the Lattice Boltzmann Method. Entropy. 2015; 17(11):7406-7419. https://doi.org/10.3390/e17117406
Chicago/Turabian StyleChen, Qing, Hongping Zhou, Xuesong Jiang, Linyun Xu, Qing Li, and Yu Ru. 2015. "Extension of the Improved Bounce-Back Scheme for Electrokinetic Flow in the Lattice Boltzmann Method" Entropy 17, no. 11: 7406-7419. https://doi.org/10.3390/e17117406