Investigation of Mixed Electro-Osmotic/Poiseuille Slip Flows of Viscoelastic Fluids in Rectangular Microchannels with Hydrophobic Surfaces

Document Type : Research Article

Authors

Mechanical Engineering Department, Center of Excellence in Energy Conversion (CEEC), Sharif University of Technology, Tehran, Iran

Abstract

In this paper, we conduct a numerical study of mixed electro-osmotic/Poiseuille slip flows of
viscoelastic fluids in microchannels with rectangular cross sections by means of second order finite difference
method. In this regard, the complete form of the PTT-constitutive equation is used to describe the rheological
behavior of the fluid. The numerical results being validated by the same simplified theoretical study reveal an
excellent accuracy with relative error less than 0.3%. Afterward, the extended numerical study is used to investigate the 2D velocity distribution and volumetric flow rate in the presence of wall surface hydrophobicity through rectangular microchannels. In addition, in this investigation, the exact solution of unidirectional electroosmotic flow of PTT-viscoelastic fluids is derived for slit hydrophobic microchannels, and after validating, the solution is used to investigate the rheological behavior of viscoelastic fluids in the range of operating parameters. The results exhibit a uniform effect of hydrophobicity in increasing the profile of 1D velocity distribution in slit microchannels. Finally, in order to determine the stability of the grid network, various under relaxation factors are applied to determine the speed of convergence of finite difference method, and then, by using the analytical procedure, the critical Weissenberg number is introduced as a function of velocity scale ratio and Debye–Hückel parameter. The evaluation of the numerical method in the critical area indicates the stability of viscoelastic fluid flow for the values of the Weissenberg number less than the corresponding critical value in the theoretical analysis.

Highlights

[1] A. Afonso, L. Ferrás, J. Nóbrega, M. Alves, F. Pinho, Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels, Microfluidics and nanofluidics, 16(6) (2014) 1131-1142.

[2] F. Brochard, P. De Gennes, Shear-dependent slippage at a polymer/solid interface, Langmuir, 8(12) (1992) 3033- 3037.

[3] Y. Inn, S.-Q. Wang, Hydrodynamic slip: Polymer adsorption and desorption at melt/solid interfaces, Phys. Rev. Lett., 76(3) (1996) 467.

[4] K. Migler, H. Hervet, L. Leger, Slip transition of a polymer melt under shear stress, Phys. Rev. Lett., 70(3) (1993) 287.

[5] V. Marry, J.-F. Dufrêche, M. Jardat, P. Turq, Equilibrium and electrokinetic phenomena in charged porous media from microscopic and mesoscopic models: electro-osmosis in montmorillonite, Molecular Physics, 101(20) (2003) 3111-3119.

[6] A. Herr, J. Molho, J. Santiago, M. Mungal, T. Kenny, M. Garguilo, Electroosmotic capillary flow with nonuniform zeta potential, Anal. Chem., 72(5) (2000) 1053-1057.

[7] H.A. Stone, A.D. Stroock, A. Ajdari, Engineering flows in small devices: microfluidics toward a lab-on-a-chip, Annu. Rev. Fluid Mech., 36 (2004) 381-411.

[8] M. Gad-el-Hak, The fluid mechanics of microdevices— the Freeman scholar lecture, Journal of Fluids Engineering, 121(1) (1999) 5-33.

[9] Y.L. Zhang, R.V. Craster, O.K. Matar, Surfactant driven flows overlying a hydrophobic epithelium: film rupture in the presence of slip, J. Colloid Interface Sci., 264(1) (2003) 160-175.

[10] D.J. Beebe, G.A. Mensing, G.M. Walker, Physics and applications of microfluidics in biology, Annual review of biomedical engineering, 4(1) (2002) 261-286.

[11] J.C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philosophical Transactions of the royal society of London, 170 (1879) 231-256.

[12] Y. Zhu, S. Granick, Rate-dependent slip of Newtonian liquid at smooth surfaces, Phys. Rev. Lett., 87(9) (2001) 096105.

[13] V.S. Craig, C. Neto, D.R. Williams, Shear-dependent boundary slip in an aqueous Newtonian liquid, Phys. Rev. Lett., 87(5) (2001) 054504.

[14] C. Soong, P. Hwang, J. Wang, Analysis of pressure-driven electrokinetic flows in hydrophobic microchannels with slip-dependent zeta potential, Microfluidics and Nanofluidics, 9(2-3) (2010) 211-223.

[15] J. Jamaati, H. Niazmand, M. Renksizbulut, Pressure-driven electrokinetic slip-flow in planar microchannels, International Journal of Thermal Sciences, 49(7) (2010) 1165-1174.

[16] C. Navier, Mémoire sur les lois du mouvement des fluides, Mémoires de l’Académie Royale des Sciences de l’Institut de France, 6 (1823) 389-440.

[17] D.C. Tretheway, C.D. Meinhart, Apparent fluid slip at hydrophobic microchannel walls, Phys. Fluids, 14(3) (2002) L9-L12.

[18] B.-H. Jo, L.M. Van Lerberghe, K.M. Motsegood, D.J. Beebe, Three-dimensional micro-channel fabrication in polydimethylsiloxane (PDMS) elastomer, Journal of Microelectromechanical Systems, 9(1) (2000) 76-81.

[19] N. Phan‐Thien, A nonlinear network viscoelastic model, J. Rheol., 22(3) (1978) 259-283.

[20] N.P. Thien, R.I. Tanner, A new constitutive equation derived from network theory, J. Non-Newtonian Fluid Mech., 2(4) (1977) 353-365.

[21] D. Li, Electrokinetics in microfluidics, Academic Press, 2004.

[22] M. Chatzimina, G.C. Georgiou, K. Housiadas, S.G. Hatzikiriakos, Stability of the annular Poiseuille flow of a Newtonian liquid with slip along the walls, J. Non- Newtonian Fluid Mech., 159(1) (2009) 1-9.

[23] R.H. Pletcher, J.C. Tannehill, D. Anderson, Computational fluid mechanics and heat transfer, CRC Press, 2012.

[24] A. Afonso, M. Alves, F. Pinho, Analytical solution of mixed electro-osmotic/pressure driven flows of viscoelastic fluids in microchannels, J. Non-Newtonian Fluid Mech., 159(1) (2009) 50-63.

[25] S. Dhinakaran, A. Afonso, M. Alves, F. Pinho, Steady viscoelastic fluid flow between parallel plates under electro-osmotic forces: Phan-Thien–Tanner model, J. Colloid Interface Sci., 344(2) (2010) 513-520.

Keywords

References

[1] A. Afonso, L. Ferrás, J. Nóbrega, M. Alves, F. Pinho, Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels, Microfluidics and nanofluidics, 16(6) (2014) 1131-1142.
[2] F. Brochard, P. De Gennes, Shear-dependent slippage at a polymer/solid interface, Langmuir, 8(12) (1992) 3033- 3037.
[3] Y. Inn, S.-Q. Wang, Hydrodynamic slip: Polymer adsorption and desorption at melt/solid interfaces, Phys. Rev. Lett., 76(3) (1996) 467.
[4] K. Migler, H. Hervet, L. Leger, Slip transition of a polymer melt under shear stress, Phys. Rev. Lett., 70(3) (1993) 287.
[5] V. Marry, J.-F. Dufrêche, M. Jardat, P. Turq, Equilibrium and electrokinetic phenomena in charged porous media from microscopic and mesoscopic models: electro-osmosis in montmorillonite, Molecular Physics, 101(20) (2003) 3111-3119.
[6] A. Herr, J. Molho, J. Santiago, M. Mungal, T. Kenny, M. Garguilo, Electroosmotic capillary flow with nonuniform zeta potential, Anal. Chem., 72(5) (2000) 1053-1057.
[7] H.A. Stone, A.D. Stroock, A. Ajdari, Engineering flows in small devices: microfluidics toward a lab-on-a-chip, Annu. Rev. Fluid Mech., 36 (2004) 381-411.
[8] M. Gad-el-Hak, The fluid mechanics of microdevices— the Freeman scholar lecture, Journal of Fluids Engineering, 121(1) (1999) 5-33.
[9] Y.L. Zhang, R.V. Craster, O.K. Matar, Surfactant driven flows overlying a hydrophobic epithelium: film rupture in the presence of slip, J. Colloid Interface Sci., 264(1) (2003) 160-175.
[10] D.J. Beebe, G.A. Mensing, G.M. Walker, Physics and applications of microfluidics in biology, Annual review of biomedical engineering, 4(1) (2002) 261-286.
[11] J.C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philosophical Transactions of the royal society of London, 170 (1879) 231-256.
[12] Y. Zhu, S. Granick, Rate-dependent slip of Newtonian liquid at smooth surfaces, Phys. Rev. Lett., 87(9) (2001) 096105.
[13] V.S. Craig, C. Neto, D.R. Williams, Shear-dependent boundary slip in an aqueous Newtonian liquid, Phys. Rev. Lett., 87(5) (2001) 054504.
[14] C. Soong, P. Hwang, J. Wang, Analysis of pressure-driven electrokinetic flows in hydrophobic microchannels with slip-dependent zeta potential, Microfluidics and Nanofluidics, 9(2-3) (2010) 211-223.
[15] J. Jamaati, H. Niazmand, M. Renksizbulut, Pressure-driven electrokinetic slip-flow in planar microchannels, International Journal of Thermal Sciences, 49(7) (2010) 1165-1174.
[16] C. Navier, Mémoire sur les lois du mouvement des fluides, Mémoires de l’Académie Royale des Sciences de l’Institut de France, 6 (1823) 389-440.
[17] D.C. Tretheway, C.D. Meinhart, Apparent fluid slip at hydrophobic microchannel walls, Phys. Fluids, 14(3) (2002) L9-L12.
[18] B.-H. Jo, L.M. Van Lerberghe, K.M. Motsegood, D.J. Beebe, Three-dimensional micro-channel fabrication in polydimethylsiloxane (PDMS) elastomer, Journal of Microelectromechanical Systems, 9(1) (2000) 76-81.
[19] N. Phan‐Thien, A nonlinear network viscoelastic model, J. Rheol., 22(3) (1978) 259-283.
[20] N.P. Thien, R.I. Tanner, A new constitutive equation derived from network theory, J. Non-Newtonian Fluid Mech., 2(4) (1977) 353-365.
[21] D. Li, Electrokinetics in microfluidics, Academic Press, 2004.
[22] M. Chatzimina, G.C. Georgiou, K. Housiadas, S.G. Hatzikiriakos, Stability of the annular Poiseuille flow of a Newtonian liquid with slip along the walls, J. Non- Newtonian Fluid Mech., 159(1) (2009) 1-9.
[23] R.H. Pletcher, J.C. Tannehill, D. Anderson, Computational fluid mechanics and heat transfer, CRC Press, 2012.
[24] A. Afonso, M. Alves, F. Pinho, Analytical solution of mixed electro-osmotic/pressure driven flows of viscoelastic fluids in microchannels, J. Non-Newtonian Fluid Mech., 159(1) (2009) 50-63.
[25] S. Dhinakaran, A. Afonso, M. Alves, F. Pinho, Steady viscoelastic fluid flow between parallel plates under electro-osmotic forces: Phan-Thien–Tanner model, J. Colloid Interface Sci., 344(2) (2010) 513-520.
AUT Journal of Mechanical Engineering
Volume 1, Issue 1
June 2017
Pages 3-12

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