Numerical Simulation of Non-Newtonian Pseudo-Plastic Fluid in a Micro-Channel Using the Lattice Boltzmann Method

ABSTRACT

In this paper, the power-law model for a non-Newtonian (pseudo-plastic) flow is investigated numerically. The D2Q9 model of Lattice Boltzmann method is used to simulate the micro-channel flow with expansion geometries. This geometry is made by two squared or trapezoid cavities at the bottom and top of the channel which can simulate an artery with local expansion. The cavities are displaced along the channel and the effects of the displacements are investigated for inline structures and staggered ones (anti-symmetric expansion). The method is validated by a Poiseuille flow of the power-law fluid in a duct. Validation is performed for two cases: The Newtonian fluid and the shear thinning fluid (pseudo-plastic) with n = 0.5. The results are discussed in four parts: 1) Pressure drop; It is shown that the pressure drop along the channel for inline cavities is much more than the pressure drop along the staggered structures. 2) Velocity profiles; the velocity profiles are sketched at the centerline of the cavities. The effects of pseudo-plasticity are discussed. 3) Shear stress distribution; the shear stress is computed and shown in the domain. The Newtonian and non-Newto- nian fluids are discussed and the effect of the power n on shear stress is argued. 4) Generated vortices in the cavities are also presented. The shape of the vortices is depicted for various cases. The results for these cases are talked over and it is found that the vortices will be removed for flows with n smaller than 0.5.

In this paper, the power-law model for a non-Newtonian (pseudo-plastic) flow is investigated numerically. The D2Q9 model of Lattice Boltzmann method is used to simulate the micro-channel flow with expansion geometries. This geometry is made by two squared or trapezoid cavities at the bottom and top of the channel which can simulate an artery with local expansion. The cavities are displaced along the channel and the effects of the displacements are investigated for inline structures and staggered ones (anti-symmetric expansion). The method is validated by a Poiseuille flow of the power-law fluid in a duct. Validation is performed for two cases: The Newtonian fluid and the shear thinning fluid (pseudo-plastic) with n = 0.5. The results are discussed in four parts: 1) Pressure drop; It is shown that the pressure drop along the channel for inline cavities is much more than the pressure drop along the staggered structures. 2) Velocity profiles; the velocity profiles are sketched at the centerline of the cavities. The effects of pseudo-plasticity are discussed. 3) Shear stress distribution; the shear stress is computed and shown in the domain. The Newtonian and non-Newto- nian fluids are discussed and the effect of the power n on shear stress is argued. 4) Generated vortices in the cavities are also presented. The shape of the vortices is depicted for various cases. The results for these cases are talked over and it is found that the vortices will be removed for flows with n smaller than 0.5.

Cite this paper

nullH. Hamedi and M. Rahimian, "Numerical Simulation of Non-Newtonian Pseudo-Plastic Fluid in a Micro-Channel Using the Lattice Boltzmann Method,"*World Journal of Mechanics*, Vol. 1 No. 5, 2011, pp. 231-242. doi: 10.4236/wjm.2011.15029.

nullH. Hamedi and M. Rahimian, "Numerical Simulation of Non-Newtonian Pseudo-Plastic Fluid in a Micro-Channel Using the Lattice Boltzmann Method,"

References

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[2] J. Boyd, J. Buick and S. Green, “A Second-Order Accurate Lattice Boltzmann Non-Newtonian Flow Model,” Journal of Physics A: Mathematical and General, Vol. 39, 2006, pp. 14241-14247. doi:10.1088/0305-4470/39/46/001

[3] S. P. Sullivan, L. F. Gladden and M. L. Johns, “Simulation of Power-Law Fluid Flow through Porous Media Using Lattice Boltzmann Techniques,” Journal of Non-Newtonian Fluid Mechanics, Vol. 133, No. 2-3, 2006, pp. 91-98. doi:10.1016/j.jnnfm.2005.11.003

[4] J. Psihogios, M. E. Kainourgiakis, A. G. Yiotis, A. Th. Papaioannou and A. K. Stubos, “A Lattice Boltzmann Study of Non-Newtonian Flow in Digitally Reconstructed Porous Domains,” Transport in Porous Media, Vol. 70, No. 2, 2007, pp. 279-292. doi:10.1007/s11242-007-9099-2

[5] M. Yoshino, Y. Hotta, T. Hirozane and M. Endob, “A Numerical Method for Incompressible Non-Newtonian Fluid Flows Based on the Lattice Boltzmann Method,” Journal of Non-Newtonian Fluid Mechanics, Vol. 147, No. 1-2, 2007, pp. 69-78. doi:10.1016/j.jnnfm.2007.07.007

[6] M. Ashrafizaadeh and H. Bakhshaei, “A Comparison of Non-Newtonian Models for Lattice Boltzmann Blood Flow Simulations,” Computers and Mathematics with Applications, Vol. 58, No. 5, 2009, pp. 1045-1054. doi:10.1016/j.camwa.2009.02.021

[7] P. H. Kao and R. J. Yang, “An Investigation into Curved and Moving Boundary Treatments in the Lattice Boltzmann Method,” Journal of Computational Physics, Vol. 227, No. 11, 2008, pp. 5671-5690. doi:10.1016/j.jcp.2008.02.002

[8] Y Sui, Y. T. Chew, P. Roy and H. T. Low, “A Hybrid Immersed-Boundary and Multi-Block Lattice Boltzmann Method for Simulating Fluid and Moving-Boundaries Interactions,” International Journal for Numerical Methods in Fluids, Vol. 53, No. 11, 2007, pp. 1727-1754. doi:10.1002/fld.1381

[9] J.Wu, C. Shu and Y. H. Zhang, “Simulation on Incompressible Viscous Flows around Moving Objects by a Variant of Immersed Boundary-Lattice Boltzmann Method,” International Journal for Numerical Methods in Fluids, Vol. 62, 2009, pp. 327-354.

[10] Ch. W. Macosko, “Rheology Principles, Measurements and Aplications,” 1st Edition, Wiley-VCH, 1994.

[11] Y. H. Qian and S. Chen, “Finite Size Effect in Lattice-BGK Models,” International Journal of Modern Physics C (IJMPC), Vol. 8, No. 4, 1977, pp. 763-771. doi:10.1142/S0129183197000655

[12] M. C. Sukop and D. T. Thorne, “Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers,” Springer, Heidelberg, 2006.

[13] S. Succi, “The Lattice Boltzmann Equation for Fluid Dynamics and Beyond,” Oxford, 2001.

[14] Q. Zou and X. He, “On Pressure and Velocity Flow Boundary Conditions and Bounceback for the Lattice Boltzmann BGK Model,” Physics of Fluids, Vol. 9, 1997, pp. 1591-1598. doi:10.1063/1.869307

[1] S. Gabbanelli, G. Drazer and J. Koplik, “Lattice Boltzmann Method for Non-Newtonian (Power-Law) Fluids,” Physical Review E, Vol. 72, 2005, p. 046312. doi:10.1103/PhysRevE.72.046312

[2] J. Boyd, J. Buick and S. Green, “A Second-Order Accurate Lattice Boltzmann Non-Newtonian Flow Model,” Journal of Physics A: Mathematical and General, Vol. 39, 2006, pp. 14241-14247. doi:10.1088/0305-4470/39/46/001

[3] S. P. Sullivan, L. F. Gladden and M. L. Johns, “Simulation of Power-Law Fluid Flow through Porous Media Using Lattice Boltzmann Techniques,” Journal of Non-Newtonian Fluid Mechanics, Vol. 133, No. 2-3, 2006, pp. 91-98. doi:10.1016/j.jnnfm.2005.11.003

[4] J. Psihogios, M. E. Kainourgiakis, A. G. Yiotis, A. Th. Papaioannou and A. K. Stubos, “A Lattice Boltzmann Study of Non-Newtonian Flow in Digitally Reconstructed Porous Domains,” Transport in Porous Media, Vol. 70, No. 2, 2007, pp. 279-292. doi:10.1007/s11242-007-9099-2

[5] M. Yoshino, Y. Hotta, T. Hirozane and M. Endob, “A Numerical Method for Incompressible Non-Newtonian Fluid Flows Based on the Lattice Boltzmann Method,” Journal of Non-Newtonian Fluid Mechanics, Vol. 147, No. 1-2, 2007, pp. 69-78. doi:10.1016/j.jnnfm.2007.07.007

[6] M. Ashrafizaadeh and H. Bakhshaei, “A Comparison of Non-Newtonian Models for Lattice Boltzmann Blood Flow Simulations,” Computers and Mathematics with Applications, Vol. 58, No. 5, 2009, pp. 1045-1054. doi:10.1016/j.camwa.2009.02.021

[7] P. H. Kao and R. J. Yang, “An Investigation into Curved and Moving Boundary Treatments in the Lattice Boltzmann Method,” Journal of Computational Physics, Vol. 227, No. 11, 2008, pp. 5671-5690. doi:10.1016/j.jcp.2008.02.002

[8] Y Sui, Y. T. Chew, P. Roy and H. T. Low, “A Hybrid Immersed-Boundary and Multi-Block Lattice Boltzmann Method for Simulating Fluid and Moving-Boundaries Interactions,” International Journal for Numerical Methods in Fluids, Vol. 53, No. 11, 2007, pp. 1727-1754. doi:10.1002/fld.1381

[9] J.Wu, C. Shu and Y. H. Zhang, “Simulation on Incompressible Viscous Flows around Moving Objects by a Variant of Immersed Boundary-Lattice Boltzmann Method,” International Journal for Numerical Methods in Fluids, Vol. 62, 2009, pp. 327-354.

[10] Ch. W. Macosko, “Rheology Principles, Measurements and Aplications,” 1st Edition, Wiley-VCH, 1994.

[11] Y. H. Qian and S. Chen, “Finite Size Effect in Lattice-BGK Models,” International Journal of Modern Physics C (IJMPC), Vol. 8, No. 4, 1977, pp. 763-771. doi:10.1142/S0129183197000655

[12] M. C. Sukop and D. T. Thorne, “Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers,” Springer, Heidelberg, 2006.

[13] S. Succi, “The Lattice Boltzmann Equation for Fluid Dynamics and Beyond,” Oxford, 2001.

[14] Q. Zou and X. He, “On Pressure and Velocity Flow Boundary Conditions and Bounceback for the Lattice Boltzmann BGK Model,” Physics of Fluids, Vol. 9, 1997, pp. 1591-1598. doi:10.1063/1.869307