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June  2011, 15(4): 1019-1044. doi: 10.3934/dcdsb.2011.15.1019

## Boundary integral equation approach for stokes slip flow in rotating mixers

 1 Grupo de Energía y Termodinámica, Escuela de Ingenierías, Universidad Pontificia Bolivariana, Medellín, Circular 1 No. 73-34, Colombia, Colombia 2 Division of Energy and Sustainability, The University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

Received  March 2010 Revised  June 2010 Published  March 2011

In order to employ continuum models in the analysis of the flow behaviour of a viscous Newtonian fluid at micro scale devices, it is necessary to consider at the wall surfaces appropriate slip boundary conditions instead of the classical non-slip condition. To account for the slip condition at the nano-scale, we used the Navier's type boundary condition that relates the tangential fluid velocity at the boundaries to the tangential shear rate. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the Stokes velocity field, which directly incorporates into the integral equations the local tangential shear rate at the wall surfaces, is presented. This formulation is used to numerically simulate concentric and eccentric rotating Couette mixers and a Single rotor mixer, including the effect of thermal creep in cases of rarefied gases. The numerical results obtained for the Couette mixers, concentric and eccentric, are validated again the corresponding analytical solutions, showing excellent agreements.
Citation: César Nieto, Mauricio Giraldo, Henry Power. Boundary integral equation approach for stokes slip flow in rotating mixers. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 1019-1044. doi: 10.3934/dcdsb.2011.15.1019
##### References:
 [1] A. Frangi, G. Spinola and B. Vigna, On the evaluation of damping in MEMS in the slipflow regime, Int. J. Numer. Meth. Engng., 68 (2006), 1031-1051. doi: doi:10.1002/nme.1749. [2] A. J. Burton and G. F. Miller, The application of integral methods for the numerical solution of boundary value problems, Proc. R. Soc. A, 232 (2008), 201-210. [3] C. Neto, D. R. Evans, E. Bonaccurso, H. Butt and V. Craig, Boundary slip in Newtonian liquids: A review of experimental studies, Rep. Progr. Phys., 68 (2005), 2859-2897. doi: doi:10.1088/0034-4885/68/12/R05. [4] D. Lockerby, J. M. Reese, D. R. Emerson and R. W. Barber, Velocity boundary condition at solid walls in rarefied gas calculations, Physical Review E, 70 (2004), 017303-1 - 017303-9. [5] D. C. Tretheway and C. D. Meinhart, A generating mechanism for apparent fluid slip in hydrophobic microchannels, Phys. Fluids, 16 (2004), 1509-1515. doi: doi:10.1063/1.1669400. [6] E. A. Mansur, Y. Mingxing, W. Yundong and D. Youyuan, A state-of-the-art review of mixing in microfluidic mixers, Chin. J. Chem. Eng., 16 (2008), 503-516. doi: doi:10.1016/S1004-9541(08)60114-7. [7] G. Hu and D. Li, Multiscale phenomena in microfluidics and nanofluidics, Chemical Engineering Science, 62 (2007), 3443-3454. doi: doi:10.1016/j.ces.2006.11.058. [8] G. Karniadakis, A. Beskok and N. Aluru, "Microflows and Nanoflows: Fundamentals and Simulation," 1st, edition, (). [9] H. Chen, J. Jin, P. Zhang and P. Lu, Multi-variable non-singular BEM for 2-D potential problems, Tsinghua Science and Technology, 10 (2005), 43-50. doi: doi:10.1016/S1007-0214(05)70007-9. [10] H. Luo and C. Pozrikidis, Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall, J. Engrg. Math, 62 (2008), 1-21. doi: doi:10.1007/s10665-007-9170-6. [11] H. Power and L. C. Wrobel, "Boundary Integral Methods in Fluid Mechanics," 1st, edition, (). [12] I. Ashino and K. Yoshida, Slow motion between eccentric rotating cylinders, Bull. JSME, 18 (1975), 280-285. [13] I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, edition, (). [14] Jian Ding and Wenjing Ye, A fast integral approach for drag force calculation due to oscillatory slip stokes flows, Int. J. Numer. Meth. Engng., 60 (2004), 1535-1567. doi: doi:10.1002/nme.1013. [15] J. Maureau, M. C. Sharatchandra, M. Sen and M. Gad-el-Hak, Flow and load characteristics of microbearings with slip, J. Micromech. Microeng., 7 (1997), 55-64. doi: doi:10.1088/0960-1317/7/2/003. [16] J. Telles, A self-adaptative coordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. Numer. Methods Engrg., 24 (1987), 959-973. doi: doi:10.1002/nme.1620240509. [17] K. F. Lei and W. J. Li, A novel in-plane microfluidic mixer using vortex pumps for fluidic discretization, JALA, 13 (2008), 227-236. [18] Long-Sheng Kuo and Ping-Hei Chen, A unified approach for nonslip and slip boundary conditions in the lattice Boltzmann method, Comput. & Fluids, 38 (2009), 883-887. doi: doi:10.1016/j.compfluid.2008.09.008. [19] M. Gad-el-Hak, "MEMS: Introduction and Fundamentals," 2nd, edition, (). [20] M. T. Matthews and J. M. Hill, Newtonian flow with nonlinear Navier boundary condition, Acta Mechanica, 191 (2007), 195-217. doi: doi:10.1007/s00707-007-0454-8. [21] N. Nguyen and S. Wereley, "Fundamentals and Applications of Microfluidics," 2nd, edition, (). [22] O. Aydin, M, Avci, Heat and fluid flow characteristics of gases in micropipes, International Journal of Heat and Mass Transfer, 49 (2006), 1723-1730. doi: doi:10.1016/j.ijheatmasstransfer.2005.10.020. [23] O. I. Vinogradova, Slippage of water over hydrophobic surfaces, Int. Journal of Miner. Process, 56 (1999), 31-60. doi: doi:10.1016/S0301-7516(98)00041-6. [24] P. A. Thompson and S. M. Troian, A general boundary condition for liquid flow at solid surfaces, Nature, 389 (1997), 360-362. doi: doi:10.1038/39475. [25] R. Courant and D. Hilbert, "Methods of Mathematical Physics," 3rd, edition, (). [26] R. W. Barber, Y. Sun, X. J. Gu and D. R. Emerson, Isothermal slip flow over curved surfaces, Vacuum, 76 (2004), 73-81. doi: doi:10.1016/j.vacuum.2004.05.012. [27] S. G. Mikhlin, "Multidimensional Singular Integrals and Integral Equations," 1st, edition, (). [28] S. Yuhong, R. W. Barber and D. R. Emerson, Inverted velocity profiles in rarefied cylindrical Couette gas flow and the impact of the accommodation coefficient, Phys. Fluids, 17 (2005), 047102-1-047102-7. doi: doi:10.1063/1.1868034. [29] T. Glatzel, C. Littersta, C. Cupelli, T. Lindemann, C. Moosmann, R. Niekrawietz, W. Streule, R. Zengerle and P. Koltay, Computational fluid dynamics (CFD) software tools for microfluidic applications ?A case study, Comput. & Fluids, 37 (2008), 218-235. doi: doi:10.1016/j.compfluid.2007.07.014. [30] V. Hessel, H. Lwe and F. Schnfeld, Micromixers: A review on passive and active mixing principles, Chemical Engineering Science, 60 (2005), 2479-2501. doi: doi:10.1016/j.ces.2004.11.033. [31] W. F. Florez and H. Power, Multi-domain mass conservative dual reciprocity method for the solution of the non-Newtonian Stokes equations, Appl. Math. Modelling, 26 (2002), 397-419. doi: doi:10.1016/S0307-904X(01)00044-0. [32] Xiaolin Li and Jialin Zhu, Meshless Galerkin analysis of Stokes slip flow with boundary integral equations, Int. J. Numer. Meth. Fluids, 61 (2009), 1201-1226. doi: doi:10.1002/fld.1991. [33] Xiaojin Wei and Yogendra Joshi, Experimental and numerical study of sidewall profile effects on flow and heat transfer inside microchannels, International Journal of Heat and Mass Transfer, 50 (2007), 4640-4651. doi: doi:10.1016/j.ijheatmasstransfer.2007.03.020.

show all references

##### References:
 [1] A. Frangi, G. Spinola and B. Vigna, On the evaluation of damping in MEMS in the slipflow regime, Int. J. Numer. Meth. Engng., 68 (2006), 1031-1051. doi: doi:10.1002/nme.1749. [2] A. J. Burton and G. F. Miller, The application of integral methods for the numerical solution of boundary value problems, Proc. R. Soc. A, 232 (2008), 201-210. [3] C. Neto, D. R. Evans, E. Bonaccurso, H. Butt and V. Craig, Boundary slip in Newtonian liquids: A review of experimental studies, Rep. Progr. Phys., 68 (2005), 2859-2897. doi: doi:10.1088/0034-4885/68/12/R05. [4] D. Lockerby, J. M. Reese, D. R. Emerson and R. W. Barber, Velocity boundary condition at solid walls in rarefied gas calculations, Physical Review E, 70 (2004), 017303-1 - 017303-9. [5] D. C. Tretheway and C. D. Meinhart, A generating mechanism for apparent fluid slip in hydrophobic microchannels, Phys. Fluids, 16 (2004), 1509-1515. doi: doi:10.1063/1.1669400. [6] E. A. Mansur, Y. Mingxing, W. Yundong and D. Youyuan, A state-of-the-art review of mixing in microfluidic mixers, Chin. J. Chem. Eng., 16 (2008), 503-516. doi: doi:10.1016/S1004-9541(08)60114-7. [7] G. Hu and D. Li, Multiscale phenomena in microfluidics and nanofluidics, Chemical Engineering Science, 62 (2007), 3443-3454. doi: doi:10.1016/j.ces.2006.11.058. [8] G. Karniadakis, A. Beskok and N. Aluru, "Microflows and Nanoflows: Fundamentals and Simulation," 1st, edition, (). [9] H. Chen, J. Jin, P. Zhang and P. Lu, Multi-variable non-singular BEM for 2-D potential problems, Tsinghua Science and Technology, 10 (2005), 43-50. doi: doi:10.1016/S1007-0214(05)70007-9. [10] H. Luo and C. Pozrikidis, Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall, J. Engrg. Math, 62 (2008), 1-21. doi: doi:10.1007/s10665-007-9170-6. [11] H. Power and L. C. Wrobel, "Boundary Integral Methods in Fluid Mechanics," 1st, edition, (). [12] I. Ashino and K. Yoshida, Slow motion between eccentric rotating cylinders, Bull. JSME, 18 (1975), 280-285. [13] I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, edition, (). [14] Jian Ding and Wenjing Ye, A fast integral approach for drag force calculation due to oscillatory slip stokes flows, Int. J. Numer. Meth. Engng., 60 (2004), 1535-1567. doi: doi:10.1002/nme.1013. [15] J. Maureau, M. C. Sharatchandra, M. Sen and M. Gad-el-Hak, Flow and load characteristics of microbearings with slip, J. Micromech. Microeng., 7 (1997), 55-64. doi: doi:10.1088/0960-1317/7/2/003. [16] J. Telles, A self-adaptative coordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. Numer. Methods Engrg., 24 (1987), 959-973. doi: doi:10.1002/nme.1620240509. [17] K. F. Lei and W. J. Li, A novel in-plane microfluidic mixer using vortex pumps for fluidic discretization, JALA, 13 (2008), 227-236. [18] Long-Sheng Kuo and Ping-Hei Chen, A unified approach for nonslip and slip boundary conditions in the lattice Boltzmann method, Comput. & Fluids, 38 (2009), 883-887. doi: doi:10.1016/j.compfluid.2008.09.008. [19] M. Gad-el-Hak, "MEMS: Introduction and Fundamentals," 2nd, edition, (). [20] M. T. Matthews and J. M. Hill, Newtonian flow with nonlinear Navier boundary condition, Acta Mechanica, 191 (2007), 195-217. doi: doi:10.1007/s00707-007-0454-8. [21] N. Nguyen and S. Wereley, "Fundamentals and Applications of Microfluidics," 2nd, edition, (). [22] O. Aydin, M, Avci, Heat and fluid flow characteristics of gases in micropipes, International Journal of Heat and Mass Transfer, 49 (2006), 1723-1730. doi: doi:10.1016/j.ijheatmasstransfer.2005.10.020. [23] O. I. Vinogradova, Slippage of water over hydrophobic surfaces, Int. Journal of Miner. Process, 56 (1999), 31-60. doi: doi:10.1016/S0301-7516(98)00041-6. [24] P. A. Thompson and S. M. Troian, A general boundary condition for liquid flow at solid surfaces, Nature, 389 (1997), 360-362. doi: doi:10.1038/39475. [25] R. Courant and D. Hilbert, "Methods of Mathematical Physics," 3rd, edition, (). [26] R. W. Barber, Y. Sun, X. J. Gu and D. R. Emerson, Isothermal slip flow over curved surfaces, Vacuum, 76 (2004), 73-81. doi: doi:10.1016/j.vacuum.2004.05.012. [27] S. G. Mikhlin, "Multidimensional Singular Integrals and Integral Equations," 1st, edition, (). [28] S. Yuhong, R. W. Barber and D. R. Emerson, Inverted velocity profiles in rarefied cylindrical Couette gas flow and the impact of the accommodation coefficient, Phys. Fluids, 17 (2005), 047102-1-047102-7. doi: doi:10.1063/1.1868034. [29] T. Glatzel, C. Littersta, C. Cupelli, T. Lindemann, C. Moosmann, R. Niekrawietz, W. Streule, R. Zengerle and P. Koltay, Computational fluid dynamics (CFD) software tools for microfluidic applications ?A case study, Comput. & Fluids, 37 (2008), 218-235. doi: doi:10.1016/j.compfluid.2007.07.014. [30] V. Hessel, H. Lwe and F. Schnfeld, Micromixers: A review on passive and active mixing principles, Chemical Engineering Science, 60 (2005), 2479-2501. doi: doi:10.1016/j.ces.2004.11.033. [31] W. F. Florez and H. Power, Multi-domain mass conservative dual reciprocity method for the solution of the non-Newtonian Stokes equations, Appl. Math. Modelling, 26 (2002), 397-419. doi: doi:10.1016/S0307-904X(01)00044-0. [32] Xiaolin Li and Jialin Zhu, Meshless Galerkin analysis of Stokes slip flow with boundary integral equations, Int. J. Numer. Meth. Fluids, 61 (2009), 1201-1226. doi: doi:10.1002/fld.1991. [33] Xiaojin Wei and Yogendra Joshi, Experimental and numerical study of sidewall profile effects on flow and heat transfer inside microchannels, International Journal of Heat and Mass Transfer, 50 (2007), 4640-4651. doi: doi:10.1016/j.ijheatmasstransfer.2007.03.020.
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