No Access Submitted: 19 February 2018 Accepted: 10 April 2018 Published Online: 30 April 2018
Physics of Fluids 30, 042007 (2018); https://doi.org/10.1063/1.5026334
We consider the steady, pressure driven flow of a viscous fluid through a microfluidic device having the geometry of a planar spiral duct with a slowly varying curvature and height smaller than width. For this problem, it is convenient to express the Navier–Stokes equations in terms of a non-orthogonal coordinate system. Then, after applying appropriate scalings, the leading order equations admit a relatively simple solution in the central region of the duct cross section. First-order corrections with respect to the duct curvature and aspect ratio parameters are also obtained for this region. Additional correction terms are needed to ensure that no slip and no penetration conditions are satisfied on the side walls. Our solutions allow for a top wall shape that varies with respect to the radial coordinate which allows us to study the flow in a variety of cross-sectional shapes, including trapezoidal-shaped ducts that have been studied experimentally. At leading order, the flow is found to depend on the local height and slope of the top wall within the central region. The solutions are compared with numerical approximations of a classical Dean flow and are found to be in good agreement for a small duct aspect ratio and a slowly varying and small curvature. We conclude that the slowly varying curvature typical of spiral microfluidic devices has a negligible impact on the flow in the sense that locally the flow does not differ significantly from the classical Dean flow through a duct having the same curvature.
This research was supported under Australian Research Council’s Discovery Projects funding scheme (Project No. DP160102021).
  1. 1. J. M. Martel and M. Toner, “Inertial focusing in microfluidics,” Annu. Rev. Biomed. Eng. 16, 371–396 (2014). https://doi.org/10.1146/annurev-bioeng-121813-120704, Google ScholarCrossref, ISI
  2. 2. G. Segre and A. Silberberg, “Radial particle displacements in Poiseuille flow of suspensions,” Nature 189, 209–210 (1961). https://doi.org/10.1038/189209a0, Google ScholarCrossref, ISI
  3. 3. H. L. Goldsmith and S. G. Mason, “Axial migration of particles in Poiseuille flow,” Nature 190, 1095–1096 (1961). https://doi.org/10.1038/1901095a0, Google ScholarCrossref
  4. 4. F. P. Bretherton, “Slow viscous motion round a cylinder in a simple shear,” J. Fluid Mech. 12, 591–613 (1962). https://doi.org/10.1017/s0022112062000415, Google ScholarCrossref
  5. 5. P. G. Saffman, “The lift on a small sphere in a slow shear flow,” J. Fluid Mech. 22, 385–400 (1965). https://doi.org/10.1017/s0022112065000824, Google ScholarCrossref, ISI
  6. 6. R. G. Cox and H. Brenner, “The lateral migration of solid particles in Poiseuille flow—I theory,” Chem. Eng. Sci. 23, 147–173 (1968). https://doi.org/10.1016/0009-2509(68)87059-9, Google ScholarCrossref
  7. 7. B. P. Ho and L. G. Leal, “Inertial migration of rigid spheres in two-dimensional unidirectional flows,” J. Fluid Mech. 65, 365–400 (1974). https://doi.org/10.1017/s0022112074001431, Google ScholarCrossref
  8. 8. P. Vasseur and R. G. Cox, “The lateral migration of a spherical particle in two-dimensional shear flows,” J. Fluid Mech. 78, 385–413 (1976). https://doi.org/10.1017/s0022112076002498, Google ScholarCrossref
  9. 9. P. Ganatos, S. Weinbaum, and R. Pfeffer, “A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion,” J. Fluid Mech. 99, 739–753 (1980). https://doi.org/10.1017/s0022112080000870, Google ScholarCrossref
  10. 10. D. Leighton and A. Acrivos, “The shear-induced migration of particles in concentrated suspensions,” J. Fluid Mech. 181, 415–439 (1987). https://doi.org/10.1017/s0022112087002155, Google ScholarCrossref, ISI
  11. 11. J. A. Schonberg and E. J. Hinch, “Inertial migration of a sphere in Poiseuille flow,” J. Fluid Mech. 203, 517–524 (1989). https://doi.org/10.1017/s0022112089001564, Google ScholarCrossref
  12. 12. E. S. Asmolov, “The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number,” J. Fluid Mech. 381, 63–87 (1999). https://doi.org/10.1017/s0022112098003474, Google ScholarCrossref
  13. 13. K. Hood, S. Lee, and M. Roper, “Inertial migration of a rigid sphere in three-dimensional Poiseuille flow,” J. Fluid Mech. 765, 452–479 (2015). https://doi.org/10.1017/jfm.2014.739, Google ScholarCrossref
  14. 14. D. Di Carlo, “Inertial microfluidics,” Lab Chip 9, 3038–3046 (2009). https://doi.org/10.1039/b912547g, Google ScholarCrossref, ISI
  15. 15. M. E. Warkiani, G. Guan, K. B. Luan, W. C. Lee, A. A. S. Bhagat, P. Kant Chaudhuri, D. S.-W. Tan, W. T. Lim, S. C. Lee, P. C. Y. Chen, C. T. Lim, and J. Han, “Slanted spiral microfluidics for the ultra-fast, label-free isolation of circulating tumor cells,” Lab Chip 14, 128–137 (2014). https://doi.org/10.1039/c3lc50617g, Google ScholarCrossref
  16. 16. C. Liu, C. Xue, J. Sun, and G. Hu, “A generalized formula for inertial lift on a sphere in microchannels,” Lab Chip 16, 884–892 (2016). https://doi.org/10.1039/c5lc01522g, Google ScholarCrossref
  17. 17. A. Shamloo and A. Mashhadian, “Inertial particle focusing in serpentine channels on a centrifugal platform,” Phys. Fluids 30, 012002 (2018). https://doi.org/10.1063/1.5002621, Google ScholarScitation, ISI
  18. 18. M. R. Maxey and J. J. Riley, “Equation of motion for a small rigid sphere in a nonuniform flow,” Phys. Fluids 26, 883–889 (1983). https://doi.org/10.1063/1.864230, Google ScholarScitation, ISI
  19. 19. T. R. Auton, J. C. R. Hunt, and M. Prud’Homme, “The force exerted on a body in inviscid unsteady non-uniform rotational flow,” J. Fluid Mech. 197, 241–257 (1988). https://doi.org/10.1017/s0022112088003246, Google ScholarCrossref
  20. 20. E. Loth and A. J. Dorgan, “An equation of motion for particles of finite Reynolds number and size,” Environ. Fluid Mech. 9, 187–206 (2009). https://doi.org/10.1007/s10652-009-9123-x, Google ScholarCrossref
  21. 21. J. M. Martel and M. Toner, “Inertial focusing dynamics in spiral microchannels,” Phys. Fluids 24, 032001 (2012). https://doi.org/10.1063/1.3681228, Google ScholarScitation, ISI
  22. 22. J. Zhou and I. Papautsky, “Fundamentals of inertial focusing in microchannels,” Lab Chip 13, 1121–1132 (2013). https://doi.org/10.1039/c2lc41248a, Google ScholarCrossref
  23. 23. W. R. Dean, “Note on the motion of fluid in a curved pipe,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 4, 208–223 (1927). https://doi.org/10.1080/14786440708564324, Google ScholarCrossref
  24. 24. S. A. Berger, L. Talbot, and L.-S. Yao, “Flow in curved pipes,” Annu. Rev. Fluid Mech. 15, 461–512 (1983). https://doi.org/10.1146/annurev.fl.15.010183.002333, Google ScholarCrossref, ISI
  25. 25. W. R. Dean and J. M. Hurst, “Note on the motion of fluid in a curved pipe,” Mathematika 6, 77–85 (1959). https://doi.org/10.1112/s0025579300001947, Google ScholarCrossref
  26. 26. C. Y. Wang, “On the low-Reynolds-number flow in a helical pipe,” J. Fluid Mech. 108, 185–194 (1981). https://doi.org/10.1017/s0022112081002073, Google ScholarCrossref
  27. 27. M. Germano, “On the effect of torsion on a helical pipe flow,” J. Fluid Mech. 125, 1–8 (1982). https://doi.org/10.1017/s0022112082003206, Google ScholarCrossref
  28. 28. M. Germano, “The dean equations extended to a helical pipe flow,” J. Fluid Mech. 203, 289–305 (1989). https://doi.org/10.1017/s0022112089001473, Google ScholarCrossref
  29. 29. H. C. Kao, “Torsion effect on fully developed flow in a helical pipe,” J. Fluid Mech. 184, 335–356 (1987). https://doi.org/10.1017/s002211208700291x, Google ScholarCrossref
  30. 30. E. R. Tuttle, “Laminar flow in twisted pipes,” J. Fluid Mech. 219, 545–570 (1990). https://doi.org/10.1017/s002211209000307x, Google ScholarCrossref
  31. 31. L. Zabielski and A. J. Mestel, “Steady flow in a helically symmetric pipe,” J. Fluid Mech. 370, 297–320 (1998). https://doi.org/10.1017/s0022112098002006, Google ScholarCrossref
  32. 32. D. G. Lynch, S. L. Waters, and T. J. Pedley, “Flow in a tube with non-uniform, time-dependent curvature: Governing equations and simple examples,” J. Fluid Mech. 323, 237–265 (1996). https://doi.org/10.1017/s0022112096000900, Google ScholarCrossref
  33. 33. D. Gammack and P. E. Hydon, “Flow in pipes with non-uniform curvature and torsion,” J. Fluid Mech. 433, 357–382 (2001). https://doi.org/10.1017/s0022112001003548, Google ScholarCrossref
  34. 34. K. H. Winters, “A bifurcation study of laminar flow in a curved tube of rectangular cross-section,” J. Fluid Mech. 180, 343–369 (1987). https://doi.org/10.1017/s0022112087001848, Google ScholarCrossref
  35. 35. D. Manoussaki and R. S. Chadwick, “Effects of geometry on fluid loading in a coiled cochlea,” SIAM J. Appl. Math. 61, 369–386 (2000). https://doi.org/10.1137/s0036139999358404, Google ScholarCrossref
  36. 36. C. Pozrikidis, “Stokes flow through a coiled tube,” Acta Mech. 190, 93–114 (2007). https://doi.org/10.1007/s00707-006-0425-5, Google ScholarCrossref
  37. 37. M. Norouzi and N. Biglari, “An analytical solution for dean flow in curved ducts with rectangular cross section,” Phys. Fluids 25, 053602 (2013). https://doi.org/10.1063/1.4803556, Google ScholarScitation
  38. 38. Y. M. Stokes, B. R. Duffy, S. K. Wilson, and H. Tronnolone, “Thin-film flow in helically wound rectangular channels with small torsion,” Phys. Fluids 25, 083103 (2013). https://doi.org/10.1063/1.4818628, Google ScholarScitation, ISI
  39. 39. S. Lee, Y. M. Stokes, and A. L. Bertozzi, “Behavior of a particle-laden flow in a spiral channel,” Phys. Fluids 26, 043302 (2014). https://doi.org/10.1063/1.4872035, Google ScholarScitation, ISI
  40. 40. D. J. Arnold, Y. M. Stokes, and J. E. F. Green, “Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015). https://doi.org/10.1017/jfm.2014.703, Google ScholarCrossref
  41. 41. D. J. Arnold, Y. M. Stokes, and J. E. F. Green, “Thin-film flow in helically wound shallow channels of arbitrary cross-sectional shape,” Phys. Fluids 29, 013102 (2017). https://doi.org/10.1063/1.4973670, Google ScholarScitation, ISI
  42. 42. J. M. Hill and Y. M. Stokes, “A note on Navier–Stokes equations with nonorthogonal coordinates,” ANZIAM J. 59, 335–348 (2018). https://doi.org/10.1017/S144618111700058X, Google ScholarCrossref
  43. 43. R. Aris, Vectors, Tensors: And the Basic Equations of Fluid Mechanics, Prentice-Hall International Series in the Physical and Chemical Engineering Sciences (Prentice-Hall, 1962). Google Scholar
  44. 44. P. N. Shankar, “On the use of biorthogonality relations in the solution of some boundary value problems for the biharmonic equation,” Curr. Sci. 85, 975–979 (2003). Google Scholar
  45. 45. S. D. Conte, “The numerical solution of linear boundary value problems,” SIAM Rev. 8, 309–321 (1966). https://doi.org/10.1137/1008063, Google ScholarCrossref
  46. 46. D. D. Joseph, “The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part I,” SIAM J. Appl. Math. 33, 337–347 (1977). https://doi.org/10.1137/0133021, Google ScholarCrossref
  47. 47. D. D. Joseph and L. Sturges, “The convergence of biorthogonal series for biharmonic and Stokes flow edge problems: Part II,” SIAM J. Appl. Math. 34, 7–26 (1978). https://doi.org/10.1137/0134002, Google ScholarCrossref
  48. 48. C. Wang, “Stokes flow in a curved duct—A Ritz method,” Comput. Fluids 53, 145–148 (2012). https://doi.org/10.1016/j.compfluid.2011.10.010, Google ScholarCrossref
  49. 49. E. H. Georgoulis and P. Houston, “Discontinuous Galerkin methods for the biharmonic problem,” IMA J. Numer. Anal. 29, 573–594 (2009). https://doi.org/10.1093/imanum/drn015, Google ScholarCrossref
  1. © 2018 Author(s). Published by AIP Publishing.