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We present a Rayleigh–Ritz method for the approximation of fluid flow in a curved duct, including the secondary cross-flow, which is well known to develop for nonzero Dean numbers. Having a straightforward method to estimate the cross-flow for ducts with a variety of cross-sectional shapes is important for many applications. One particular example is in microfluidics where curved ducts with low aspect ratio are common, and there is an increasing interest in nonrectangular duct shapes for the purpose of size-based cell separation. We describe functionals which are minimized by the axial flow velocity and cross-flow stream function which solve an expansion of the Navier–Stokes model of the flow. A Rayleigh–Ritz method is then obtained by computing the coefficients of an appropriate polynomial basis, taking into account the duct shape, such that the corresponding functionals are stationary. Whilst the method itself is quite general, we describe an implementation for a particular family of duct shapes in which the top and bottom walls are described by a polynomial with respect to the lateral coordinate. Solutions for a rectangular duct and two nonstandard duct shapes are examined in detail. A comparison with solutions obtained using a finite-element method demonstrates the rate of convergence with respect to the size of the basis. An implementation for circular cross-sections is also described, and results are found to be consistent with previous studies.

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[1] Alnæs, M. S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M. E. and Wells, G. N., “The FEniCS project version 1.5”, Arch. Numer. Softw. 3 (2015) 923; doi:10.11588/ans.2015.100.20553.
[2] Brown, R. E. and Stone, M. A., “On the use of polynomial series with the Rayleigh–Ritz method”, Compos. Struct. 39 (1997) 191196; doi:10.1016/S0263-8223(97)00113-X.
[3] Dean, W. R. XVI, “Note on the motion of fluid in a curved pipe”, Lond. Edinb. Dublin Philos. Mag. J. Sci. 4 (1927) 208223; doi:10.1080/14786440708564324.
[4] Dean, W. R. and Hurst, J. M., “Note on the motion of fluid in a curved pipe”, Mathematika 6 (1959) 7785; doi:10.1112/S0025579300001947.
[5] Di Carlo, D., “Inertial microfluidics”, Lab Chip 9 (2009) 30383046; doi:10.1039/B912547G.
[6] Fan, Y., Tanner, R. I. and Phan-Thien, N., “Fully developed viscous and viscoelastic flows in curved pipes”, J. Fluid Mech. 440 (2001) 327357; doi:10.1017/S0022112001004785.
[7] Galdi, G. P. and Robertson, A. M., “On flow of a Navier–Stokes fluid in curved pipes. Part I. Steady flow”, Appl. Math. Lett. 18 (2005) 11161124; doi:10.1016/j.aml.2004.11.004.
[8] Geislinger, T. M. and Franke, T., “Hydrodynamic lift of vesicles and red blood cells in flow – From Fåhræus and Lindqvist to microfluidic cell sorting”, Adv. Colloid Interface. Sci. 208 (2014) 161176; doi:10.1016/j.cis.2014.03.002.
[9] Georgoulis, E. H. and Houston, P., “Discontinuous Galerkin methods for the biharmonic problem”, IMA J. Numer. Anal. 29 (2009) 573594; doi:10.1093/imanum/drn015.
[10] Germano, M., “The Dean equations extended to a helical pipe flow”, J. Fluid Mech. 203 (1989) 289305; doi:10.1017/S0022112089001473.
[11] Harding, B. and Stokes, Y. M., “Fluid flow in a spiral microfluidic duct”, Phys. Fluids 30 (2018) 042007; doi:10.1063/1.5026334.
[12] Hood, K., Lee, S. and Roper, M., “Inertial migration of a rigid sphere in three-dimensional Poiseuille flow”, J. Fluid Mech. 765 (2015) 452479; doi:10.1017/jfm.2014.739.
[13] Liew, K. M. and Wang, C. M., “pb-2 Rayleigh–Ritz method for general plate analysis”, Eng. Struct. 15 (1993) 5560; doi:10.1016/0141-0296(93)90017-X.
[14] Martel, J. M. and Toner, M., “Particle focusing in curved microfluidic channels”, Sci. Rep. 3 (2013) 3340; doi:10.1038/srep03340.
[15] Robertson, A. M. and Muller, S. J., “Flow of Oldroyd-B fluids in curved pipes of circular and annular cross-section”, Int. J. Non-Lin. Mech. 31 (1996) 120doi:10.1016/0020-7462(95)00040-2.
[16] Wang, C. Y., “Stokes flow in a curved duct – A Ritz method”, Comput. Fluids 53 (2012) 145148; doi:10.1016/j.compfluid.2011.10.010.
[17] Wang, C. Y., “Ritz method for slip flow in curved micro-ducts and application to the elliptic duct”, Meccanica 51 (2016) 10691076; doi:10.1007/s11012-015-0288-8.
[18] Warkiani, M. E., Guan, G., Luan, K. B., Lee, W. C., Bhagat, A. A. S., Kant Chaudhuri, P., Tan, D. S.-W., Lim, W. T., Lee, S. C., Chen, P. C. Y., Lim, C. T. and Han, J., “Slanted spiral microfluidics for the ultra-fast, label-free isolation of circulating tumor cells”, Lab Chip 14 (2014) 128137; doi:10.1039/C3LC50617G.
[19] Yamamoto, K., Wu, X., Hyakutake, T. and Yanase, S., “Taylor–dean flow through a curved duct of square cross section”, Fluid Dyn. Res. 35 (2004) 6786; doi:10.1016/j.fluiddyn.2004.04.003.
[20] Yanase, S., Goto, N. and Yamamoto, K., “Dual solutions of the flow through a curved tube”, Fluid Dyn. Res. 5 (1989) 191201; doi:10.1016/0169-5983(89)90021-X.
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