Convergence analysis of inertial lift force estimates using the finite element method

Abstract

We conduct a convergence analysis for the estimation of inertial lift force on a spherical particle suspended in flow through a straight square duct using the finite element method. Specifically, we consider the convergence of an inertial lift force approximation with respect to a range of factors including the truncation of the domain, the resolution of the tetrahedral mesh and the boundary conditions imposed at the (truncated) ends of the domain. Additionally, we compare estimates obtained via the Lorentz reciprocal theorem with those obtained via a direct integration of fluid stress over the particle surface. References

• M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells. The FEniCS project version 1.5. Arch. Numer. Software, 3(100):9–23, 2015. doi:10.11588/ans.2015.100.20553.
• D. Di Carlo. Inertial microfluidics. Lab Chip, 21:3038–3046, 2009. doi:10.1039/B912547G.
• C. Geuzaine and J.-F. Remacle. Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng., 79(11):1309–1331, 2009. doi:10.1002/nme.2579.
• B. Harding. A study of inertial particle focusing in curved microfluidic ducts with large bend radius and low flow rate. In Proc. 21st Australasian Fluid Mechanics Conference, number 603, 2018. URL https://people.eng.unimelb.edu.au/imarusic/proceedings/21/Contribution_603_final.pdf.
• B. Harding, Y. M. Stokes, and A. L. Bertozzi. Effect of inertial lift on a spherical particle suspended in flow through a curved duct. J. Fluid Mech., accepted, 2019. URL https://arxiv.org/abs/1902.06848.
• A. J. Hogg. The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows. J. Fluid Mech., 272:285–318, 1994. doi:10.1017/S0022112094004477.
• 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. doi:10.1017/jfm.2014.739.
• N. Nakagawa, T. Yabu, R. Otomo, A. Kase, M. Makino, T. Itano, and M. Sugihara-Seki. Inertial migration of a spherical particle in laminar square channel flows from low to high reynolds numbers. J. Fluid Mech., 779:776–793, 2015. doi:10.1017/jfm.2015.456.
• T.-W. Pan and R. Glowinski. Direct simulation of the motion of neutrally buoyant balls in a three-dimensional poiseuille flow. C. R. Mecanique, 333(12):884–895, 2005. doi:10.1016/j.crme.2005.10.006.
• C. Taylor and P. Hood. A numerical solution of the navier-stokes equations using the finite element technique. Comput. Fluids, 1(1):73–100, 1973. doi:10.1016/0045-7930(73)90027-3.
• 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, 1:128–137, 2014. doi:10.1039/C3LC50617G.
• B. H. Yang, J. Wang, D. D. Joseph, H. H. Hu, T.-W. Pan, and R. Glowinski. Migration of a sphere in tube flow. J. Fluid Mech., 540:109–131, 2005. doi:10.1017/S0022112005005677.
• L. Zeng, S. Balachandar, and P. Fischer. Wall-induced forces on a rigid sphere at finite reynolds number. J. Fluid Mech., 536:1–25, 2005. doi:10.1017/S0022112005004738.