Abstract
Numerical simulations of the two-dimensional motion of multiple paramagnetic particles suspended in a viscous fluid subjected to a uniform magnetic field are presented. Both the magnetic field and flow field can be described efficiently with simple series in local coordinates attached to each particle. The coefficients of the series can be obtained with fast convergence when only a few leading coefficients are treated implicitly. Numerical results for the flow field are validated by comparing the data with those given by an asymptotic solution for a pair of particles separated by a small distance. The numerical results of the magnetic field are validated by comparison with the solutions in bipolar coordinates. Simulations of the motion of multiple particles reveal interesting phenomena and shed light on the fundamental mechanism of particles clustering into a straight chain. The data presented in this paper can be used as a benchmark solution for verifying codes for simulating the motion of paramagnetic particles in a magnetic field.
Similar content being viewed by others
References
Mirowski E, Moreland J, Russek S, Donahue M, Hsieh K (2007) Manipulation of magnetic particles by patterned arrays of magnetic spin-valve traps. J Magn Magn Mater 311: 401–404
Choi JW, Ahn CH, Bhansali S, Henderson HT (2000) A new magnetic bead-based filterless bio-separator with planar electromagnet surfaces for integrated bio-detection systems. Sens Actuators B 68: 34–39
Tondra M, Granger M, Fuerst R, Porter M, Nordman C, Taylor J, Akou S (2001) Design of integrated microfluidic device for sorting magnetic beads in biological assays. IEEE Trans Magn 37: 2621–2623
Voltairas PA, Fotiadis DI, Michalis LK (2002) Hydrodynamics of magnetic drug targeting. J Biomech 35: 813–821
Suzuki H, Ho C-M (2004) A chaotic mixer for magnetic bead-based micro cell sorter. J MEMS 13(5): 779–790
Biswal SL, Gast AP (2004) Micromixing with linked chains of paramagnetic particles. Anal Chem 76: 6448–6455
Rida A, Gijs MAM (2004) Manipulation of self-assembled structures of magnetic beads for microfluidic mixing and assaying. Anal Chem 76: 6239–6246
Calhoun R, Yadav A, Vuppu PPA, Garcia A, Hayes M (2006) Paramagnetic particles and mixing in micro-scale flows. Lab Chip 6: 247–257
Lund-Olsen T, Buus BB, Howalt JG, Hansen MF (2008) Magnetic beads micromixer: influence of magnetic element geometry and field amplitude. J Appl Phys 103: 07E902
Wang Y, Zhe J, Chung BTF, Dutta P (2008) A rapid magnetic particle driven micromixer. Microfluid Nanofluidics 4: 375–389
Schotter J, Shoshi A, Brueckl H (2009) Development of a magnetic lab-on-a-chip for point-of-care sepsis diagnosis. J Magn Magn Mater 321: 1671–1675
Pamme N (2006) Magnetism and microfluidics. Lab Chip 6: 24–38
Rosenweig RE (1997) Ferrohydrodynamics. Dover Publications Inc, Mineola
Ly HV, Reitich F, Jolly MR, Banks HT, Ito K (1999) Simulations of particle dynamics in magnetorheological fluids. J Comput Phys 155: 160–177
Liu D, Maxey MR, Karniadakis GE (2005) Simulations of dynamic self-assembly of paramagnetic microspheres in confined microgeometries. J MEMS 15: 2298–2308
Krishnamurthy S, Yadav A, Phelan PE, Calhoun R, Vuppu AK, Garcia AA, Hays MA (2005) Dynamics of rotating paramagnetic particle chains simulated by particle dynamics, Stokesian dynamics and lattice Boltzmann methods. Microfluid Nanofluidics 5: 33–41
Peng X, Min Y, Ma T, Luo W, Man M (2009) Two-dimensional Monte Carlo simulations of structures of a suspension comprised of magnetic and nanomagnetic particles in uniform magnetic fields. J Magn Magn Mater 321: 1221–1226
From Wikipedia, http://en.wikipedia.org/wiki/Paramagnetism
Kang TG, Hulsen MA, den Toonder JMJ, Anderson PD, Meijer HEH (2008) A direct simulation method for flows with suspended paramagnetic particles. J Comput Phys 227: 4441–4458
Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Martinus Nijhoff Publishers, The Hague
Gradshteyn IS, Ryzhik IM (1980) Table of integrals, series, and products. Academic Press, San Diego
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Suh, Y.K., Kang, S. Motion of paramagnetic particles in a viscous fluid under a uniform magnetic field: benchmark solutions. J Eng Math 69, 25–58 (2011). https://doi.org/10.1007/s10665-010-9364-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-010-9364-1