Hostname: page-component-857557d7f7-qdphv Total loading time: 0 Render date: 2025-11-26T09:09:41.081Z Has data issue: false hasContentIssue false
  • English
  • Français

Experiments on a sphere settling towards a boundary in a viscous liquid under the influence of a magnetic force

Published online by Cambridge University Press:  26 November 2025

John Sebastian Open the ORCID record for John Sebastian [Opens in a new window] ,
Alexander Lukinych Schødt  and
Kaare H. Jensen Open the ORCID record for Kaare H. Jensen [Opens in a new window]
John Sebastian
Affiliation:
Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark
Alexander Lukinych Schødt
Affiliation:
Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark
Kaare H. Jensen*
Affiliation:
Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark
*
Corresponding author: Kaare H. Jensen, khjensen@fysik.dtu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

Magnets have been utilised widely for their ability to induce rapid contact – such as snapping between magnets and ferromagnetic materials. Yet, how such interactions proceed under immersion in a viscous fluid remains poorly understood. Here, we study this problem using the classical configuration of a smooth solid sphere approaching a plane in a quiescent fluid. Induced magnetic attraction, a spatially varying force analogous to short-range dispersion forces, offers a plausible route to overcome the constraint of a diverging hydrodynamic drag, which is well understood using the framework of classical lubrication theory. Instinctively, one might expect it to enable finite-time contact. However, our experiments reveal a counterintuitive result: while magnetic forces accelerate the sphere towards the surface, reducing the approach time by two orders of magnitude compared with gravity, they ultimately fail to effectuate contact in finite time, as induced magnetic interactions are unable to mitigate lubrication drag, which is singular at the thin gap limit, and transitions to an exponential descent characteristic of constant forcing. We support these findings with a simple theoretical model that accurately predicts the magnetic force law from purely kinematic observations. Finally, we outline the conditions under which spatially varying forces can enable true finite-time contact and discuss future experimental directions.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory Multiphase and Particle-laden Flows: Particle/fluid flow

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1024 , 10 December 2025 , A12
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abelmann, L., Gwag, E. & Sung, B. 2024 Permanent magnet systems to study the interaction between magnetic nanoparticles and cells in microslide channels. J. Magn. Magn. Mater. 591, 171696.10.1016/j.jmmm.2023.171696CrossRefGoogle Scholar
Agarwal, P.K. 2006 Enzymes: an integrated view of structure, dynamics and function. Microbial Cell Factories 5 (1), 2.10.1186/1475-2859-5-2CrossRefGoogle ScholarPubMed
Birwa, S.K., Rajalakshmi, G., Govindarajan, R. & Menon, N. 2018 Solid-on-solid contact in a sphere-wall collision in a viscous fluid. Phys. Rev. Fluids 3 (4), 044302.10.1103/PhysRevFluids.3.044302CrossRefGoogle Scholar
Brangbour, C., du Roure, O., Helfer, E., Démoulin, D., Mazurier, A., Fermigier, M., Carlier, M.-F., Bibette, J., Baudry, J. & Spudich, J.A. 2011 Force–velocity measurements of a few growing actin filaments. PLoS Biol. 9 (4), e1000613.10.1371/journal.pbio.1000613CrossRefGoogle ScholarPubMed
Casadevall, A., Nosanchuk, J.D., Williamson, P. & Rodrigues, M.L. 2009 Vesicular transport across the fungal cell wall. Trends Microbiol. 17 (4), 158–162.10.1016/j.tim.2008.12.005CrossRefGoogle ScholarPubMed
Cawthorn, C.J. & Balmforth, N.J. 2010 Contact in a viscous fluid. Part 1. A falling wedge. J. Fluid Mech. 646, 327–338.10.1017/S0022112009993090CrossRefGoogle Scholar
Chastel, T. & Mongruel, A. 2019 Sticking collision between a sphere and a textured wall in a viscous fluid. Phys. Rev. Fluids 4 (1), 014301.10.1103/PhysRevFluids.4.014301CrossRefGoogle Scholar
Choi, J.S. & Yoo, J. 2008 Design of a Halbach magnet array based on optimization techniques. IEEE Trans. Magn. 44 (10), 2361–2366.10.1109/TMAG.2008.2001482CrossRefGoogle Scholar
Cox, R.G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface—II small gap widths, including inertial effects. Chem. Engng Sci. 22 (12), 1753–1777.10.1016/0009-2509(67)80208-2CrossRefGoogle Scholar
Cui, Y., Wang, J., Liang, J. & Qiu, H. 2023 Molecular engineering of colloidal atoms. Small 19 (20), 2207609.10.1002/smll.202207609CrossRefGoogle ScholarPubMed
Davis, R.H., Serayssol, J.-M. & Hinch, E.J. 1986 The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163, 479–497.10.1017/S0022112086002392CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds number hydrodynamics: with special applications to particulate media. In Mechanics of Fluids and Transport Processes, 1st edn, vol. 1, M. Nijhoff. Distributed by Kluwer Boston.Google Scholar
Hasimoto, H. 1996 Lorentz’s theorem on the Stokes equation. J. Engng Maths 30 (1), 215–224.10.1007/BF00118831CrossRefGoogle Scholar
Israelachvili, J.N. 2011 Intermolecular and Surface Forces, 3rd edn. Academic Press.Google Scholar
Ku, J., Lei, Z., Lin, H., Yan, Q., Chen, H. & Guo, B. 2022 Interaction of magnetic spheres in magnetic fields from the view of magnetic energy density: a 3D finite element analysis (FEA). Intl J. Mining Sci. Technol. 32 (6), 1341–1350.10.1016/j.ijmst.2022.08.007CrossRefGoogle Scholar
Leckband, D. & Israelachvili, J. 2001 Intermolecular forces in biology. Q. Rev. Biophys. 34 (2), 105–267.10.1017/S0033583501003687CrossRefGoogle ScholarPubMed
Lorentz, H.A. 1897 A general theorem concerning the motion of a viscous fluid and a few consequences derived from it. Verh. K. Akad. Wet. Amsterdam 5, 168–175.Google Scholar
Michelin, S.& Lauga, E. 2014 Phoretic self-propulsion at finite Péclet numbers. J. Fluid Mech. 747, 572–604.10.1017/jfm.2014.158CrossRefGoogle Scholar
Mongruel, A., Lamriben, C., Yahiaoui, S. & Feuillebois, F. 2010 The approach of a sphere to a wall at finite Reynolds number. J. Fluid Mech. 661, 229–238.10.1017/S0022112010003459CrossRefGoogle Scholar
Niu, R., Du, C.X., Esposito, E., Ng, J., Brenner, M.P., McEuen, P.L. & Cohen, I. 2019 Magnetic handshake materials as a scale-invariant platform for programmed self-assembly. Proc. Natl Acad. Sci. 116 (49), 24402–24407.10.1073/pnas.1910332116CrossRefGoogle ScholarPubMed
Phillips, R. 2013 Physical Biology of the Cell, 2nd edn. Garland Science.Google Scholar
Purcell, E.M. 2013 Electricity and Magnetism, 3rd edn. Cambridge University Press.10.1017/CBO9781139012973CrossRefGoogle Scholar
Serayssol, J.-M. & Davis, R.H. 1986 The influence of surface interactions on the elastohydrodynamic collision of two spheres. J. Colloid Interface Sci. 114 (1), 54–66.10.1016/0021-9797(86)90240-7CrossRefGoogle Scholar
Skotheim, J.M. & Mahadevan, L. 2004 Soft lubrication. Phys. Rev. Lett. 92 (24), 245509.10.1103/PhysRevLett.92.245509CrossRefGoogle Scholar
Smythe, W.R. 1989 Static and Dynamic Electricity, 3rd edn. Hemisphere publ.Google Scholar
Squires, T.M., Messinger, R.J. & Manalis, S.R. 2008 Making it stick: convection, reaction and diffusion in surface-based biosensors. Nat. Biotechnol. 26 (4), 417–426.10.1038/nbt1388CrossRefGoogle ScholarPubMed
Stone, H.A. 2005 On lubrication flows in geometries with zero local curvature. Chem. Engng Sci. 60 (17), 4838–4845.10.1016/j.ces.2005.03.021CrossRefGoogle Scholar
Thiele, K., Wanner, G., Kindzierski, V., Jürgens, G., Mayer, U., Pachl, F. & Assaad, F.F. 2009 The timely deposition of callose is essential for cytokinesis in Arabidopsis. Plant J. 58 (1), 13–26.10.1111/j.1365-313X.2008.03760.xCrossRefGoogle ScholarPubMed
Van Saarloos, W., Vitelli, V. & Zeravcic, Z. 2023 Soft Matter: Concepts, Phenomena, and Applications. Princeton University Press.Google Scholar
Verma, D.P.S. 2001 Cytokinesis and building of the cell plate in plants. Ann. Rev. Plant Physiol. Plant Mol. Biol. 52 (1), 751–784.10.1146/annurev.arplant.52.1.751CrossRefGoogle ScholarPubMed
Vogt, C. & Weckhuysen, B.M. 2022 The concept of active site in heterogeneous catalysis. Nat. Rev. Chem. 6 (2), 89–111.10.1038/s41570-021-00340-yCrossRefGoogle ScholarPubMed
Wang, Z., Miksis, M.J. & Vlahovska, P.M. 2022 Particle-surface interactions in a uniform electric field. Phys. Rev. E 106 (3), 034607.10.1103/PhysRevE.106.034607CrossRefGoogle Scholar
Xu, Y., Choi, K.H., Nagella, S.G. & Takatori, S.C. 2023 Dynamic interfaces for contact-time control of colloidal interactions. Soft Matt. 19 (30), 5692–5700.10.1039/D3SM00673ECrossRefGoogle ScholarPubMed
Yariv, E. 2016 Wall-induced self-diffusiophoresis of active isotropic colloids. Phys. Rev. Fluids 1 (3), 032101.10.1103/PhysRevFluids.1.032101CrossRefGoogle Scholar
Zeichner, G.R. & Schowalter, W.R. 1979 Effects of hydrodynamic and colloidal forces on the coagulation of dispersions. J. Colloid Interface Sci. 71 (2), 237–253.10.1016/0021-9797(79)90235-2CrossRefGoogle Scholar
Zhou, H.-X. & Pang, X. 2018 Electrostatic interactions in protein structure, folding, binding, and condensation. Chem. Rev. 118 (4), 1691–1741.10.1021/acs.chemrev.7b00305CrossRefGoogle ScholarPubMed