Analysis of magnetic fluid displacement in capillaries

  • Douglas Daniel de Carvalho
  • Francisco Ricardo Cunha
  • Rafael Gabler GontijoEmail author
Technical Paper
Part of the following topical collections:
  1. 17th Brazilian Congress of Thermal Sciences and Engineering


This work presents a theoretical, numerical and experimental investigation on the shape and vertical displacement of a free surface formed in the interface between a magnetic fluid and a non-magnetic one. The magnetic fluid is confined between two parallel vertical flat plates. The presence of an external magnetic field applied by a permanent magnet arbitrarily positioned in space is considered. A new mathematical model is proposed, leading to a nonlinear differential equation that governs both the shape and the vertical displacement of the free surface due to capillary and magnetic effects. The applied field considered in this work satisfies the Ampère–Maxwell equation in the magnetostatic limit. This equation is numerically solved by direct integration using a fourth-order Runge–Kutta method coupled with a Newton–Raphson scheme. The numerical code is validated by means of some analytical solutions valid on specific asymptotic limits and by experimental results. Experimental measurements of surface tension, density and vertical displacement of Newtonian fluids in capillaries are also presented in order to provide a new methodology to estimate the contact angle combining experiments and numerical integration results. The influence of the variation of the physical variables concerning the physics of the problem on the shape and on the vertical displacement is evaluated. An increase in the intensity of the magnetic effects resulting in an increase in the vertical displacement and in the asymmetry degree of the free surfaces is observed. Finally, relevant physical discussions exploring how this long range interaction between the applied field and the magnetic liquid could be applied to promote fluid displacement in capillaries are presented.


Magnetic fluids Capillary pressure Magnetic pressure Meniscus shape and displacement Contact angle 



The authors wish to acknowledge the support of this work by FAPESP—Brazil, under the Grant Number 2017/05643-8. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. R. G. Gontijo and F. R. Cunha would like to acknowledge for the support of CNPq-Brazil researcher fellowships (PQ).


  1. 1.
    Rosensweig R, Elborai S, Lee S-H, Zahn M (2005) Ferrofluid meniscus in a horizontal or vertical magnetic field. J Magn Magn Mater 289:192–195CrossRefGoogle Scholar
  2. 2.
    John T, May K, Stannarius R (2011) Meniscus of a ferrofluid around a vertical cylindrical wire carrying electric current. Phys Rev E 83(5):056308CrossRefGoogle Scholar
  3. 3.
    Scherer C, Figueiredo Neto A M (2005) Ferrofluids: properties and applications. Braz J Phys 35(3A):718–727CrossRefGoogle Scholar
  4. 4.
    Cunha F (2012) Fundamentos da hidrodinâmica de fluidos magnéticos. In: Carmo BS, Assi GR, Meneghini JR, Aranha JAP, Volpe EV (eds) Turbulência, vol. 8, pp 257–339Google Scholar
  5. 5.
    Gontijo R, Cunha F (2012) Experimental investigation on thermo-magnetic convection inside cavities. J Nanosci Nanotechnol 12(12):9198–9207CrossRefGoogle Scholar
  6. 6.
    Cunha F, Sobral Y (2004) Characterization of the physical parameters in a process of magnetic separation and pressure-driven flow of a magnetic fluid. Physica A 343:36–64CrossRefGoogle Scholar
  7. 7.
    Rosa A, Gontijo R, Cunha F (2016) Laminar pipe flow with drag reduction induced by a magnetic field gradient. Appl Math Model 40(5–6):3907–3918MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guedes M, Guedes M, Morais P, Da Silva M, Santos T, Alves J Jr, Bertelli C, Azevedo R, Lacava Z (2004) Proposal of a magnetohyperthermia system: preliminary biological tests. J Magn Magn Mater 272:2406–2407CrossRefGoogle Scholar
  9. 9.
    Boudouvis AG, Puchalla JL, Scriven L (1988) Interaction of capillary wetting and fringing magnetic field in ferrofluid systems. J Colloid Interface Sci 124(2):677–687CrossRefGoogle Scholar
  10. 10.
    Bragard J, Lebon G (1994) Capillary ascension in porous media: a scaling law. Transp Porous Media 16(3):253–261CrossRefGoogle Scholar
  11. 11.
    Bashtovoi V, Kuzhir P, Reks A (2002) Capillary ascension of magnetic fluids. J Magn Magn Mater 252:265–267CrossRefGoogle Scholar
  12. 12.
    Bashtovoi V, Bossis G, Kuzhir P, Reks A (2005) Magnetic field effect on capillary rise of magnetic fluids. J Magn Magn Mater 289:376–378CrossRefGoogle Scholar
  13. 13.
    Polevikov V, Tobiska L (2005) Instability of magnetic fluid in a narrow gap between plates. J Magn Magn Mater 289:379–381CrossRefGoogle Scholar
  14. 14.
    Eissmann P, Lange A, Odenbach S (2011) Meniscus of a magnetic fluid in the field of a current-carrying wire: two-dimensional numerical simulations. Magnetohydrodynamics 47(2):149–157CrossRefGoogle Scholar
  15. 15.
    Gontijo R, Malvar S, Sobral Y, Cunha F (2017) The influence of a magnetic field on the mechanical behavior of a fluid interface. Meccanica 52(6):1309–1327MathSciNetCrossRefGoogle Scholar
  16. 16.
    Çengel YA, Cimbala JM (2015) Mecânica dos fluidos-3. Amgh EditoraGoogle Scholar
  17. 17.
    Griffiths D (2017) Introduction to electrodynamics, vol v. 2. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. 18.
    Rosensweig R (1997) Ferrohydrodynamics. Dover Books on Physics, Dover PublicationsGoogle Scholar
  19. 19.
    McCaig M, Clegg AG (1987) Permanent magnets in theory and practice, 2nd edn. Wiley, New YorkGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Douglas Daniel de Carvalho
    • 1
  • Francisco Ricardo Cunha
    • 2
  • Rafael Gabler Gontijo
    • 1
    Email author
  1. 1.Departamento de Energia, Faculdade de Engenharia MecânicaUniversidade Estadual de CampinasCampinasBrazil
  2. 2.Departamento de Engenharia Mecânica, Fluid Mechanics of Complex Flow (VORTEX Group)Universidade de BrasíliaBrasíliaBrazil

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