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Annular flow instabilities and large-scale vortices in electromagnetically driven horizontal soap films

Andrey Pototsky1,*, Aldo Figueroa2, José Olvera-Orozco3, Misael Álvarez-Jiménez4, Sergio Cuevas3, and Sergey A. Suslov1

  • *Contact author: apototskyy@swin.edu.au
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Phys. Rev. Fluids 10, 034802 – Published 24 March, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.034802

Abstract

We study the conditions under which large-scale vortex flow patterns appear in electromagnetically driven conducting horizontal soap films created between two coaxial cylindrical electrodes and placed in a vertical magnetic field. For large Marangoni and Peclet numbers, the two-dimensional flow in a flat free film is effectively incompressible and the surface concentration of an insoluble surfactant plays the role of the pressure away from the boundaries. If the direct radial electric current flows through the film, the Lorentz force arising in a uniform field drives the azimuthal flow. Its stability is investigated in cases of slip and no-slip boundary conditions at the edges of the film in the limit of a small Hartmann number. Flow stability changes nontrivially depending on the slip length, and the flow stabilizes for perfect-slip boundaries. In contrast, if the vertical axisymmetric field is strongly nonuniform, the flow may become linearly unstable at much smaller Reynolds numbers than the critical value for no-slip boundaries,- leading to the formation of large-scale circulation cells. Our experiments conducted using a piecewiseconstant magnetic field detect the formation of vortex structures that are in qualitative agreement with those predicted by the performed analysis.

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References (46)

  1. G. R. Flierl, Isolated eddy models in geophysics, Annu. Rev. Fluid Mech. 19, 493 (1987).
  2. M. V. Melander, N. J. Zabusky, and J. C. Mcwilliams, Symmetric vortex merger in two dimensions: Causes and conditions, J. Fluid Mech. 195, 303 (1988).
  3. X. J. Carton, G. R. Flierl, and L. M. Polvani, The generation of tripoles from unstable axisymmetric isolated vortex structures, Europhys. Lett. 9, 339 (1989).
  4. V. I. Arnold and L. D Meshalkin, The seminar of A. N. Kolmogorov on selected topics in analysis (1958–1959), Usp. Mat. Nauk 15, 247 (1960).
  5. Y. Couder, The observation of a shear flow instability in a rotating system with a soap membrane, J. Phys. Lett. 42, 429 (1981).
  6. V. M Ponomarev, Stability of a class of axisymmetric flows of an incompressible liquid, Fluid Dyn. 15, 1 (1980).
  7. V. A. Dovzhenko, A. M. Obukhov, and V. M Ponomarev, Generation of vortices in an axisymmetric shear flow, Fluid Dyn. 16, 510 (1981).
  8. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, U.K., 1967).
  9. V. A. Romanov, Stability of plane-parallel Couette flow, Funct. Anal. Appl. 7, 137 (1973).
  10. G. I. Taylor, VIII. Stability of a viscous liquid contained between two rotating cylinders, Philos. Trans. R. Soc. London, Ser. A 223, 289 (1923).
  11. A. M. Obukhov, Kolmogorov flow and laboratory simulation of it, Russ. Math. Surv. 38, 113 (1983).
  12. E. Fylladitakis, Kolmogorov flow: Seven decades of history, J. Appl. Math. Phys. 6, 2227 (2018).
  13. P. R. Gent and J. C. Mcwilliams, The instability of barotropic circular vortices, Geophys. Astrophys. Fluid Dyn. 35, 209 (1986).
  14. C. E. Leith, Minimum enstrophy vortices, Phys. Fluids 27, 1388 (1984).
  15. B. Legras, P. Santangelo, and R. Benzi, High-resolution numerical experiments for forced two-dimensional turbulence, Europhys. Lett. 5, 37 (1988).
  16. H. Lamb, Hydrodynamics, Cambridge Mathematical Library (Cambridge University Press, Cambridge, U.K., 1993).
  17. E. J. Hopfinger and G. J. F. van Heijst, Vortices in rotating fluids, Annu. Rev. Fluid Mech. 25, 241 (1993).
  18. G. R. Flierl, On the instability of geostrophic vortices, J. Fluid Mech. 197, 349 (1988).
  19. V. B. Levin, Free rotating layer of an electrically conducting fluid in an axial magnetic field, Magnetohydrodynamics 16, 86 (1980).
  20. A. A. Klyukin and V. B. Levin, The stability of a free rotating layer in an electrically conducting fluid in an axial field, Fluid Dyn. 19, 815 (1984).
  21. Kh. E. Kalis and Y. B. Kolesnikov, Stability of a rotating transverse axisymmetric flow in an axial magnetic field, Magnetohydrodynamics 3, 72 (1983).
  22. A. Potherat, J. Sommeria, and R. Moreau, An effective two-dimensional model for MHD flows with transverse magnetic field, J. Fluid Mech. 424, 75 (2000).
  23. N. F. Bondarenko, M. Z. Gak, and F. V. Dolzhanskiy, Laboratory and theoretical models of plane periodic flows, Izv. Akad. Nauk SSSR. Fyz. Atmos. Okeana 15, 1017 (1979).
  24. A. M. Batchaev and V. M. Ponomarev, Experimental and theoretical investigation of Kolmogorov flow on a cylindrical surface, Fluid Dyn. 24, 675 (1990).
  25. J. Sommeria, Experimental study of the two-dimensional inverse energy cascade in a square box, J. Fluid Mech. 170, 139 (1986).
  26. Y. Couder, Two-dimensional grid turbulence in a thin liquid film, J. Phys. Lett. 45, 353 (1984).
  27. Y. Couder, J.-M. Chomaz, and M. Rabaud, On the hydrodynamics of soap films, Physica D 37, 384 (1989).
  28. J.-M. Chomaz, The dynamics of a viscous soap film with soluble surfactant, J. Fluid Mech. 442, 387 (2001).
  29. R. C. C. Gomez, Response of a soap film to a continuous electromagnetic forcing, J. Visualization 20, 87 (2017).
  30. H. Manikantan and T. M. Squires, Surfactant dynamics: Hidden variables controlling fluid flows, J. Fluid Mech. 892, P1 (2020).
  31. I. Kim and S. Mandre, Marangoni elasticity of flowing soap films, Phys. Rev. Fluids 2, 082001(R) (2017).
  32. N. S. Lalli and A. Giusti, Thin film modelling of magnetic soap films, J. Fluid Mech. 986, A7 (2024).
  33. A. Figueroa, F. Demiaux, S. Cuevas, and E. Ramos, Electrically driven vortices in a weak dipolar magnetic field in a shallow electrolytic layer, J. Fluid Mech. 641, 245 (2009).
  34. J. Pérez-Barrera, A. Ortiz, and S. Cuevas, Analysis of an annular MHD stirrer for microfluidic applications, in Recent Advances in Fluid Dynamcis with Environmental Applications, edited by J. Klapp, L. D. G. Sigalotti, A. Medina Ovando, A. López Villa, and G. Ruíz Chavarría (Springer, Berlin, 2016), pp. 275–288.
  35. A. Pototsky and S. A. Suslov, Electromagnetically driven flow in unsupported electrolyte layers: Lubrication theory and linear stability of annular flow, J. Fluid Mech. 984, A75 (2024).
  36. A. Pototsky and S. A. Suslov, Effect of the dynamic contact angle on electromagnetically driven flows in free liquid films, Phys. Fluids 36, 072111 (2024).
  37. K. J. Mysels, Soap Films: Studies of Their Thinning and a Bibliography (Pergamon Press, New York, 1959).
  38. M. Barigou and J. F. Davidson, Soap film drainage: Theory of experiment, Chem. Eng. Sci. 49, 1807 (1994).
  39. E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, and X. Wang, AUTO97: continuation and bifurcation software for ordinary differential equations (with HomCont), Technical report, (Concordia University, Montreal, Canada, 1999).
  40. B. Krauskopf, H. M. Osinga, and J. Galan-Vioque, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems (Springer, Dordrecht, 2014).
  41. S. A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50, 689 (1971).
  42. P. Drazin and W. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, U.K., 1981).
  43. Q. He and X.-P. Wang, The effect of the boundary slip on the stability of shear flow, Z. Angew. Math. Mech. 88, 729 (2008).
  44. J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng. 195, 6011 (2006).
  45. J. Olvera-Orozco, O. Suárez, G. Ramírez, J. Pérez-Barrera, A. Figueroa, and S. Cuevas, Electromagnetically driven flow in a soap film in an annular configuration (unpublished).
  46. W. Thielicke and E. J. Stamhuis, PIVlab—towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB, J. Open Res. Softw. 2, e30 (2014).

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Phys. Rev. Fluids 10, 034802– Published 24 March, 2025
  • Received 1 November 2024
  • Accepted 11 March 2025
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