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Fingering patterns in magnetic fluids: Perturbative solutions and the stability of exact stationary shapes

Pedro H. A. Anjos1, Sérgio A. Lira2,3,*, and José A. Miranda1,†

  • 1Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
  • 2Instituto de Física, Universidade Federal de Alagoas, Maceió Alagoas 57072-900 Brazil
  • 3Institut Curie, PSL Research University, CNRS, Physical Chemistry Curie, F-75005, Paris, France

  • *sergio@fis.ufal.br
  • †jme@df.ufpe.br
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Phys. Rev. Fluids 3, 044002 – Published 17 April, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.044002

Abstract

We examine the formation of interfacial patterns when a magnetic liquid droplet (ferrofluid, or a magnetorheological fluid), surrounded by a nonmagnetic fluid, is subjected to a radial magnetic field in a Hele-Shaw cell. By using a vortex-sheet formalism, we find exact stationary solutions for the fluid-fluid interface in the form of n-fold polygonal shapes. A weakly nonlinear, mode-coupling method is then utilized to find time-evolving perturbative solutions for the interfacial patterns. The stability of such nonzero surface tension exact solutions is checked and discussed, by trying to systematically approach the exact stationary shapes through perturbative solutions containing an increasingly larger number of participating Fourier modes. Our results indicate that the exact stationary solutions of the problem are stable, and that a good matching between exact and perturbative shape solutions is achieved just by using a few Fourier modes. The stability of such solutions is substantiated by a linearization process close to the stationary shape, where a system of mode-coupling equations is diagonalized, determining the eigenvalues which dictate the stability of a fixed point.

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Phys. Rev. Fluids 3, 044002– Published 17 April, 2018
  • Received 7 December 2017
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