Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T17:05:35.998Z Has data issue: false hasContentIssue false
  • English
  • Français

Travelling wave solutions on an axisymmetric ferrofluid jet

Published online by Cambridge University Press:  19 February 2019

A. Doak Open the ORCID record for A. Doak [Opens in a new window]  and
J.-M. Vanden-Broeck
A. Doak*
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
J.-M. Vanden-Broeck
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a potential flow model of axisymmetric waves travelling on a ferrofluid jet. The ferrofluid coats a copper wire, through which an electric current is run. The induced azimuthal magnetic field magnetises the ferrofluid, which in turn stabilises the well known Plateau–Rayleigh instability seen in axisymmetric capillary jets. This model is of interest because the stabilising mechanism allows for axisymmetric magnetohydrodynamical solitary waves. A numerical scheme capable of computing steady periodic, solitary and generalised solitary wave solutions is presented. It is found that the solution space for the model is very similar to that of the classical problem of two-dimensional gravity–capillary waves.

JFM classification

Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 414 - 439
Copyright
© 2019 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.)

References

Akers, B. F., Ambrose, D. M., Pond, K. & Wright, J. D. 2016 Overturned internal capillary–gravity waves. Eur. J. Mech. B 57, 143–151.10.1016/j.euromechflu.2015.12.006Google Scholar
Akylas, T. R. 1993 Envelope solitons with stationary crests. Phys. Fluids A 5 (4), 789–791.10.1063/1.858626Google Scholar
Arkhipenko, V. I., Barkov, Y. D., Bashtovoi, V. G. & Krakov, M. S. 1980 Investigation into the stability of a stationary cylindrical column of magnetizable liquid. Fluid Dyn. 15 (4), 477–481.10.1007/BF01089602Google Scholar
Bashtovoi, V. G. & Krakov, M. S. 1978 Stability of an axisymmetric jet of magnetizable fluid. J. Appl. Mech. Tech. Phys. 19 (4), 541–545.10.1007/BF00859405Google Scholar
Batchelor, G. K. 1994 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. 1982 The solitary wave with surface tension. Q. Appl. Maths 40 (2), 231–234.10.1090/qam/666677Google Scholar
Blyth, M. G. & Părău, E. I. 2014 Solitary waves on a ferrofluid jet. J. Fluid Mech. 750, 401–420.10.1017/jfm.2014.275Google Scholar
Bourdin, E., Bacri, J. C. & Falcon, E. 2010 Observation of axisymmetric solitary waves on the surface of a ferrofluid. Phys. Rev. Lett. 104 (9), 094502.10.1103/PhysRevLett.104.094502Google Scholar
Byatt-Smith, J. G. B. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves. Proc. R. Soc. Lond. A 350 (1661), 175–189.10.1098/rspa.1976.0102Google Scholar
Champneys, A. R., Vanden-Broeck, J.-M. & Lord, G. J. 2002 Do true elevation gravity–capillary solitary waves exist? A numerical investigation. J. Fluid Mech. 454, 403–417.10.1017/S0022112001007200Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (6), 532–540.10.1017/S0022112057000348Google Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31 (1), 301–346.10.1146/annurev.fluid.31.1.301Google Scholar
Dias, F., Menace, D. & Vanden-Broeck, J. M. 1996 Numerical study of capillary–gravity solitary waves. Eur. J. Mech. 15, 17–36.Google Scholar
Doak, A. & Vanden-Broeck, J.-M. 2018 Solution selection of axisymmetric Taylor bubbles. J. Fluid Mech. 843, 518–535.10.1017/jfm.2018.156Google Scholar
Gao, T. & Vanden-Broeck, J.-M. 2014 Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves. Phys. Fluids 26 (8), 087101.10.1063/1.4893677Google Scholar
Gao, T., Wang, Z. & Vanden-Broeck, J.-M. 2017 Investigation of symmetry breaking in periodic gravity–capillary waves. J. Fluid Mech. 811, 622–641.10.1017/jfm.2016.751Google Scholar
Grandison, S., Vanden-Broeck, J.-M., Papageorgiou, D. T., Miloh, T. & Spivak, B. 2008 Axisymmetric waves in electrohydrodynamic flows. J. Engng Maths 62 (2), 133–148.10.1007/s10665-007-9183-1Google Scholar
Groves, M. & Nilsson, D. 2018 Spatial dynamics methods for solitary waves on a ferrofluid jet. J. Math. Fluid Mech. 20 (4), 1427–1458.10.1007/s00021-018-0370-9Google Scholar
Guyenne, P. & Părău, E. I. 2016 An operator expansion method for computing nonlinear surface waves on a ferrofluid jet. J. Comput. Phys. 321, 414–434.10.1016/j.jcp.2016.05.055Google Scholar
Hunter, J. K. & Vanden-Broeck, J.-M. 1983 Solitary and periodic gravity–capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219.10.1017/S0022112083003316Google Scholar
Jeppson, R. W. 1970 Inverse formulation and finite difference solution for flow from a circular orifice. J. Fluid Mech. 40 (01), 215–223.10.1017/S0022112070000137Google Scholar
Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77 (2), 229–241.10.1017/S0022112076002085Google Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary wave. Phil. Mag. 39, 422–443.10.1080/14786449508620739Google Scholar
Laget, O. & Dias, F. 1997 Numerical computation of capillary–gravity interfacial solitary waves. J. Fluid Mech. 349, 221–251.10.1017/S0022112097006861Google Scholar
Raj, K., Moskowitz, B. & Casciari, R. 1995 Advances in ferrofluid technology. J. Magn. Magn. Mater. 149 (1–2), 174–180.10.1016/0304-8853(95)00365-7Google Scholar
Rannacher, D. & Engel, A. 2006 Cylindrical Korteweg–de Vries solitons on a ferrofluid surface. New J. Phys. 8 (6), 108.10.1088/1367-2630/8/6/108Google Scholar
Rayleigh, J. W. S. 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.10.1112/plms/s1-10.1.4Google Scholar
Rosensweig, R. E. 1985 Ferrohydrodynamics. Dover.Google Scholar
Vanden-Broeck, J. M. 1991 Elevation solitary waves with surface tension. Phys. Fluids A 3 (11), 2659–2663.10.1063/1.858155Google Scholar
Vanden-Broeck, J.-M. 2010 Gravity–Capillary Free-Surface Flows. Cambridge University Press.10.1017/CBO9780511730276Google Scholar
Vanden-Broeck, J.-M. & Dias, F. 1992 Gravity-capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240, 549–557.10.1017/S0022112092000193Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1980 A new family of capillary waves. J. Fluid Mech. 98 (1), 161–169.10.1017/S0022112080000080Google Scholar
Vanden-Broeck, J. M., Miloh, T. & Spivack, B. 1998 Axisymmetric capillary waves. Wave Motion 27 (3), 245–256.10.1016/S0165-2125(97)80078-9Google Scholar
Wilton, J. R. 1915 Lxxii. On ripples. Lond. Edinb. Dublin Phil. Mag. J. Sci. 29 (173), 688–700.10.1080/14786440508635350Google Scholar
Woods, L. C. 1951 A new relaxation treatment of flow with axial symmetry. Q. J. Mech. Appl. Maths 4 (3), 358–370.10.1093/qjmam/4.3.358Google Scholar