The present study considers axially symmetrical waves on the surface of a ferromagnetic viscous fluid film flowing down a cylindrical conductor with a non-constant current. In addition to the gravitational force, the film is affected by a uniform magnetic field. Using the long-wave approximation, a model equation is obtained for the evolution of the film thickness from its undisturbed value. By showing that solutions with and without a constant magnetic field are self-similar, scaling parameters are identified and presented. Numerical solutions of the time-dependent non-linear partial differential equation are presented when current in the conductor is described by a ramp function (i.e., magnetic field onset).
REFERENCES
1.
S. V.
Alekseenko
, V. E.
Nakoryakov
, and B. G.
Pokusaev
, Wave Flow of Liquid Films
(Begell House
, New York
, 1994
).2.
M. I.
Shliomis
, “Magnetic fluids
,” Sov. Phys. Usp.
17
, 153
–169
(1974
).3.
V. G.
Bashtovoi
, B. M.
Berkovskii
, and A. N.
Vislovich
, An Introduction to Thermomechanics of Ferrofluids
(Institute of High Temperatures, USSR Academy of Science
, Moscow
, 1985
) (in Russian).4.
E. A.
Demekhin
, M. A.
Kaplan
, and R. A.
Foigel
, “Nonlinear waves in a discharging layer of a viscous magnetic field
,” Magnetohydrodynamics
1
, 17
–24
(1988
).5.
T.
Kuramoto
and T.
Tsuzuki
, “Persistent propagation of concentration waves in dissipative media far from thermal equilibrium
,” Prog. Theor. Phys.
55
, 356
–369
(1976
).6.
A. A.
Nepomnyashchii
, “Stability of wave regimes in a film flowing down an inclined plane
,” Fluid Dyn.
9
, 354
–359
(1974
).7.
G.
Sivashinsky
, “Nonlinear analysis of hydrodynamic instability in laminar flame. Part I: Derivation of basic equation
,” Acta Astronaut.
4
, 1177
–1206
(1977
).8.
H.-C.
Chang
and E. A.
Demekhin
, in Complex Wave Dynamics on Thin Films
, edited by D.
Mobius
and R.
Miller
(Elsevier
, Amsterdam
, 2002
).9.
S.
Kalliadasis
, C.
Ruyer-Quil
, B.
Scheid
, and M. G.
Velarde
, Falling Liquid Films
(Springer
, London, Dordrecht, Heidelberg, New York
, 2012
).10.
O. Y.
Tsvelodub
, “Stationary travelling waves on a film flowing down an inclined plane
,” Fluid Dyn.
4
, 591
–594
(1980
).11.
O. Y.
Tsvelodub
and Y. Y.
Trifonov
, “On steady-state travelling solutions of an evolution equation describing the behaviour of disturbances in active dissipative media
,” Physica D
39
, 336
–351
(1989
).12.
Y. Y.
Trifonov
, “Two-periodical and quasi-periodical wave solutions of the Kuramoto-Sivashinsky equation and their stability and bifurcation
,” Physica D
54
, 311
–330
(1992
).13.
O. Y.
Tsvelodub
, and S. N.
Samatov
, “VDI Fortschritt-Berichte
,” in Proceedings of the 5th International Berlin Workshop on Transport Phenomena with Moving Boundaries, Berlin, Germany, 8–9 October 2009
, edited by F.-P.
Schinder
(VDI-Verlag
, Düsseldorf
, 2011
).14.
O.
Tsvelodub
and S.
Samatov
, “Effect of the electric field on the wave flow regimes of a thin film of a viscous dielectric fluid
,” J. Appl. Mech. Tech. Phys.
51
, 359
–368
(2010
).15.
W.
Rohlfs
, G. F.
Dietze
, H. D.
Haustein
, O.
Tsvelodub
, and R.
Kneer
, “Experimental investigation into three-dimensional wavy liquid films under the influence of electrostatic forces
,” Exp. Fluids
53
, 1045
–1056
(2012
).16.
See supplementary material at http://dx.doi.org/10.1063/1.4819895 for an executable MATLAB program to reproduce the results presented. In addition, contours of the instantaneous wave profile can be reconstructed on the basis of the calculated harmonics Hn. The program requires the 64-bit version of the MATLAB Compiler Runtime (MCR) version 8.0 (R2012b) which is available free of charge.
© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
