Effect of magnetic field on parametrically driven surface waves (original) (raw)
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Surface waves created by low-frequency magnetic fields
European Journal of Mechanics - B/Fluids, 2005
This paper analyses the effects of a low frequency A.C. magnetic field on the free surface of a liquid metal. The action of the vertical and uniform magnetic field is twofold. First it creates forced standing surface waves which generally exhibit symmetry related to that of the container; second it triggers non-symmetric free surface instabilities superimposed on the forced regime. A previous paper considered the case of a circular cylindrical tank where axisymmetric forced standing waves caused an electric current perturbation which then excited non-axisymmetric waves at a critical A.C. field intensity. Nonlinear interaction between the symmetric and non-symmetric modes was not taken into account. The present work treats the problem from a more general standpoint. Equilibrium perturbations are developed systematically to order N 2 (where N is the magnetic interaction parameter) and at this level of approximation we also need to consider nonlinear mode interactions and electromagnetic damping. The theory applies to tanks of arbitrary shape and the O(N) irrotational motion may be described by the torsion function for the particular pool cross-section. For circular and annular tanks we then derive a system of coupled Mathieu-Hill equations for the time-development of non-symmetric surface modes. Two main types of parametric resonance are predicted, namely the single or combination mode, and the particular type observed may depend on the geometry of the tank. Results of the stability analysis are confirmed by experimental work carried out in mercury pools.
Study of small-amplitude magnetohydrodynamic surface waves on liquid metal
Physics of Plasmas, 2005
Magnetohydrodynamic (MHD) surface waves on liquid metal are studied theoretically and experimentally in the small magnetic Reynolds number limit. A linear dispersion relation is derived when a horizontal magnetic field and a horizontal electric current is imposed. Waves always damp in the deep liquid limit with a magnetic field parallel to the propagation direction. When the magnetic field is weak, waves are weakly damped and the real part of the dispersion is unaffected, while in the opposite limit waves are strongly damped with shortened wavelengths. In a table-top experiment, planar MHD surface waves on liquid gallium are studied in detail in the regime of weak magnetic field and deep liquid. A noninvasive diagnostic accurately measures surface waves at multiple locations by reflecting an array of lasers off the surface onto a screen, which is recorded by an intensified-CCD (charge-coupled device) camera. The measured dispersion relation is consistent with the linear theory with a reduced surface tension likely due to surface oxidation. In excellent agreement with linear theory, it is observed that surface waves are damped only when a horizontal magnetic field is imposed parallel to the propagation direction. No damping is observed under a perpendicular magnetic field. The existence of strong wave damping even without magnetic field suggests the importance of the surface oxide layer. Implications to the liquid metal wall concept in fusion reactors, especially on the wave damping and a Rayleigh-Taylor instability when the Lorentz force is used to support liquid metal layer against gravity, are discussed.
Parametrically Excited Surface Waves in Magnetic Fluids: Observation of Domain Structures
Physical Review Letters, 1998
Observations of parametrically excited surface waves in a magnetic fluid are presented. Under the influence of a magnetic field these waves have a non-monotonic dispersion relation, which leads to a richer behavior than in ordinary liquids. We report observation of three novel effects, namely: i) domain structures, ii) oscillating defects and iii) relaxational phase oscillations.
Surface waves in ferrofluids under vertical magnetic field
The European Physical Journal B, 1999
We present here new experimental results about the waves at the horizontal free surface of a magnetic fluid submitted to a normal magnetic field. The waves are generated by a small modulation at frequency ω of the vertical field H e . Using a shadowgraph method, we are able to measure the wavevector k of the 2D waves for a given value of ω and H e . The dispersion relation of the surface waves is established experimentally. On the other hand, we propose a theoretical derivation of the dispersion equation which includes a more complete treatment of the magnetic term than the previous works. Finally, we conclude that a linear and inviscid analysis is sufficient to fit well the experimental data, except in the vicinity of the critical field where a surface instability occurs. PACS. 75.50.Mm Magnetic liquids -47.35.+i Hydrodynamic waves -68.10.-m Fluid surfaces and fluid-fluid interface
Applied Thermal Engineering, 2015
The actual requirements of high-grade materials lead to a constant increase in energy consumption, creating the necessity of new efficient and environmentally friendly technologies. In the past decades, the Electromagnetic Processing of Materials (EPM) has been established as an alternative for the manipulation, monitoring and control of conducting materials such as liquid metals, semiconductors, molten salts, and electrolytes, during the processing stages. Besides the application of nonintrusive methods that allow a convenient handling of the material while keeping it free from contamination of external agents, EPM offers the possibility of a the rational use of energy and the search for environmentally friendly innovation technologies. This paper presents a specific application for the control of liquid metal free surface flows, namely, the damping of surface waves with a uniform magnetic field. The attention is mainly focused on the orientation of the applied magnetic field relative to the free surface, the effects of the depth of the fluid layer, and the surface tension.
EPL, 1998
In a magnetic fluid parametrically driven surface waves can be excited by an external oscillating magnetic field. A static magnetic field changes the restoring forces and damping coefficients of the various surface waves. This property enables the excitation of both subharmonic and harmonic responses of the standing waves.
Edge pinch instability of liquid metal sheet in a transverse high-frequency ac magnetic field
Physical Review E, 2006
We analyze the linear stability of the edge of a thin liquid metal layer subject to a transverse high-frequency ac magnetic field. The layer is treated as a perfectly conducting liquid sheet that allows us to solve the problem analytically for both a semi-infinite geometry with a straight edge and a thin disk of finite radius. It is shown that the long-wave perturbations of a straight edge are monotonically unstable when the wave number exceeds the critical value k c = F 0 / ͑␥l 0 ͒, which is determined by the linear density of the electromagnetic force F 0 acting on the edge, the surface tension ␥, and the effective arclength of edge thickness l 0 . Perturbations with wavelength shorter than critical are stabilized by the surface tension, whereas the growth rate of long-wave perturbations reduces as ϳk for k → 0. Thus, there is the fastest growing perturbation with the wave number k max =2/3k c . When the layer is arranged vertically, long-wave perturbations are stabilized by the gravity, and the critical perturbation is characterized by the capillary wave number k c = ͱ g / ␥, where g is the acceleration due to gravity and is the density of metal. In this case, the critical linear density of electromagnetic force is F 0,c =2k c l 0 ␥, which corresponds to the critical current amplitude I 0,c =4 ͱ k c l 0 L␥ / 0 when the magnetic field is generated by a straight wire at the distance L directly above the edge. By applying the general approach developed for the semi-infinite sheet, we find that a circular disk of radius R 0 placed in a transverse uniform high-frequency ac magnetic field with the induction amplitude B 0 becomes linearly unstable with respect to exponentially growing perturbation with the azimuthal wave number m = 2 when the magnetic Bond number exceeds Bm c = B 0 2 R 0 2 / ͑2 0 l 0 ␥͒ =3. For BmϾ Bm c , the wave number of the fastest growing perturbation is m max = ͓2Bm/ ͑3͔͒. These theoretical results agree well with the experimental observations.
Instabilities in liquid metals controlled by constant magnetic field—Part I: vertical magnetic field
Journal of Crystal Growth, 2002
We consider the e ect of a constant magnetic ÿeld on buoyant ows generated by temperature gradients. We focus on the domain of weak magnetic ÿelds, i.e., small values of the Hartmann number Ha, for which general scaling laws can be derived. Concerning the braking of these buoyant ows, it was found to scale at small Ha as even powers of Ha. Concerning the damping of the oscillations, it can be shown that the instability characteristics, critical threshold expressed through the critical Grashof number Gr c , critical eigenvector, and critical pulsation also scale as even powers of Ha. In particular, this gives an initial MHD stabilization e ect at small Ha of the form Gr c − Gr c0 ∼ Ha 2 where Gr c0 is the critical Grashof number at Ha = 0. These ÿndings have been illustrated by results obtained in the case of the ow in an inÿnite layer.