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Effect of magnetic field on parametrically driven surface waves. (English) Zbl 1129.76024

Summary: We present a stability analysis of parametrically driven surface waves in liquid metals in the presence of a uniform vertical magnetic field. Floquet analysis gives various subharmonic and harmonic instability zones. The magnetic field stabilizes the onset of parametrically excited surface waves. The minima of all instability zones are raised by a different amount as Chandrasekhar number is raised. The increase in the magnetic field leads to a series of bicritical points at a primary instability in thin layers of a liquid metal. The bicritical points involve one subharmonic and another harmonic solution of different wavenumbers. A tricritical point may also be triggered as a primary instability by tuning the magnetic field.

MSC:

76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Arbell, H. 2002 Pattern formation in two-frequency forced parametric waves. <i>Phys. Rev. E</i>&nbsp;<b>65</b>, 036224. (doi:10.1103/PhysRevE.65.036224).
[2] Besson, T., Edwards, W.S. & Tuckerman, L.S. 1996 Two-frequency parametric excitation of surface waves. <i>Phys. Rev. E</i>&nbsp;<b>54</b>, 507–513, (doi:10.1103/PhysRevE.54.507).
[3] Cerda, E. & Tirapegui, E. 1997 Faraday’s instability for viscous fluids. <i>Phys. Rev. Lett.</i>&nbsp;<b>78</b>, 859–862, (doi:10.1103/PhysRevLett.78.859).
[4] Chandrasekhar, S. 1961 Hydrodynamic and hydromagnetic stability. Oxford, UK: Clarendon. · Zbl 0142.44103
[5] Edwards, W.S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. <i>J. Fluid Mech.</i>&nbsp;<b>278</b>, 123–148, (doi:10.1017/S0022112094003642).
[6] Faraday, M. 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. <i>Phil. Trans. R. Soc. A</i>&nbsp;<b>52</b>, 319–340.
[7] Ji, H., Fox, W. & Pace, D. 2005 Study of magnetohydrodynamic surface waves in liquid gallium. <i>Phys. Plasma</i>&nbsp;<b>12</b>, 012102. (doi:10.1063/1.1822933).
[8] Kudrolli, A., Abraham, M.C. & Gollub, J.P. 2001 Scarred patterns in surface waves. <i>Phys. Rev. E</i>&nbsp;<b>63</b>, 026208. (doi:10.1103/PhysRevE.63.026208).
[9] Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. <i>Proc. R. Soc. A</i>&nbsp;<b>452</b>, 1113–1126. · Zbl 0885.76035
[10] Kumar, K. & Tuckerman, L.S. 1994 Parametric instability of the interface between two fluids. <i>J. Fluid Mech.</i>&nbsp;<b>279</b>, 49–68, (doi:10.1017/S0022112094003812).
[11] Kumar, K., Bandyopadhay, A. & Mondal, G.C. 2004 Parametric instability in a fluid with temperature-dependent surface tension. <i>Europhys. Lett.</i>&nbsp;<b>65</b>, 330–336, (doi:10.1209/epl/i2003-10085-3).
[12] Lamb, H. 1932 Hydrodynamics. Cambridge, UK: Cambridge University Press. · JFM 58.1298.04
[13] Mahr, T. & Rehberg, I. 1998 Magnetic Faraday instability. <i>Europhys. Lett.</i>&nbsp;<b>43</b>, 23–28, (doi:10.1209/epl/i1998-00313-4).
[14] Meneguzzi, M., Sulem, C., Sulem, P.L. & Thual, O. 1987 Three-dimensional numerical simulation of convection in low Prandtl number fluids. <i>J. Fluid Mech.</i>&nbsp;<b>182</b>, 169–191, (doi:10.1017/S0022112087002295). · Zbl 0633.76049
[15] Miles, J.W. 1999 On Faraday resonance of a viscous fluid. <i>J. Fluid Mech.</i>&nbsp;<b>395</b>, 321–325, (doi:10.1017/S0022112099005935). · Zbl 0965.76019
[16] Mondal, G.C. & Kumar, K. 2004 The effect of Coriolis force on Faraday waves. <i>Proc. R. Soc. A</i>&nbsp;<b>469</b>, 897. (doi:10.1098/rspa.2003.1259). · Zbl 1041.76022
[17] Mondal, G.C. & Kumar, K. 2006 Effect of Marangoni and Coriolis forces on multicritical points in Faraday experiments. <i>Phys. Fluids</i>&nbsp;<b>18</b>, 032101. (doi:10.1063/1.2167994).
[18] Müller, H.W. 1998 Parametrically driven surface waves on viscous ferrofluids. <i>Phys. Rev. E</i>&nbsp;<b>58</b>, 6199–6205.
[19] Pétrélis, F., Falcon, É. & Fauve, S. 2000 Parametric stabilization of the Rosensweig instability. <i>Eur. Phys. J. B</i>&nbsp;<b>15</b>, 3–6, (doi:10.1007/s100510051092).
[20] Porter, J. & Silber, M. 2002 Broken symmetries and pattern formation in two-frequency forced faraday waves. <i>Phys. Rev. Lett.</i>&nbsp;<b>89</b>, 084501. (doi:10.1103/PhysRevLett.89.084501).
[21] Silber, M. & Skeldon, A.C. 1999 Parametrically excited surface waves: two-frequency forcing, normal form symmetries, and pattern. <i>Phys. Rev. E</i>&nbsp;<b>59</b>, 5446–5456, (doi:10.1103/PhysRevE.59.5446).
[22] Wagner, C., Müller, H.W. & Knorr, K. 2003 Pattern selection at the bicritical point of the Faraday instability. <i>Phys. Rev. E</i>&nbsp;<b>68</b>, 066204. (doi:10.1103/PhysRevE.68.066204).
[23] Westra, M.T., Binks, D.J. & van de Water, W. 2003 Pattern of Faraday waves. <i>J. Fluid Mech.</i>&nbsp;<b>496</b>, 1–32, (doi:10.1017/S0022112003005895). · Zbl 1068.76033
[24] Zhang, W. 1997 Pattern formation in weakly damped parametric surface waves. <i>J. Fluid Mech.</i>&nbsp;<b>336</b>, 301. (doi:10.1017/S0022112096004764).
[25] Zhang, W. 1997 Pattern formation in weakly damped parametric surface waves driven by two frequency components. <i>J. Fluid Mech.</i>&nbsp;<b>341</b>, 225. (doi:10.1017/S0022112097005387).
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