Effect of magnetic field on parametrically driven surface waves BY SUPRIYO PAUL AND KRISHNA KUMAR* Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur 721 302, India Stability analysis of parametrically driven surface waves in liquid metals in the presence of a uniform vertical magnetic field is presented. Floquet analysis gives various subharmonic and harmonic instability zones. The magnetic field stabilizes the onset of parametrically excited surface waves. The minima of all the instability zones are raised by a different amount as the Chandrasekhar number Q 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. Keywords: driven surface waves; magnetic field; multicritical points 1. Introduction Parametrically driven surface waves, known since the pioneering experiments of Faraday (1831), have attracted considerable interests (Zhang & Vin ˜als 1997a,b; Mahr & Rehberg 1998; Mu ¨ller 1998; Miles 1999; Pe ´tre ´lis et al. 2000; Kudrolli et al. 2001; Arbell & Fineberg 2002; Porter & Silber 2002; Wagner et al. 2003; Westra et al. 2003). Thin layers of viscous fluids show an interesting possibility of a bicritical point (Kumar 1996; Cerda & Tirapegui 1997; Kumar et al. 2004; Mondal & Kumar 2004, 2006) at the onset of driven surface waves. They are also seen in the presence of a two-frequency vertical vibration (Edwards & Fauve 1994; Besson et al. 1996; Silber & Skeldon 1999). The magnetic field is known to have a stabilizing effect on convective instability (e.g. Chandrasekhar 1961) in liquid metals. The role of the horizontal magnetic field on surface waves in liquid metals was investigated recently by Ji et al. (2005). However, the effects of the magnetic field on parametrically driven surface waves in liquid metals are not investigated. The presence of the magnetic field increases dissipation in fluids due to Joule heating. The selection of wavenumber in thin layers of viscous fluids is strongly affected by the dissipation. The presence of the magnetic field is therefore likely to modify multicritical points, which occur in thin layers of fluids. This may significantly affect pattern selection close to the onset of driven surface waves. Proc. R. Soc. A (2007) 463, 711–722 doi:10.1098/rspa.2006.1789 Published online 5 December 2006 * Author for correspondence (kumar@phy.iitkgp.ernet.in). Received 18 July 2006 Accepted 26 October 2006 711 This journal is q 2006 The Royal Society on August 15, 2017 http://rspa.royalsocietypublishing.org/ Downloaded from