DOI 10.1140/epjp/i2011-11007-y Regular Article Eur. Phys. J. Plus (2011) 126:7 T HE EUROPEAN P HYSICAL JOURNAL PLUS Magnetohydrodynamic stability of an oscillating fluid cylinder A.E. Radwan 1, a , W.M. Hydia 2 , and A.A. Hasan 3, b 1 Mathematics Department, Faculty of Science, Ain-Shams University Cairo, Egypt 2 Mathematics Department, Women University College, Ain-Shams University Cairo, Egypt 3 Basic and Applied Sciences Department, College of Engineering and Technology, Arab Academy for Science & Technology and Maritime Transport (AAST), P.O. Box 2033, Cairo, Egypt Received: 7 June 2010 Published online: 19 January 2011 – c  Societ`a Italiana di Fisica / Springer-Verlag 2011 Abstract. The magnetohydrodynamic stability of an oscillating streaming fluid cylinder pervaded by lon- gitudinal magnetic fields has been discussed. The problem is formulated, the basic equations are solved, upon appropriate boundary conditions the non-singular solutions are identified. Moreover, a second-order integro-differential equation in the amplitude of the deflection of perturbation, of Mathieu equation type is derived. The magnetic fields interior and exterior to the fluid have strong stabilizing influences for all perturbation modes. The capillary force is destabilizing only in the small axisymmetric domain while it is stabilizing in all the other domains of perturbation modes. The oscillating streaming of the fluid has a strong stabilizing tendency. The destabilizing character of the model due to the capillary force could be reduced and suppressed, then the stability sets in. 1 Introduction The process of a liquid jet capillary instability that leads to breaking it into droplets or, more precisely, into spray is, in reality, a critical feature of many engineering applications. E.g. fuel injection in engines, ink jet printing, spray painting processes. See Gavrilyuk and Teshukov [1]. The stability of a fluid cylinder under the action of the capillary force and other factors has received the attention of several scientists (cf. Donnelly and Glaberson [2], Yuen [3], Nayfeh [4], Nayfeh and Hasan [5], Kakutani [6]), since the pioneering works of Rayleigh [7]. The effect of the electromagnetic force (for axisymmetric perturbation) on the capillary instability of a liquid jet has been discussed by Chandrasekhar [8]. Several extensions of this problem have been recently carried out by Radwan [9–11]. In all foregoing studies the fluid is assumed to be at rest or even streaming uniformly (see also Radwan and Hasan [12]). Here, we intend to investigate the magnetohydrodynamic instability of an oscillating streaming fluid jet. Such assumption of oscillating streaming is vital for the stability of the present model and for its mechanical applications. 2 Formulation of the problem Consider a perfectly conducting fluid cylinder of uniform cross-section (radius R o ) of permeability coefficient μ. The fluid is assumed to be incompressible, non-viscous, and penetrated by a uniform longitudinal magnetic field H (i) o = (0, 0,H o ). (1) The surrounding medium exterior to the fluid cylinder is pervaded by a magnetic field H (e) o = (0, 0, αH o ), (2) where α is parameter while H o is the intensity of the magnetic field interior to the fluid. a e-mail: ahmed16853@yahoo.com b e-mail: alfaisal772001@yahoo.com