Mechanics and Mechanical Engineering Vol. 11, No 1 (2007) 87–96 c  Technical University of Lodz Stability of Compressible Magnetized Hollow Cylinder Pervaded by Azimuthally Field Ahmed E. Radwan Mathematics Department, Faculty of Science, Ain-Shams University, Cairo, Egypt Nasser E. Elazab and Nahed S. Hussien Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt Received (05 March 2006) Revised (25 March 2007) Accepted (8 May 2007) The stability of compressible magnetized hollow cylinder (gas jet embedded into a liquid) pervaded by varying azimuthally magnetic field has been developed for all symmetric m = 0 and asymmetric m = 0 pertubation modes (m transverse wavenumber). The problem is formulated well, apart from the singular solutions the different variables are determined, the stability criterion is derived and discussed. The axial field in the liquid region is stabilizing for all short and long wavelengths and that effect is independent of m values. In contrast, the azimuthal field in the gas is destabilizing or not according to restrictions and m values. The capillary force along the gas – liquid interface is destabilizing only for m = 0, for small longitudinal wavenumber and stabilizing in the rest. The compressibility has a strong stabilizing effect for all m ≥ 0 disturbance modes. Here the instability due to azimuthal field and the capillary force could be completely suppressed under restrictions and stability sets in. Keywords : magnetic field, capillary force, magnetohydrodynamic stability 1. Introduction The stability of a (gas cylinder immersed into a liquid) hollow jet under the effect of the capillary force is indicated for first time by Chandrasekhar [2], see also Rayleigh [12]. That is as the perturbation is axisymmetric for azimuthal direction. Drazin and Reid [4] gave the dispersion relation in a classical method for such model. In such model it is proposed that the liquid inertia force is predominate over that of the gas jet during perturbation. But one has to keep in mind that the gas pressure in the unperturbed state must be greater than other stresses of the model, otherwise the model will collapse and the gas will be distributed in the liquid layers. Cheng [3] discussed analytically the capillary instability of unbounded hollow jet. Kendall [5] performed very interesting experiments with modern equipment for studying the