Journal of Magnetism and Magnetic Materials 111 (i992)313-324 North-Holland /HN _1 MHD instability of a streaming jet having a solid mantle acted upon by a transverse magnetic field Ahmed E. Radwan Department of Mathematics, Faculty of Science, Ain-Shams Unit'ersity, Cairo, Eg3'pt Received 6 January 1991; in revised form 6 March 1991 The magnetohydrodynamic (MHD) stability of a streaming annular jet having a solid rod as a mantle, subjected to the capillarity, inertia and electromagnetic forces is presented. An eigenvalue relation valid to all axisymmetric and non-axisym- me,ric disturbances is derived. The (in-)stability characteristics of that mode! are identified analytically and confirmed numerically for numerous forms of the (interior and exterior) magnetic field, the radii ratio of the solid rod to the fluid cylinder and the basic flow vel~cit3,. There were unstable states to non-axisymmetric disturbances due to the ~trong destabilizing influences of the streaming and the vacuum varying magnetic field. The required restrictions for the stability are determined and several special cases are studied. The thicker the solid cylinder so the greater its stabilizing influe~ce and the MHD non-streaming annular jet is absolutely stable. However, the streaming destabilizing effect will upset that situation and there will be unstable states not only to axisymmetric bui also to non-axisymmetric disturbances. Duc to the strong stabilizing influences of the interior field and the curvature pressure on the very thin annular jet, around a very thic' solid rod, :he (ia-)stabilities are slowed and more easier observations are expected to be allowed than in the case of the full jet and also than in the case of the annular jet having a gas-core jet. !. Introduction The experimental and theoretical hydrody- namic stability of a full liquid jet has been studied comprehensively since more than a century. This is not only from the academic viewpoint but also for its crucial applications in miscellaneous do- mains. Research in this field intensified when it became apparent that the physical properties ef liquid jets play a fundamental role in a rapidly growing number of applications such as: the spin- ning of synthetic fibers, fuel atomization, spray drying and even the diagnosis of certain abnor- malities of the human urinary tract. Plateau [1] was the first to obtain the capillary critical wavelength experimentally and theoreti- cally using a naive approach. Rayleigh [2] derived the dispersion relation, developed the most im- portant concept of maximum mode of instability and laid the basic foundations for the theoretical treatment of such problems. The hydrodynamic and hydromagnetic stability of the full jet and those of various configurations were given by Chandrasekhar [3] also with miscellaneous exten- sions. The nonlinear capillary instability of a full liquid jet has been investigated using different approaches by Wang [4], Yuep, [5], Nayfeh [6] and the complete analysis was finally elaborated by Kabutani, Indue and Kan [7] using the ~:..rivative expansion approaches developed by Kawahara [8]. The hydrodynamic and hydromagnetic stability of an annular liquid jet is also important to be investigated. The response of an incompressible- inviscid annular liquid jet (.gas cylinder ambient with liquid) subjected to capillary and inertia forces (as the inertia ' .... "~ '~ ..... : . . . . . . ~ : . . . . llqUlkl ltol~.c 13 pt~uullllilvtlLt over th,t of the gas) for small axisymmetric dis- turbances was given by Chandrasekh~r [3] (p. 540). Recently Kendall [9] has performed very inter- esting experiments with modern equipment on the capillary instability of an annular liquid jet for 0304-8853/92/$05.00 © 1992 -- Elsevier Science Publishers B.V. All rights reserved