Journal of Magnetism and Magnetic Materials 87 (1990) 311-322 North-Holland 311 NONAXISYMMETRIC CAPILLARY INSTABILrFY OF A STREAMING JET HAVING A SOLID MANTLE EMBEDDED IN UNIFORM AXIAL MAGNETIC FIELDS Ahmed E. RADWAN 1 Department of Mathematics, Faculty of Science, Ain-Shams University, Abbassia, Cairo, Egypt Received 15 January 1990 The magnetohydrodynamic (in-)stability of a streaming annular liquid jet having a solid rod as a mantle pervaded by and submerged in longitudinal magnetic fields of different intensity is presented. A general eigenvalue relation is derived and studied analytically and the results are confirmed nmnerieally for numerous states. The model is capillary unstable for small axisymmetric disturbances. The thicker the solid rod so the greater is its stabilizing influence. The magnetic fields have strong stabilizing effects for all wavelengths in all axisymmetric and nonaxisymmetric disturbances. Above a certain value of the applied basic magnetic field, the capillary instability is completely suppressed and stability sets in. The streaming has a strong destabilizing influence and that influence increases with increasing streaming velocity. The streaming has the influence of enlarging the MHD unstable domains and decreasing the stable states not only in the axisymmetrie mode but also in the nonaxisymmetrie disturbances. The streaming destabilizing influence can only be suppressed if the Alfv~n wave velocity is superior to the streaming velocity; in particular it manifests itself when the fluid layers are very thin around a very thick rod. 1. Introduction The linear [1-3] and noxdinear [4-7] (in-)stabil- ity of a full liquid jet subject to the capillarity or/and other forces has been aiscussed and in- vestigated over many years by means of different techniques. Observation has shown that the capil- larity instability, driven by surface tension forces on the liquid jet interface, leads to the breakup of the jet into droplets. This knowledge has been essential for numerous crucial applications rang- ing from the design of sprays to the design of inkjet printers. Recently the hydrodynamic stability study of an annular liquid jet has flourished experimentally a,xu ,l.w,,~u,.,~y..,, "~Sew,.,x"¢" .,..a,.,,.ol on,-,l~oa_ tions in the astrophysical domain, Kendall [8] has performed some neat experiments to produce an annular jet (a liquid jet having a gas jet as a ! Present address: Department of Mathematics, Centre of Science aria Maths. and T.T.J. College, Dammam, Saudi Arabia. mantle) and to study its stability for different amplitudes wavelengths such that the liquid in- ertia is superior to that of the gas. Radwan and Elazab [9] examined the stability of a viscous hollow jet for small axisymmetric disturbances. The capillary instability of a streaming hollow jet is discussed for all kinds of disturbances by Radwan [10]. However, Kendall [8] did attract and draw attention by investigating the stability of the annular liquid jet in general. The hydrodynamic stability of an annular liquid jet (radius R 0) hav- ing a solid axis (radius qRo) as a mantle, using the energy principle, has recently been elaborated ana- lyrically and numerically by Radwan [11]. It is found, for all q values, that the model is stable to nen~-dsym_metric modes for all wavelengths and also stable (unstable) to axisymmetric dis- turbances if the perturbed wavelength is equal to or shorter (longer) than 2~rR 0. The region of insta- bility is decreasing with increasing q values (0 ~<q < 1) but the instability is never suppressed how- ever large the value of q. The purpose of the present paper is to examine 0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)