IL NUOVO CIMENTO VOL. 106 B, N. 9 Settembre 1991 Gravitating Instability of a Compound Liquid Jet. AHMED E. RADWAN Department of Mathematics and Computer Science, Faculty of Science Ain-Shams University -Abbassia, Cairo, Egypt (ricevuto il 20 Ottobre 1988; approvato il 14 Gennaio 1991) Summary. -- The stability of a self-gravitating gas jet ambient with a self- gravitating liquid is discussed analytically and the results are confwmed numeri- cally. A general eigenvalue relation describing the characteristics of the gas-core liquid jet, based on the linear perturbation techniques, is derived by employing the energy principle. It is found that the fluid densities ratio S plays an important role in the (de-)stabilizing of the present model. If 0 <~ S < 1 (S = s2/sl, where s2 is the liquid density and sl is the gas density), the model is unstable for certain values of the longitudinal wave number x (mainly 0 ~<x < 1.0668) and stable for the rest. However with increasing S values provided that 0 < S < 1 the unstable domain is fastly decreasing but never vanishing. As S> 1, unexpected results have been obtained according to which the model is unstable gravitationally not only for long wavelengths but also for very short wavelengths. These analytical results are interpreted physically and confLrmed numerically and the disturbance wave numbers at which stability as well as instability are tabulated. If S = 0 we recover the reported works in the literature. PACS 03.40.Gc - Fluid dynamics: general mathematical aspects. 1. - Introduction. The stability of a full liquid jet has been studied since a long time ago, for its important applications in several domains in physics. It was Plateau[l] who first obtained the capillary critical wavelength experimentally and theoretically. Rayteigh [2] derived the dispersion relation and developed the important concept of maximum mode of instability based on the linear theory. By extending Rayleigh's theory, Weber [3] studied the capillary instability of a viscous liquid jet. These and other extensions were summarized by Rayleigh [4]; see also Chandrasekhar [5]. The effect of nonlinearities on the capillary instability of a full liquid jet was considered by Yuen [6], Wang [7], Nayfeh [8], Nayfeh and Hassan [9] and a complete analysis was given by Kakutani et al. [10]. The response of a self-gravitating incompressible cylinder to small axisymmetric disturbances was investigated by Chandrasekhar and Fermi[ll] by means of an energy principle. Soon afterwards, Oganesian [12] was the first to perform a detailed 969