Instability of a gas jet of zero inertia dispersed in a liquid using Lagrangian’s non-linear differential equation

摘要

The stability of a gas jet of zero inertia (of radius R) dispersed in a liquid (i.e. hollow jet) has been developed. The potential and kinetic energies of the system are computed, Lagrangian’s function is constructed and the desired stability criterion is derived on using Lagrangian’s non-linear differential equation. The eigenvalue relation is discussed and the results are analysed numerically for all wavelengths in various modes of disturbance. It is found that the temporal amplification prevailing in the full liquid jet is much lower than that of the hollow jet. The latter is capillary stable in the non-axisymmetric modes m≠0 for all wavelengths and also stable in the axisymmetric mode m=0 whose wavelength λ is equal to or less than the circumference 2πR of the gas jet, where m (an integer) is the azimuthal wavenumber. The hollow jet is unstable only in the mode m=0 as λ>2πR.