Instability of a viscous liquid jet surrounded by a viscous gas in a vertical pipe

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Abstract
The instability of a cylindrical liquid jet encapsulated by a viscous gas in a pipe is analysed in a parameter space spanned by the Reynolds number, the Froude number, the Weber number, the density ratio, the viscosity ratio, and the diameter ratio. A convergent solution of the problem is constructed by a Galerkin projection with two orthogonal sets of functions. Two distinctively different modes of instability are obtained. The first is the Rayleigh mode which tends to break up the jet into drops of diameter comparable with the jet diameter. The amplification rate of the disturbance belonging to this mode depends weakly on all parameters except the Weber number which represents the ratio of the surface tension force to the inertia force at the interface. The mechanism of the instability remains that of capillary pinching even in the presence of a viscous gas and gravity. However, the surface tension is stabilizing in the other mode termed the Taylor mode. The Taylor mode instability is due to the pressure and shear fluctuations at the interface. This mode tends to produce droplets of diameters much smaller than that of the jet. It is shown that the former mode appears when the Weber number is much larger than the gas to liquid density ratio. When this ratio is of order one, the instability can be due to either modes depending on the values of the rest of the parameters. When the density ratio is much larger than the Weber number, Taylor's atomization mode replaces the Rayleigh mode.

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