Abstract
The weakly nonlinear stability due to an axisymmetric disturbance of two immiscible liquids with different viscosities in a pipe is worked out. The most important conclusions drawn from this work are the independence of the Reynolds number in the stability criterion, the existence of stable waves with sawtooth profile when the more viscous fluid is located at the core and it occupies most of the pipe, and the destabilizing effect of the surface tension, effect that can be overtaken by the stabilizing effect due to the difference in viscosity of the two fluids, when the viscosity of the ring fluid is much smaller than the viscosity of the core fluid and the thickness of the ring flow is smaller than the thickness of the core flow.
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