  We provide an extension to previous analysis of the localised beading
instability of soft slender tubes under surface tension and axial stretching.
The primary questions pondered here are: under what loading conditions, if any,
can bifurcation into circumferential buckling modes occur, and do such
solutions dominate localisation and periodic axial modes? Three distinct
boundary conditions are considered; in case 1 the tube's curved surfaces are
traction free and under surface tension, whilst in cases 2 and 3 the inner and
outer surfaces (respectively) are fixed to prevent radial displacement and
surface tension. A linear bifurcation analysis is conducted to determine
numerically the existence of circumferential mode solutions. In case 1 we focus
on the tensile stress regime given the preference of slender compressed tubes
towards Euler buckling over axial wrinkling. We show that tubes under several
loading paths are highly sensitive to circumferential modes; in contrast,
localised and periodic axial modes are absent, suggesting that the
circumferential buckling is dominant by default. In case 2, circumferential
mode solutions are associated with negative surface tension values and thus are
physically implausible. Circumferential buckling solutions are shown to exist
in case 3 for tensile and compressive axial loads, and we demonstrate for
multiple loading scenarios their dominance over localisation and periodic axial
modes within specific parameter regimes.
