Recent experiments by Pailha et al. [Phys. Fluids24, 021702 (2012)10.1063/1.3682772] uncovered a rich array of propagation modes when air displaces oil from axially uniform tubes that have local variations in flow resistance within their cross-sections. The behaviour is particularly surprising because only a single, symmetric mode has been observed in tubes of regular cross-section, e.g., circular, elliptical, rectangular, and polygonal. In this paper, we present experimental results describing a new mode, an asymmetric localised air finger, that persists in the limit of zero propagation speed. We show that the experimental observations are consistent with a model based on capillary static calculations within the tube's cross-section, and the observed bistability is a consequence of the existence of multiple solutions to the Young-Laplace equations. The model also provides an upper bound for the previously reported symmetry-breaking bifurcation [A. de Lózar, A. Heap, F. Box, A. L. Hazel, and A. Juel, Phys. Fluids21, 101702 (2009)10.1063/1.3247879].
