A cylindrical interface containing a viscous liquid set into axial motion is subject to a capillary and to a surface‐wave instability. Clues from previous studies suggest that, even though both mechanisms separately are destabilizing, under certain circumstances their mutual interaction can lead to a stable interface; shear can stabilize capillary breakup. These clues lead the authors to consider an axial flow through an annular cross section bounded on the inside by a rigid rod and on the outside by a deformable interface. The competition between the two mechanisms is studied through the temporal growth of infinitesimal axisymmetric and nonaxisymmetric disturbances. This examination of temporal stability shows that, indeed, for geometries corresponding to thin annular layers both instabilities can be completely suppressed—disturbances of all wavelengths decay.