Stability of surfactant-laden core-annular flow and rod-annular flow to non-axisymmetric modes

The linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but under certain conditions the flow may be linearly stable to axisymmetric disturbances but linearly unstable to non-axisymmetric disturbances.