Influence of variable viscosity on density-driven instabilities in capillary tubes

Abstract

A linear stability analysis is conducted for the density-driven flow of variable viscosity miscible fluids in a vertically oriented capillary tube. The main goal is to assess the competition between the axisymmetric and the first azimuthal mode as a function of Rayleigh number, viscosity ratio and interfacial thickness parameter. In the absence of a net flow, the symmetry properties of the linearized set of equations indicate that the growth rates do not depend on which of the two fluids is the more viscous, although the shape of the eigenmodes does. For most parameter combinations, the first azimuthal mode is found to have larger growth rates than the axisymmetric mode. For thin interfaces and large Rayleigh numbers, however, the axisymmetric mode dominates above a certain viscosity ratio. An unexpected result is found regarding the influence of the interface thickness on the instability. For large viscosity ratios, intermediate interface thicknesses are found to be more unstable than either very thin or very thick interfaces. The reason for this behaviour is traced to a shift of the eigenfunctions towards the less viscous fluid, which allows the instability to grow in an overall less viscous environment. In the presence of a net axial flow, the upward and downward displacements of a more viscous fluid by a less viscous one are seen to result in the same growth rate. For large viscosity ratios, the axisymmetric mode becomes destabilized by the net flow, whereas the leading azimuthal mode is stabilized. This trend is in line with experimental observations.