Density-driven instabilities of miscible fluids in a capillary tube: linear stability analysis

Abstract

A linear stability analysis is presented for the miscible interface formed by placing a heavier fluid above a lighter one in a vertically oriented capillary tube. The analysis is based on the three-dimensional Stokes equations, coupled to a convection–diffusion equation for the concentration field, in cylindrical coordinates. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenmodes as functions of the governing parameters in the form of a Rayleigh number and a dimensionless interfacial thickness. The dispersion relations show that for all values of the governing parameters the three-dimensional mode with an azimuthal wavenumber of 1 represents the most unstable disturbance. The stability results also indicate the existence of a critical Rayleigh number of about 920, below which all perturbations are stable. The growth rates are seen to reach a plateau for Rayleigh numbers in excess of $10^6$. In order to analyse the experimental observations by Kuang et al.(2002), which show that a small amount of net flow can stabilize the azimuthal instability mode and maintain an axisymmetric evolution, a base flow of Poiseuille type is included in the linear stability analysis. Results show that a weak base flow leads to a slight reduction of the growth rates of both axisymmetric and azimuthal modes. However, within the velocity interval that could be analysed in the present investigation, there is no indication that the axisymmetric mode overtakes its azimuthal counterpart.