Taylor bubble motion in stagnant and flowing liquids in vertical pipes. Part II: Linear stability analysis

Abstract

In this study, we examine the linear stability of an axisymmetric Taylor bubble moving steadily in a flowing liquid enclosed in a circular tube. Linearisation is performed about axisymmetric base states obtained in Part I of this study by Abubakar and Matar (2021). The stability is characterised by the dimensionless inverse viscosity $\left( Nf \right)$, E\"{o}tv\"{o}s $\left( Eo \right)$, and Froude numbers $\left( U_m \right)$, the latter being based on the centreline liquid velocity. The analysis shows that there exist regions of $(Nf,Eo,U_m)$ space within which the bubble is unstable and assumes an asymmetric shape. To elucidate the mechanisms underlying the instability, an energy budget analysis is carried out which reveals that perturbation growth is driven by the bubble pressure for $Eo \geq 100$, and by the tangential interfacial stress for $Eo < 100$. Examples of the asymmetric bubble shapes and their associated flow fields are also provided near the onset of instability for a wide range of $Nf$, $Eo$, and $U_m$.