A pair of oblate bubbles rising in-line: a linear stability analysis

TL;DR

This study reveals that inclination-induced rotational feedback, rather than deformation, primarily stabilizes in-line rising oblate bubbles, highlighting the complex hydrodynamic interactions governing their stability and oscillations.

Contribution

It demonstrates that inclination-induced lift, not deformation, is the main stabilizing mechanism for in-line oblate bubbles, providing a unified stability framework.

Findings

Inclination-shear coupling governs stability recovery with aspect ratio.

Unstable drafting-kissing-tumbling mode results from short-range bubble interactions.

Oscillatory mode acts as a hydrodynamic spring linking the bubbles.

Abstract

The stability of two bubbles rising initially in-line through a viscous liquid is revisited using a global linear stability analysis formulated within an Arbitrary Lagrangian-Eulerian framework, complemented by fully resolved Embedded Boundary Method simulations. Whereas previous studies attributed the promoted in-line stability of oblate bubbles to a deformation-enhanced wake entrainment, the present analysis demonstrates that the dominant stabilizing mechanism arises instead from an inclination-induced rotational feedback generated as the trailing bubble experiences the asymmetric shear of the leading bubble's wake. This inclination-shear coupling, rather than deformation itself, governs the recovery of stability with increasing aspect ratio. Furthermore, the results reveal that the unstable drafting-kissing-tumbling mode originates from short-range, two-way coupling between the bubbles, whereas the asymmetric side-escape mode corresponds to a long-range, one-way interaction dominated by the trailing bubble response. In addition, a previously unreported oscillatory global mode emerges from the unsteady recirculation linking the two bubbles, acting as a hydrodynamic spring whose effective stiffness and damping govern the oscillation frequency and growth rate. Together, these findings identify inclination-induced lift as the primary mechanism controlling the stability of rising bubble pairs and provide a unified framework for interpreting their stationary and oscillatory transitions across a broad range of inertial and deformable regimes.