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

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.

Figures (25)

• Figure 1: Sketch of the bubble-pair configuration(detailed descriptions are provided in the main text). $(a)$ Steady, axi-symmetric base flow and definition of the principal geometric parameters. $(b)$ Typical unstable configurations following an infinitesimally imposed lateral perturbation. $(c)$ Computational domain and spatial discretisation adopted for the ALE–LSA simulations.
• Figure 2: Phase map associated with the formation of a standing eddy connecting the two bubbles for $\chi = 1.3$. $(a)$ Critical curve in the $(S,Re)$ (or $(\delta S,Re)$) plane marking the onset of a connected recirculation between the two bubbles, open circle denotes the marginal configuration, corresponding to $Re \simeq 180$. $(b)$ Streamlines illustrating representative flow structures for the cases indicated in $(a)$: A $(\Rey, S) = (30, 0.98)$, B $(100, 0.98)$, C $(100, 1.20)$, and D $(100, 1.25)$. The connected eddy appears at points B and C, whereas points A and D correspond to the uni-direction flow structure.
• Figure 3: Same as \ref{['large_aspect_ratio-chi1.3']}, but for a more oblate shape with $\chi = 1.9$. $(a)$ Critical curve in the $(S, Re)$ (or $(\delta S,Re)$) plane marking the onset of a connected recirculation between the two bubbles. $(b)$ Streamlines illustrating the corresponding flow structures for the cases indicated: A $(Re, S) = (100, 2.0)$, B $(100, 2.5)$, C $(50, 2.5)$, and D $(200, 2.5)$. The connected eddy appears at point A, whereas points B--D correspond to the detached eddy structure.
• Figure 4: Equilibrium centre-to-centre distance $S_e$ of the bubble pair in the steady, axi-symmetric base state, as a function of the bubble aspect ratio ($\chi$) and Reynolds number ($\Rey$). $(a)$ Variation of $S_e$ with $\chi$ for different Reynolds numbers. $(b)$ Variation of $S_e$ with $\Rey$ for different aspect ratios. Closed symbols in $(b)$ mark the minimum $S_e$ at which the two bubbles come into contact, corresponding to $S_e = S_{\min} = \chi^{-2/3}$ ($\delta S = 0$).
• Figure 5: Comparison between the present ALE–LSA results with rotational degrees of freedom suppressed, the VOF–DNS data of zhang2021three, and the supporting results obtained from the EBM–DNS framework. $(a)$ Schematic of the ALE–LSA configuration. Blue arrows indicate that only lateral (transverse) translational motion is permitted, while rotations are constrained. $(b)$ Neutral curve of the stationary mode obtained from the non-rotating ALE–LSA () compared with the VOF–DNS results of zhang2021three (their figure 21). The dashed line marks the transition boundary between the stable () and unstable (green,0;blue,128, draw=rgb,255:red,128;green,0;blue,128] (0,0) -- (4pt,0) -- (4pt,4pt) -- (0,4pt) -- cycle;) regimes observed in the VOF–DNS. Symbols: neutral curve from the non-rotating ALE–LSA; stable bubble coalescence; green,0;blue,128, draw=rgb,255:red,128;green,0;blue,128] (0,0) -- (4pt,0) -- (4pt,4pt) -- (0,4pt) -- cycle; unstable lateral escape of the trailing bubble (TB). $(c)$ Lift force experienced by the TB computed using the EBM–DNS framework, showing the total ($C_L$), pressure ($C_{L, p}$), and viscous ($C_{L, \mu}$) lift coefficients as functions of $\Rey$ for $(\chi, S, S_r) = (1.6, 3, 0.1)$, corresponding to a horizontally aligned LB and TB ($\Theta_{LB} = \Theta_{TB} = 0^{\circ}$). Additional results for bubble pairs with $\chi = 1.0$ (open square) and $1.5$ (open triangle) are included for comparison.