Deformation and stability of a gas bubble in a biaxial straining flow

TL;DR

This work uses the L-ALE framework to map the full bifurcation structure and linear stability of a gas bubble at the stagnation point of a biaxial straining flow across a range of $We$ and $Oh$. It uncovers a saddle-node bifurcation with strongly $Oh$-dependent equilibrium shapes, including both oblate and prolate morphologies and two sets of disconnected branches. The linear stability analysis reveals symmetry-preserving and drift modes, plus a novel azimuthal $m=2$ instability on oblate shapes, suggesting possible breakup scenarios in turbulence. The findings offer insights into bubble deformation and breakup in isotropic turbulence and extend prior uniaxial results to biaxial configurations, with potential implications for modeling bubble transport and fragmentation in practical flows.

Abstract

Taking advantage of the recently developed L-ALE framework [Sierra-Ausin \textit{et al.}, Phys. Rev. Fluids {\bf{7}}, 113603 (2022)], we characterize the linear dynamics of an incompressible gas bubble immersed in a biaxial straining flow. We show that the system undergoes a saddle-node bifurcation with strongly different equilibrium shapes when varying the Ohnesorge number, $\Oh$, which compares viscous and capillary effects. Equilibrium shapes are found to be oblate for sufficiently large $\Oh$ while, counter-intuitively, they are prolate for low-enough $\Oh$. The bifurcation diagram is found to contain also two sets of disconnected branches that cannot be obtained by continuation starting from a spherical shape. One set corresponds to bubble shapes expected to be unstable, while the second set comprises a wide region exhibiting stable shapes that might be observed in practice. We then characterize the linear stability of the various branches. In addition to the unstable axisymmetric mode arising at the saddle-node bifurcation, two non-oscillating drift modes are also identified, together with a new unstable non-oscillating mode with azimuthal wave number $m=2$. This mode might be responsible for some type of bubble breakup observed in experiments.

Figures (13)

• Figure 1: (a) Schematic representation of a bubble in a biaxial straining flow in a $(r, \theta,z)$ cylindrical coordinate system. Black arrows schematize the streamlines of the undisturbed flow. Lengths $L_r$ and $L_z$ quantify the bubble lengths in the $z=0$ plane and along the $r=0$ axis, respectively. (b) Sketch of the numerical domain $\Omega$, not to scale. $\Gamma_b(t)$ denotes the time-dependent interface position, while $\Gamma_r(t)$ and $\Gamma_z(t)$ are the boundaries of the domain along the $z=0$ symmetry plane and the $r=0$ symmetry axis, respectively. Finally, $\Gamma_\infty$ represents the outer boundary that closes the computational domain.
• Figure 2: Equilibrium shapes. (a) Variations of the stationary bubble shape with $\textrm{We}$ for various $\textrm{Oh} \in \{3.10^{-3}, 5.10^{-3}, 6\cdot 10^{-3}, 7\cdot 10^{-3}, 8\cdot 10^{-3}, 0.01,0.015, 0.025, 0.05, 0.0625 , 0.1, 0.25, 0.5\}$ (color-coded, from yellow to black). Solid curves correspond to states computed by continuation, starting from a spherical shape $\Psi = 0$ – $\textrm{We} = 0$, while dash-dotted lines correspond to disconnected branches. Values of $\textrm{Oh}$ corresponding to transitions between the regimes discussed in Sec. \ref{['sec:eqpos']} are indicated on the corresponding curves. Symbols along the curves correspond to the parameters for which the steady solutions are represented on figures \ref{['fig:BF_Oh0v05']}-\ref{['fig:BF_otherbranchesOh0v008']}. Some typical shapes are also shown. (b) Variations of the shape factor $\Psi$ for the six largest $\textrm{Oh}$ as a function of the capillary number $\textrm{Ca}$. As $\textrm{Oh}$ increases, branches converge toward the creeping-flow solutions of kang1989biaxial (black dotted line) and zabarankin2013 (black dashed line).
• Figure 3: Example of steady solutions obtained at $\textrm{Oh}=0.05$, shown with bullets in Fig. \ref{['fig:phase diagram']}(a). The upper right quadrant is computed numerically; the other three quadrants are reconstructed by symmetry. The bubble-induced pressure disturbance with respect to the base flow, $(P - P_{BSF})/S^2$, is color-coded. Some streamlines are depicted with thin black lines and arrows. As $\textrm{We}$ increases on the primary branch (P), the bubble becomes more oblate. At the critical point (C), $\textrm{We} = 9.58$, the bubble becomes concave in the vicinity of the symmetry axis, a trend that further increases on the folded branch (F), until $L_z$ vanishes.
• Figure 4: Example of steady solutions obtained for $\textrm{Oh}=0.008$ (squares in Fig. \ref{['fig:phase diagram']}(a)). For details see the caption in Fig. \ref{['fig:BF_Oh0v05']}. On the primary branch, the bubble elongates along the axial direction, forming a prolate cylinder. At $\textrm{We}=5.27$ (P$_M$), the extension is maximal and then decreases, yielding oblate bubble shapes with relatively sharp edges along the folded branch (F).
• Figure 5: Example of steady solutions obtained for $\textrm{Oh}=0.007$ (pentagons in Fig. \ref{['fig:phase diagram']}(a)). For details see the caption of Fig. \ref{['fig:BF_Oh0v05']}. At this $\textrm{Oh}$, the bubble exhibits a prolate shape on the primary branch (P), turns approximately into a cylinder at the critical point (C), and becomes increasingly squeezed in the symmetry plane $z=0$ along the folded branch (F), until it completely necks down and forms a dumbbell.