Dynamics of a gas bubble in a straining flow: Deformation, oscillations, self-propulsion

We revisit from a dynamical point of view the classical problem of the deformation of a gas bubble suspended in an axisymmetric uniaxial straining flow. Thanks to a recently developed Linearized Arbitrary Lagrangian-Eulerian approach, we compute the steady equilibrium states and associated bubble shapes. Considering perturbations that respect the symmetries of the imposed carrying flow, we show that the bifurcation diagram is made of a stable and an unstable branch of steady states separated by a saddle-node bifurcation, the location of which is tracked throughout the parameter space. We characterize the most relevant global mode along each branch, namely, an oscillatory mode that becomes neutrally stable in the inviscid limit along the stable branch, and an unstable nonoscillating mode eventually leading to the breakup of the bubble along the unstable branch. Next, considering perturbations that break the symmetries of the carrying flow, we identify two additional unstable nonoscillating modes associated with the possible drift of the bubble centroid away from the stagnation point of the undisturbed flow. One of them corresponds merely to a translation of the bubble along the elongational direction of the flow. The other is counterintuitive, as it corresponds to a drift of the bubble in the symmetry plane of the undisturbed flow, where this flow is compressional. We confirm the existence and characteristics of this mode by computing analytically the corresponding leading-order disturbance in the inviscid limit, and show that the observed dynamics are made possible by a specific self-propulsion mechanism that we explain qualitatively.

Automated summary

In this paper, the deformation problem of a gas bubble suspended in an axisymmetric uniaxial straining flow is revisited from a dynamical perspective. By utilizing a newly developed Linearized Arbitrary Lagrangian-Eulerian approach, the steady equilibrium states and bubble shapes are computed. Perturbations respecting the symmetries of the imposed carrying flow are considered, revealing the bifurcation diagram consisting of a stable and an unstable branch of steady states separated by a saddle-node bifurcation. The location of this bifurcation is tracked in the parameter space. The most relevant global mode along each branch is characterized, including an oscillatory mode that becomes neutrally stable in the inviscid limit along the stable branch, and an unstable nonoscillating mode eventually leading to the breakup of the bubble along the unstable branch. Additional unstable nonoscillating modes associated with the possible drift of the bubble centroid away from the stagnation point of the undisturbed flow are identified when perturbations breaking the symmetries of the carrying flow are considered. The existence and characteristics of these modes are confirmed analytically in the inviscid limit, and a qualitative explanation is provided for the observed dynamics through a specific self-propulsion mechanism.