Bubble breakup reduced to a one-dimensional nonlinear oscillator

Breaking dynamics of bubbles in turbulence produce a wide range of bubble sizes, which mediates gas transfer, in particular, at the ocean/atmosphere interface. At the scales close to the stability limit of bubbles torn away by inertial forces, a typical geometry that induces bubble breakup is the uniaxial straining flow. In this configuration, the bubble shapes and their limit of stability have been studied theoretically and numerically near their equilibrium. Using numerical simulations, we investigate the bubble dynamics and breakup in such flows, starting from initial shapes far from equilibrium. We show that the breakup threshold is significantly smaller than the previous linear predictions and evidence that the breakup threshold depends on both the Reynolds number at the bubble size, and the initial bubble shape (ellipsoids). To rationalize the bubble dynamics and the observed thresholds, we propose a reduced model for the oblate/prolate oscillations (second Rayleigh mode) based on an effective potential that depends on the control parameters and the initial bubble shape. Our model successfully reproduces bubble oscillations, the maximal deformation below the threshold, and the bubble lifetime above the threshold.

Figure 1

(a) Scheme of a bubble at the center of a uniaxial straining flow. $\left(Oz\right)$ is the axis of symmetry. Arrows show typical streamlines in the absence of bubble. (b) Enlargement around a bubble experiencing its maximum deformation at $\mathrm{Re}=400$ and $We=7.3\approx {\text{We}}_c(\mathrm{Re}=400)$. The blue line denotes the bubble interface, black lines, isocontours of the stream function.

Figure 2

(a) Critical Weber number versus $Re$ (crosses) with an error inferior to ${10}^{-2}$. The solid black line is the inviscid value ${\text{We}}_c^{\infty }$. The blue curve has expression ${\text{We}}_c^{\infty }exp(-100/\left({\text{We}}_c^{\infty }\mathrm{Re}\right))$. Open circles and dashed lines are ${\text{We}}_c^S\left(\mathrm{Re}\right)$ and its inviscid limit as computed by Sierra-Ausin et al. [14] and Miksis [11], respectively. The inset shows the viscous correction to ${\text{We}}_c^{\infty }$. The dotted line follows $100/\mathrm{Re}$. (b) Critical Weber number versus the ellipsoid shape parameter $a_0$, for inviscid simulations. The blue curve is a polynomial fit of degree two with a maximum at $a_0=1$. Initial (black) and critical (red) shapes are represented for every $a_0$.

Figure 3

(a), (b) Several typical temporal evolution of $x$ at $\mathrm{Re}=400$. Simulation data are in color, the model (4) is superimposed (black dotted line). (a) An initially spherical bubble at low $\text{We}$. (b) Three evolutions close to ${\text{We}}_c\left(a_0\right)$, for a stable sphere (blue line), an unstable sphere (gray line), and a stable ellipsoid (orange line, $a_0=0.8$).

Figure 4

(a) Evolution of the potential $V$, defined in (4), with $\text{We}$ for initially spherical bubbles at $\mathrm{Re}=400$. The range of explored $x$ values have a more intense color. Stable equilibrium positions are denoted by red diamonds (red dashed lines in a and (b). The maximum values, $x_{max}$, are denoted by black stars [same in 3 and 3]. (b) Evolution of $V$ with $\mathrm{Re}$ for a fixed of $\text{We}=5$.

Figure 5

Evolution of the coefficients of the three parameters defined in (5) with Re and We for initially spherical bubbles. Colored circles correspond to finite Re simulations and black crosses to inviscid simulations. (a) Constant coefficient $p_0$ as a function of $We$. $p_0$ evolves linearly with We. (b) Evolution of $p_0$ with Re. The dotted black line is the average inviscid value. We recover a more efficient forcing at small Re, compatible with a $1/\mathrm{Re}$ scaling (solid black line). (c) Evolution of the linear coefficient $p_1$ with Re. The black dotted line is the prediction of the pulsation from Kang and Leal [39] found from a linear development. (d) Evolution of the quadratic coefficient $p_2$ with We.

Figure 6

(a) Maximal deformation as a function of the distance to ${\text{We}}_c$. Finite $\mathrm{Re}$ simulations are denoted with circles, inviscid ellipsoids by triangles, and inviscid spheres by diamonds. As $\text{We}\to {\text{We}}_c$, the maximal deformation converges to its critical value ${\mathcal{D}}_c$. Inset plot: Rescaled ${\mathcal{D}}_{max}$ for the spheres, with two parameters that depend on $\mathrm{Re}$: ${\mathcal{D}}_c$ and $\alpha$. In (b) the black dotted line denotes the inviscid value for spheres [and correspond to $a_0=1$ in (c)]. (b), (c) Evolution of ${\mathcal{D}}_c$ with Re and $a_0$, respectively. (d), (e) Similar plots for the evolution of the slope with Re and $a_0$.

Figure 7

(a) Dimensionless lifetime, $ET$, as a function of the distance to ${\text{We}}_c$. Finite $\mathrm{Re}$ simulations are denoted with circles, inviscid ellipsoids by triangles, and inviscid spheres by diamonds. As $\text{We}\to {\text{We}}_c$, the lifetime diverges logarithmically. Inset plot: Rescaled $ET$ for the spheres, with two parameters that depend on $\mathrm{Re}$: $ET\left(2\text{We}\right)$ and $\beta$. In (b) and (c) the black dotted line denotes the inviscid value for spheres [and correspond to $a_0=1$ in (c) and (e)]. (b), (c) Evolution of $ET\left(2\text{We}\right)$ with Re and $a_0$, respectively. (d), (e) Similar plots for the evolution of the slope $\beta$ with Re and $a_0$.