Path instability of an air bubble rising in water

Abstract

It has been documented since the Renaissance that an air bubble rising in water will deviate from its straight, steady path to perform a periodic zigzag or spiral motion once the bubble is above a critical size. Yet, unsteady bubble rise has resisted quantitative description, and the physical mechanism remains in dispute. Using a numerical mapping technique, we for the first time find quantitative agreement with high-precision measurements of the instability. Our linear stability analysis shows that the straight path of an air bubble in water becomes unstable to a periodic perturbation (a Hopf bifurcation) above a critical spherical radius of R = 0.926 mm, within 2% of the experimental value. While it was previously believed that the bubble's wake becomes unstable, we now demonstrate a new mechanism, based on the interplay between flow and bubble deformation.

Keywords: boundary layers; bubbles; hydrodynamic stability; numerical methods.

Fig. 1.

(A) The rise speed V_t as function of the undeformed bubble radius R; the symbols are the data of (3), the solid line shows the simulation with ρ = 998.3 kg/m³, μ = 1.014 mPa/s, γ = 72.8 mN/m; the red dashed line is the theory of (5); bubble shapes as insets. (B) Real and imaginary parts of the frequency 2πω near the transition; first eigenvalue (largest imaginary part): black and red solid lines, second eigenvalue: dashed lines; ω_i = 0: vertical dashed lines. (C and D) The flow field at R_c = 0.926 mm and R = 1 mm. On the left, red flow lines and blue arrows to represent the flow velocity. In (D), recirculating flow lines at the rear are shown in green. On the right, azimuthal vorticity contours.

Fig. 2.

Periodic perturbations during zigzag motion, the amplitude of the linear perturbation having been chosen arbitrarily. (A) Tilt angle θ, maximum curvature, and maximum axial velocity, as function of time. Solid lines: right side of the drop, dashed lines: left side. (B) Snapshots of the perturbations to curvature, pressure, and bubble shape are plotted on the bubble surface. The unperturbed bubble is shown in black, and the perturbation is plotted in red in the normal direction on each point on the surface.