On the rise of small air bubbles in water

This paper is concerned with the motion in water of air bubbles whose equivalent spherical radii are in the range 0.5-4.0 mm. These bubbles are not spherical but are, approximately, oblate spheroids; and they may rise steadily in a vertical straight line, or along a zig-zag path, or in a uniform spiral. The rectilinear motion occurs when the radius is less than about 0.7 mm, and the other motions occur for larger bubbles. There is disagreement in the literature as to whether it is the zig-zag or the spiral motion that occurs. It was found experimentally that, when the bubbles are produced in the manner described in this paper, only the zig-zag motion occurs when the radius of the bubble is less than about 1 mm, but bubbles of larger radius either zig-zag or spiral depending upon various factors. The spiralling bubble is treated theoretically by assuming that the flow near the front of the bubble is inviscid (the Reynolds number of the motion is several hundred) and considering the distribution of pressure over the front surface. Equations are obtained relating the geometrical parameters of the spiral, the shape of the bubble and the velocity of rise. The analysis is simplified by assuming that the pitch of the spiral is large compared with its radius, and the velocity of rise and shape of the bubble are determined as functions of the radius. The experimental and theoretical values are compared, and fair agreement found. Reasons to account for the disagreement are proposed. A modification of the theory is proposed to take account of the presence of impurities or surface-active substances in the water, and the velocities of rise thus predicted are in agreement with the experimental observations. The zig-zag motion is treated in a similar way, and the analysis leads to an equation which determines the stability of the rectilinear motion. The value of the Weber number at which the rectilinear motion. The value of the Weber number at which the rectilinear motion becomes unstable is deduced, and is found to be in fair agreement with experiment. The experimental evidence on the wake behind solid bodies is described briefly, and reasons are given for suggesting that the zig-zag motion is due to an interaction between the instability of the rectilinear motion and a periodic oscillation of the wake.