Bubble or drop distortion in a straining flow in two dimensions

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Abstract
The distortion of a two‐dimensional bubble (or drop) in a straining flow of an inviscid imcompressible fluid is examined theoretically. Far from the bubble the stream function of the flow is assumed to be αxy, where α is a constant. Within the bubble the pressure is assumed to be a constant pb, and the bubble surface is assumed to have a surface tension σ. Then, the shape of the bubble depends upon the single dimensionless constant γ=2(pb−ps)/(2σα)2/3ρ1/3, where ρ is the fluid density, ps is the stagnation pressure of the flow, and the size of the bubble is proportional to (2σ/ρα2)1/3. For γ large, it is found that the bubble tends to a circle of radius (2σ/ρα2)1/3 γ−1. As γ decreases, numerical solutions show that the bubble at first becomes a square with rounded corners. Then, it develops four horns or spikes with large curvature near their ends. Finally at γ∼−1.8, the two sides of each spike touch each other near the tip and enclose a small bubble there. It is also found that there is a maximum value of the Weber number above which there is no steady solution.

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