On the shape of a two-dimensional bubble in uniform motion

Abstract

Consider a two-dimensional bubble moving with speed U through an unbounded, inviscid fluid. Let all lengths be normalized by T/ρU² where T is the surface tension. Then the shape of the bubble depends on a single parameter Γ = 2Δp/ρU² − 1, where Δp = p_b − p_∞ is the difference between the bubble pressure and the ambient pressure. We obtain solutions for the bubble shape over the whole range of Γ-values that are physically relevant. The formulation involves a mapping from an auxiliary circle plane ζ where the flow field is known. The problem then reduces to solving an infinite set of nonlinear algebraic equations for the coefficients in the mapping function.

To a first approximation, when Γ → ∞, the bubble takes an elliptical shape of aspect ratio $(1 + {\textstyle\frac{2}{3}}\Gamma^{-1})/(1 - {\textstyle\frac{2}{3}}\Gamma^{-1})$ flattened in the flow direction. The solution correct to order Γ⁻⁵ is then obtained which is fairly accurate for Γ as low as 2. When Γ = 0 the exact, nonlinear solution for the bubble shape is given by $x = \frac{1}{3}(\frac{1}{3}\cos\phi - \frac{1}{27}\cos 3\phi), y=\frac{1}{3}(\frac{5}{3}\sin\phi + \frac{1}{27}\sin 3\phi)$. We can then obtain a perturbation solution for Γ → 0 correct to order Γ⁶. This solution, useful in the range 0.75 > Γ > − 0.4537, even gives reasonable descriptions of non-convex bubble shapes for Γ < 0 down to the pinch-off limit Γ* when the bubble ceases to be simply connected. It is remarkable that a simple analytical representation correct to order Γ² analytically yields a value for Γ* of − 0.4548, i.e. within 0.3% of the correct value; naturally, the higher-order approximations are even more accurate. While the present results eliminate the need for direct numerical computations over most of the range of Γ, such results, too, are presented. Finally, the dependence of the bubble geometrical parameters, Weber number and added mass on Γ is determined.