Deformation of a bubble or drop in a uniform flow

Abstract

Steady potential flow around a two-dimensional bubble with surface tension, either free or attached to a wall, is considered. The results also apply to a liquid drop. The flow and the bubble shape are determined as functions of the contact angle β and the dimensionless pressure ratio γ = (p_b − p_s)/½ρU². Here p_b is the pressure in the bubble, p_s = p_∞ + ½ρU² is the stagnation pressure, p∞ is the pressure at infinity, ρ is the fluid density and U is the velocity at infinity. The surface tension σ determines the dimensions of the bubble, which are proportional to 2σ/ρU². As γ tends to ∞, the bubble surface tends to a circle or circular arc, and as γ decreases the bubble elongates in the direction normal to the flow. When γ reaches a certain value γ₀(β), opposite sides of the bubble touch each other. The problem is formulated as an integrodifferential equation for the bubble surface. This equation is discretized and solved numerically by Newton's method. Bubble profiles, the bubble area, the surface energy and the kinetic energy are presented for various values of β and γ. In addition a perturbation solution is given for γ large when the bubble is nearly a circular arc, and a slender-body approximation is presented for β ∼ ½π and γ ∼ γ₀(β), when the bubble is slender.