Question 5: Angles Between Coalescing Bubbles

Solution

When soap bubbles coalesce, the films (lamellae) meet at a common junction. The stability of this junction is governed by the surface tension forces.

Assuming the soap solution is uniform, the surface tension $\sigma$ is the same for all three films meeting at the line of intersection.

For the junction to be in mechanical equilibrium, the vector sum of the three surface tension forces must be zero:

$$ \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = 0 $$ $$ |\vec{F}_1| = |\vec{F}_2| = |\vec{F}_3| = \sigma \cdot l $$

Three equal vectors can only sum to zero if the angle between any two of them is $120^\circ$.

Thus, regardless of the radii of the bubbles, the angles at the junction are equal:

$$ \alpha = \beta = \gamma = 120^\circ $$