Why will two bubbles floating on water surface attract each other?

Two identical bubbles floating on water surface will form clumps, according to the "cheerio effect". But what's the detail about the force? It's necessary to calculate the shape of water surface, in order to find the force?

• An intuitive explanation lies in the idea that (unless I'm mistaken) the attraction results from the fact that the joining of the bubbles results in a reduction in the effective surface area of the system, which minimizes the total surface tension energy. Thinking of the process as a continuous deformation from initial state to final state, the driving force behind the process is this surface energy minimization. As far as finding an explicit closed form for the force, I imagine it might be a bit complicated. Mar 6 '14 at 17:20
• I'm not convinced they "attract" each other. When they bump into each other they tend to join and form a state of lower energy. Mar 7 '14 at 20:18
• May 22 '20 at 9:58

Vella and Mahadevan explain the effect as follows:

For simplicity, we consider the latter case schematically illustrated in Fig. 2, although the explanation of the clustering of many bubbles is similar. Here, the air–water interface is significantly distorted by the presence of the wall the well-known meniscus effect, and because the bubble is buoyant, there is a net upward force due to gravity, Fg , on the bubble. Because it is constrained to lie at the interface, however, the bubble cannot simply rise vertically, and instead does the next best thing by moving upward along the meniscus. [..........] A single bubble will deform the interface just as the presence of a wall does, although for a different reason and to a lesser extent. In the case of the bubble, it can only remain at the interface because the buoyancy force, which tends to push the bubble out of the liquid, is counterbalanced by the surface tension force, which opposes the deformation of the interface and hence acts to keep the bubble in the liquid. These two competing effects reach a compromise where the bubble is partially out of the liquid but the interface is slightly deformed. This deformation is sufficiently significant to influence other bubbles nearby, which move upward along the meniscus and so spontaneously aggregate. They also provide an answer for the force of interaction between two particles with a Radius $R$: $$F(l)=-2 \pi R B^{5/2}\Sigma^2 K_1\left(\frac{l}{L_c}\right)$$

With Bond number $B= \frac{R^2}{L^2_c}$, $L_c=\sqrt{\gamma/\rho g}$, $\Sigma$ a dimensionless Archimedes weight parameter and $K_1$ a first order Bessel function.