# Drops and Bubbles

One consequence of surface tension is that free liquid drops and bubbles are spherical if effects of gravity can be neglected. You must have seen this especially clearly in small drops just formed in a high-speed spray or jet, and in soap bubbles blown by most of us in childhood. Why are drops and bubbles spherical? What keeps soap bubbles stable?

As we have been saying repeatedly, a liquidair interface has energy, so for a given volume the surface with minimum energy is the one with the least area. The sphere has this property. Though it is out of the scope of this book, but you can check that a sphere is better than at least a cube in this respect! So, if gravity and other forces (e.g. air resistance) were ineffective, liquid drops would be spherical.

Another interesting consequence of surface tension is that the pressure inside a spherical drop Fig. 1(a) is more than the pressure outside. Suppose a spherical drop of radius r is in equilibrium. If its radius increase by ∆r. The extra surface energy is

[4π(r + ∆r)$^2$- 4πr$^2$] $S_{la}$ = 8πr∆r $S_{la}$  (1)

If the drop is in equilibrium this energy cost is balanced by the energy gain due to expansion under the pressure difference (P$_i$  – P$_o$) between the inside of the bubble and the outside. The work done is

W = (P$_i$  – P$_o$) 4πr$^2$∆r                                  (2)

so that

(P$_i$  – P$_o$)  = $\frac{2S_{la}}{r}$                       (3)

In general, for a liquid-gas interface, the convex side has a higher pressure than the concave side. For example, an air bubble in a liquid, would have higher pressure inside it. See Fig 1(b).

Fig. 1: Drop, cavity and bubble of radius r.

A bubble Fig 1(c) differs from a drop and a cavity; in this it has two interfaces. Applying the above argument we have for a bubble

(P$_i$  – P$_o$)  = $\frac{4S_{la}}{r}$           (4)

This is probably why you have to blow hard, but not too hard, to form a soap bubble. A little extra air pressure is needed inside!