Find the lifetime of a soap bubble of radius `R` connected with the atmosphere through a capillary of length `l` and inside radius `r`. The surface tension is `alpha`, the viscosity coefficient of the gas is `eta`.

Explanation

Suppose the radius of the bubble is `x` at some instant. Then the pressure inside is `p_0 + (4 alpha)/(x)`. The flow through the capillary is by Poiseuille's equation,
`Q = (pi r^4)/(8 eta l)(4 alpha)/(x) - 4 x^2 (dx)/(dt)`
Integrating `(pi r^4 alpha)/(2 eta l) t = pi (R^2 - x^4)` where we have used the fact that `t = 0` where `x = R`.
This gives `t = (2 eta l R^4)/(alpha r^4)` as the life time of the bubble corresponding to `x = 0`.