By conservation of mass, the inverse-square law requires that the radially outward velocity  must be inversely proportional to the square of the distance from the origin (the center of the bubble).  Therefore, letting  be some function of time,
In the case of zero mass transport across the bubble surface, the velocity at the interface must be
which gives that
In the case where mass transport occurs, the rate of mass increase inside the bubble is given by
with  being the volume of the bubble.  If  is the velocity of the liquid relative to the bubble at , then the mass entering the bubble is given by
with  being the surface area of the bubble.  Now by conservation of mass , therefore .  Hence
Therefore
In many cases, the liquid density is much greater than the vapor density, , so that  can be approximated by the original zero mass transfer form , so that