Carbon nanotubes are commonly dispersed in liquid solvents by means of sonication. This has the disadvantage, however, that it can induce the scission of the particles that are near imploding cavitation bubbles. Nanotube scission arises from the fluid friction at the surface of the nanotubes in the radial elongational flow field that forms around a cavitation bubble. An understanding of the kinetics of this phenomenon is of critical importance for controlling the length of the nantoubes in their applications yet remains elusive. We investigate this kinetics quantitatively in the present work. The strain rate of the elongational flow around a cavitation bubble is estimated experimentally using carbon microfibers of known mechanical properties. The average length L(t) of the nanotubes is measured by means of dynamic light scattering as a function of time t, and we observed that L(t) scales as t-n, with n 0.2. This scaling differs from the one predicted theoretically in the literature for the scission of flexible polymer chains. Possible origins of this difference are discussed. We believe that the reduced probability of a nanotube to be in the vicinity of a cavitation bubble if the sonication power is in some sense low and can slow down the kinetics of nanotube scission.