  In a recent paper [Eur. Phys. J {\bf C 44}, 567 (2005)] we developed a very
general formulation to take into account explicitly the effects of hydrodynamic
flow profile on the gluonic breakup of $J/\psi$'s produced in an equilibrating
quark-gluon plasma. Here we apply that formulation to the case when the medium
is undergoing cylindrically symmetric {\it transverse} expansion starting from
RHIC or LHC initial conditions. Our algebraic and numerical estimates
demonstrate that the transverse expansion causes enhancement of local gluon
number density $\ng$, affects the $\pt$-dependence of the average dissociation
rate $\tilg$ through a partial-wave interference mechanism, and makes the
survival probability $\spt$ to change with $\pt$ very slowly. Compared to the
previous case of longitudinal expansion the new graph of $\spt$ is pushed up at
LHC, but develops a rich structure at RHIC, due to a competition between the
transverse catch-up time and plasma lifetime.
