  Free surface, axially symmetric shallow flow is analysed in both the
centrifugal and centripetal directions. Referring to the inviscid steady flow
over a flat plate characterised by a unique value of specific energy, the
analytical sub- and supercritical solutions are determined. Furthermore, the
sub- and supercritical steady solutions for the flow with friction over a flat
plate are determined, provided that inertial terms are important compared with
frictional terms. As the inviscid case, the sub- and supercritical solutions
over a bottom topography are determined, provided that the bottom unevenness is
compatible with a continuous solution. In the fundamental case of inviscid flow
over a flat plate, the discontinuous solution is also analysed for a direct
hydraulic jump imposed by proper boundary conditions. In the simple scheme of
an inviscid shock of zero length, the jump position and the sequent depths are
analytically derived, thus indicating that all results are uniquely functions
of one dimensionless number, i. e. the specific energy ratio between the sub-
and supercritical flows.
