  In this work we establish constraints on the temperature dependence of the
shear viscosity $\eta$ in the superfluid phase of a dilute Fermi gas in the
unitary limit. Our results are based on analyzing experiments that measure the
aspect ratio of a deformed cloud after release from an optical trap. We discuss
how to apply the two-fluid formalism to the unitary gas, and provide a suitable
parametrization of the equation of state. We show that in expansion experiments
the difference between the normal and superfluid velocities remains small, and
can be treated as a perturbation. We find that expansion experiments favor a
shear viscosity that decreases significantly in the superfluid regime. Using an
exponential parametrization we find $\eta(T_c/(2T_F))< 0.37\eta (T_c/T_F)$,
where $T_c$ is the critical temperature, $T_F$ is the local Fermi temperature
of the gas.
