  We prove that the two fluid Landau hydrodynamic equations, when applied to a
gas interacting with infinite scattering length (unitary gas) in the presence
of harmonic trapping, admit exact scaling solutions of mixed compressional and
surface nature. These solutions are characterized by a linear dependence of the
velocity field on the spatial coordinates and a temperature independent
frequency which is calculated in terms of the parameters of the trap. Our
results are derived in the regime of small amplitude oscillations and hold both
below and above the superfluid phase transition. They apply to isotropic as
well as to deformed configurations, thereby providing a generalization of
Castin's theorem (Y. Castin, C. R. Phys. \textbf{5}, 407 (2004)) holding for
isotropic trapping. Our predictions agree with the experimental findings in
resonantly interacting atomic Fermi gases. The breathing scaling solution, in
the presence of isotropic trapping, is also used to prove the vanishing of two
bulk viscosity coefficients in the superfluid phase.
