Scaling solutions of the two-fluid hydrodynamic equations in a harmonically trapped gas at unitarity

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 [Castin, C. R. Phys. 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.