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Trapped phonons

We analyze the effect of restricted geometries on the contribution of Nambu-Goldstone bosons (phonons) to the shear viscosity, $η$, of a superfluid. For illustrative purpose we examine a simplified system consisting of a circular boundary of radius $R$, confining a two-dimensional rarefied gas of phonons. Considering the Maxwell-type conditions, we show that phonons that are not in equilibrium with the boundary and that are not specularly reflected exert a shear stress on the boundary. In this case it is possible to define an effective (ballistic) shear viscosity coefficient $η\propto ρ_{\rm ph} χR$, where $ρ_{\rm ph}$ is the density of phonons and $χ$ is a parameter which characterizes the type of scattering at the boundary. For an optically trapped superfluid our results corroborate the findings of Refs. \cite{Mannarelli:2012su, Mannarelli:2012eg}, which imply that at very low temperature the shear viscosity correlates with the size of the optical trap and decreases with decreasing temperature.