  The shear viscosity has been shown to be equal to the product of pressure and
relaxation time in normal scale-invariant fluids, but the presence of
superfluidity at low temperatures can alter the relation. By using the
mean-field BCS-Leggett theory with a gauge-invariant linear response theory for
unitary Fermi superfluids, we present an explicit relation between
thermodynamic quantities, including the pressure and chemical potential, and
transport coefficients, including the shear viscosity, superfluid density, and
anomalous shear viscosity from momentum transfer via Cooper pairs. The relation
is modified when pairing fluctuations associated with noncondensed Cooper pairs
are considered. Within a pairing fluctuation theory consistent with the
BCS-Leggett ground state, we found an approximate relation for unitary Fermi
superfluids. The exact mean-field relation and the approximate one with pairing
flucutaions advance our understanding of relations between equilibrium and
transport quantities in superfluids, and they help determine or constrain
quantities which can be otherwise difficult to measure.
