  We study the shear viscosity to entropy ratio $\eta/S$ in the boundary field
theories dual to black hole backgrounds in theories of gravity coupled to a
scalar field, and generalisations including a Maxwell field and non-minimal
scalar couplings. Motivated by the observation in simple examples that the
saturation of the $\eta/S\ge 1/(4\pi)$ bound is correlated with the existence
of a generalised Smarr relation for the planar black-hole solutions, we
investigate this in detail for the general black-hole solutions in these
theories, focusing especially on the cases where the scalar field plays a
non-trivial role and gives rise to an additional parameter in the space of
solutions. We find that a generalised Smarr relation holds in all cases, and in
fact it can be viewed as the bulk gravity dual of the statement of the
saturation of the viscosity to entropy bound. We obtain the generalised Smarr
relation, whose existence depends upon a scaling symmetry of the planar
black-hole solutions, by two different but related methods, one based on
integrating the first law of thermodynamics, and the other based on the
construction of a conserved Noether charge.

High Energy Physics - Theory;General Relativity and Quantum Cosmology