Starting from relativistic quantum field theories, Kovtun et al.(2005) have quite recently proposed a lower bound η/s⩾ ℏ/(4 π k B), where η is the shear viscosity and s the volume density of entropy for dense liquids. If their proposal can eventually be proved, then this would provide key theoretical underpinning to earlier semiempirical proposals on the relation between a transport coefficient η and a thermodynamic quantity s. Here, we examine largely experimental data on some dense liquids, the insulators nitrogen, water, and ammonia, plus the alkali metals, where the shear viscosity η (T) for the four heaviest alkalis is known to scale onto an ‘almost universal’curve, following the work of Tankeshwar and March a decade ago. So far, all known results for both insulating and metallic dense liquids correctly exceed the lower bound prediction of Kovtun et al.