The effective film viscosity coefficients of a thin floating fluid layer

We derive a constitutive relation, relating the tangential stress, tangential velocity, thickness it, and viscosity mu, for a thin layer of Newtonian fluid on top of a fluid substrate. We find that the upper layer exerts a viscous tangential sl?ear stress on the lower fluid, behaving as if it were a film with a two-dimensional shear viscosity equal to mu h, and a dilatational viscosity 3 mu h.