  We study the dynamic behaviour of two viscous fluid films confined between
two concentric cylinders rotating at a small relative velocity. It is assumed
that the fluids are immiscible and that the volume of the outer fluid film is
large compared to the volume of the inner one. Moreover, while the outer fluid
is considered to have constant viscosity, the rheological behaviour of the
inner thin film is determined by a strain-dependent power-law. Starting from a
Navier--Stokes system, we formally derive evolution equations for the interface
separating the two fluids. Two competing effects drive the dynamics of the
interface, namely, the surface tension and the shear stresses induced by the
rotation of the cylinders. When the two effects are comparable, the solutions
behave, for large times, as in the Newtonian regime. We also study the regime
in which the surface tension effects dominate the stresses induced by the
rotation of the cylinders. In this case, we prove local existence of positive
weak solutions both for shear-thinning and shear-thickening fluids. In the
latter case, we show that interfaces which are initially close to a circle
converge to a circle in finite time and keep that shape for later times.
