I will offer an insight into mathematical models describing the dynamic behaviour of non-Newtonian
thin films. The resulting PDEs are in general nonlinear, degenerate, of fourth order, and with a
possibly ‘weak’ dependence of the coefficients.
I will discuss recent results on such an evolution equation for the interface separating two viscous
immiscible fluids, confined between two concentric cylinders rotating at a small relative velocity. In
this so-called Taylor–Couette setting, two competing effects drive the dynamics of the interface – the
surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects
are comparable, solutions behave, for large times, as in the Newtonian regime. For the regime in
which surface tension effects dominate the stresses induced by the rotating cylinders, we prove local
existence of positive weak solutions for both shear-thinning and shear-thickening fluids. In the case
of a shear-thickening fluid, one observes that interfaces which are initially close to a circle converge
to a circle in finite time.
The talk is based on a joint work with Tania Pernas-Castaño and Juan Velázquez: https://arxiv.