Singularities of the thin film equation
Fluids and Materials Seminar
11th October 2018, 2:00 pm – 3:00 pm
Main Maths Building, SM3
The thin film or lubrication approximation is a method in interfacial fluid dynamics to derive simplified evolution equations (PDEs) for the thickness of a liquid film. All sorts of physical effects (gravity, surface tension, Marangoni forces, intermolecular forces, etc.) may be included in such a framework, depending on the application.
Many such models (in idealised or asymptotic circumstances) end up as special cases of a single canonical nonlinear 'thin film' equation, in which the thickness is stabilised by a fourth order term, and destabilised by a second order term, with coefficients as some power of the thickness. This equation exhibits a rich display of nonlinear phenomena, particularly related to the formation of singularities by either finite-time rupture (thickness going to zero at a point), or finite-time blow up (thickness going to infinity at a point). Such singularities can be examined through the assumption of self-similarity near the singularity.
I will discuss two aspects of singularity formation: first, the loss of stability of self-similar rupture solutions leading to the creation of discretely self-similar rupture profiles (resulting from periodic orbits in a scaled version of the problem); and second, current work being undertaken on the asymptotic structure of blowup profiles above a critical line in parameter space.