A diversity of phenomena...

Thin liquid films appear in a variety of situations in nature and in engineering applications. Examples include tear films in the eye, membranes in biophysics, linings in the lungs of animals, paints and thin films in the spin coating process, in which thin polymer layers are produced by depositing a polymer solution on a rotating disc.

...captured by lubrication theory

Despite the diversity of phenomena and applications, the mathematical modeling is quite similar if the film is sufficiently viscous. Excluding other contributions such as rotational forces, gravity, or van-der-Waals (molecular) interactions, it is reasonable to assume that the dynamics of the film are driven by only two dominant mechanisms: surface tension and viscosity. In the regime of thin films, one commonly refers to this approach as "lubrication approximation".

The mathematical model

The resulting partial differential equation is a degenerate fourth order parabolic equation that - for a thin film on a one-dimensional solid - reads

h_t + ((h³ + λhⁿ) h_zzz)_z = 0,   (1)

where h is the height of the film, t the time, λ a slippage length, 0 < n < 3 a mobility exponent, and z the spatial variable. The parameters λ and n are determined by the underlying exact model from which equation (1) was derived:

The case n = 2 can be derived by asymptotic expansions from the Stokes system as the underlying model, with a linear Navier-slip condition at the boundary. For λ = 0 the Navier-slip condition is merely a no-slip condition so that the droplet is hindered to move. This is also true for the lubrication model (1), i.e. a singularity at the triple junction is present, leading to an infinite dissipation rate. Hence a positive slippage length λ, appearing in (1) in the regularizing addend λhⁿ, is necessary to obtain a physically reasonable model. In this case, droplets can only spread but not contract.

The case n = 1, where droplets can both spread and contract, can be seen as a lubrication approximation of the two-dimensional Hele-Shaw flow. In a work with Giacomelli [1], we in fact obtained this approximation rigorously.

The contact line singularity

We are interested in the behavior of the solution close to the contact line, where the contact angle θ between liquid and solid is determined by an equilibrium of surface tensions (see figure on the left). For films sufficiently thinner than the slippage length, the term h³ is negligible and equation (1) reduces to the scaling invariant equation

h_t + (hⁿ h_zzz)_z = 0.  (2)

Using linearization and maximal regularity techniques, we have already studied the singularity of h at the triple junction for n = 1, when a zero contact angle θ between liquid and solid is imposed ([2], with Giacomelli and Knüpfer). In this case, perturbations of h from the steady state are smooth up to the boundary. For n = 2 and vanishing contact angle θ we have meanwhile managed to prove in cooperation with the same authors that the solution, a perturbation of a traveling wave, is actually non-smooth at the contact line [3]. This can be already seen in the case of similarity solutions (see figure on the right and [4], with Giacomelli). In fact Knüpfer proved a similar regularity result for the non-zero contact angle case [5].

Future questions

Future work will deal with the study of the free boundary value problem associated to (1), i.e. the investigation of films thicker than the slippage length λ, where we expect the stability of Tanner's law, i.e. h ~ z log^1/3 z, for large z. Furthermore we aim for investigating the singularity for the full Stokes problem at a triple junction and similarity solutions associated to this problem.

Up to now, the question of imposing suitable boundary conditions for the Stokes problem at a triple junction is not solved. Our hope is that a well-understood regularity theory for the thin-film equation may also lead to insights in this direction.

Apart from that we aim for investigating thin film equations in the context of the above mentioned spin coating process, where additionally centrifugal forces lead to a first order transport term.

Finally, the understanding of regularity properties at the triple junction may lead to higher order numerical schemes.