Undercompressive shocks in driven films I: Experiments, mathematical model, and numerical computations

Andrea L. Bertozzi

We consider viscous films driven by a thermal gradient with an opposing gravitational force. When the gravitational effect becomes non-negligible, the advancing front produces a very large capillary ridge which shows a remarkable tendency to remain stable. This phenomenon can be explained by a lubrication model of the experiment.
The advancing front evolves into an `undercompressive' capillary shock structure which is stable to contact line perturbations, unlike typical capillary ridges in driven film flows. We discuss experiments conducted by Ludvikkson/Lightfoot in 1971 and more experiments by A. M. Cazabat and colleagues (PRL 1998, JCIS 2000). These results are compared with numerical simulations and linear stability computations for traveling wave solutions.


M. Schneemilch, A. M. Cazabat, JCIS 2000.
A. Muench, A. L. Bertozzi, Phys. Fluids, 11(10) pp. 2818-2814, 1999.
A. L. Bertozzi, A. Muench, X. Fanton, A. M. Cazabat, Phys. Rev. Lett., 81(23), p. 5169-5172, 1998.
M. Bowen, A. L. Bertozzi, in preparation.

Undercompressive shocks in driven films II:Existence, bifurcations, and stability of traveling waves

Andrea L. Bertozzi

Equations of the type

$$h_t + (h^2-h^3)_x = -\mathbf{e}^3(h^3h_{xxx})_x$$

arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. Jump initial data, from a moderately thick film to a thin precurser layer, is shown to give rise to a double wave structure that includes an undercompressive wave.
This wave, which approaches an undercompressive shock as $\mathbf{e}\to 0,$ is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a series of bifurcations parameterized by the shock speed. At each bifurcation, a pair of traveling waves is produced, one being stable for the PDE, the other unstable.
We discuss an existence theorem for the undercompressive wave and Evans function theory for one-dimensional stability of the compressive and undercompressive waves.


A. L. Bertozzi, A. Muench, M Shearer, K. Zumbrun, EJAM to appear 2001
A. L. Bertozzi, M. Shearer, SIAM J. Math. Anal., 32(1), pp. 194-213, 2000
A. L. Bertozzi, A. Muench, M. Shearer, Physica D 134(4), 431-464, 1999