Thin liquid films present interesting scientific issues that are being addressed with mathematics, experiments and numerical simulation. In this minisymposium, we will highlight applications, some of the interesting observed phenomena, and the effectiveness of mathematical modeling. One theme is the effect of Marangoni forces induced by surface heating or surfactants.
Speakers are named first.
2:15-2:40 Roman Grigoriev, Georgia Tech
Transient growth in driven contact lines
The problem of transient growth in the distortion of driven contact lines has recently sparked a controversy as to whether this mechanism can provide an alternative route to pattern formation in the absence of linear instability. To resolve the disagreement between previous studies we conduct a generalized linear stability analysis of different lubrication models of gravity-driven spreading and compare our results with those based on direct numerical simulations. We find that linear and nonlinear theory are in reasonable qualitative agreement and show that the quantitative discrepancies in the predicted transient growth are caused by the differences in (1) the choice of initial disturbances and (2) the definition of the maximal transient amplification used in different studies. We further show by comparing the predictions of the precursor and the slip model that the latter substantially underestimates transient growth by neglecting the disturbances in the slip parameter.
2:45-3:10 Chris Kees, U.S. Army Engineer R&D Center
Infinite Speed of Propagation for Some Models of Two-Phase Flow in Porous Media
In continuum models for flow of two immiscible, incompressible fluids in a porous medium, the wetting phase saturation is usually described by a doubly degenerate nonlinear advection-diffusion equation. The degeneracies in the diffusion coefficient typically give rise to finite speed of propagation: Perturbations in saturation propagate with finite speed through regions that are either fully saturated or fully unsaturated. This qualitative property of solutions is considered physically realistic since the fluids flow at a finite speed. Under certain parameter choices for capillary pressure curves of the type described by M. Th. van Genuchten, combined with the permeability model of Y. Mualem, the finite speed of propagation property is lost, despite the fact that the equation has degenerate diffusion. We present analytical and numerical results demonstrating the loss of finite speed of propagation.
3:15-3:40 Ryan Haskett, Duke University
Localized Marangoni Forcing in Driven Thin Films
We consider the use of localized Marangoni forcing to produce a thermocapillary ``microfluidic valve'' that allows us to control the downstream flow of a thin film. Lubrication theory is used to derive a fourth order PDE for the film thickness. The long-time asymptotic solutions approach classes of either homoclinic or hetroclinic steady states. We use linearized analysis and numerical bifurcation studies to characterize the homoclinic solutions. The heteroclinic solutions describe stable stationary hydraulic jumps for thin films and have unusual scaling behavior. In the dynamics we find an interesting bi-stability between these states. We explain how to exploit the bi-stability to extend the effectiveness of the valve over a larger parameter space.
3:45-4:10 Michael Shearer and Rachel Levy, NC State
Waves in Thin Liquid Films
The lubrication approximation for the slow flow of a thin liquid film leads to a single partial differential equation (known as the thin film equation) for the height of the free surface. However, when the motion is driven by a nonuniform distribution of surfactant on the free surface, an equation for the surfactant concentration is coupled to the thin film equation. The coupled system has several important parameters, and we shall show how a collection of self similar solutions depends on some of these. Of primary interest is the limiting system, in which surface tension is neglected. The resulting equations sustain discontinuities in the surface height, and corresponding discontinuities in surfactant concentration. In this preliminary report, we show some numerical simulations and discuss their connection with a family of discontinuous solutions and a family of smooth traveling waves.