Thin Fluid Films with Surfactant Ellen Peterson*, Michael Shearer*, Rachel Levy † , Karen Daniels ‡ , Dave Fallest ‡ , T

1. Thin Fluid Films with Surfactant Ellen Peterson*, Michael Shearer*, Rachel Levy†, Karen Daniels‡, Dave Fallest‡,Tom Witelski§ *North Carolina State University, Raleigh, NC †Harvey Mudd College, Claremont, CA ‡North Carolina State University (Physics), Raleigh, NC §Duke University, Durham, NC erpeters@ncsu.edu • INTRODUCTION • The movement of a thin liquid film can be driven by various factors such as gravity or surface tension. We examine two situations: • Horizontal substrate where the liquid is driven by a surfactant • (surface tension reducing agent) • Inclined plane where the liquid is driven by both a surfactant and gravity • We examine cases both with and without the inclusion of regularizing terms which account for capillarity and surface diffusion. • STABILITY ANALYSIS • We consider the stability of the various forms the thin film system on an inclined plane under the influence of surfactant. We examine the stability of the traveling wave. • One-Dimensional Unregularized System: • Place perturbations on the various steps of the traveling wave • (h: piecewise constant, Γ: piecewise linear) • Dispersion relation of the linearized equations indicates the direction and growth rate of the perturbation • Suggests linear stability in 1-D1 • One-Dimensional Regularized System: • Perturb about traveling wave with Γ=0; linearized equations partly decouple • Examine the Evans function and stability indicator function2 • Analysis consistent with linear stability • Multi-Dimensional Regularized System: • Analyze the eigenfunction associated with translation invariance, the stability of the system depends on the size of the capillary ridge3 • Suggests linear instability • APPLICATIONS • Surfactant Replacement Therapy • Coating Flows • Food Science EQUATIONS • NUMERICAL METHOD • Examine the unregularized equations on both the horizontal surface and inclined plane. • Inclined Plane • Numerically examine the stability of the 1-D unregularized system • Place a perturbation on one of the steps of the traveling wave • Integrate the system using finite difference method • convective terms: explicit upwind • time step/parabolic terms: implicit centered • Perturbations move towards center of wave profile • Numerical results suggest stability • Horizontal Surface • Examine the unregularized 1-D system • Maintain a compact support for the surfactant • concentration • Implement a change of variables: h: height of thin film Γ: surfactant concentration • Regularized System on Inclined Plane: • -include all terms • Unregularized System on Inclined Plane: • -include red and black terms • Regularized System on Horizontal Surface: • -include blue and black terms • Unregularized System on Horizontal Surface: • -include black terms Traveling wave solution on an inclined plane h:blue, Γ:yellow Profile for the horizontal surface, h: blue, Γ: yellow Perturbations placed on the inner steps of the traveling wave of the height profile • GOALS • Analytically examine the stability of the unregularized and • regularized thin film system in 1-dimension and • multi-dimensions • Numerically examine the stability of the thin film system • Create a numerical method that can be compared to experimental data, on • horizontal substrate Top two plots: height and surfactant profiles in variable ξ Bottom two plots: height and surfactant profiles in the variable x • REFERENCES • E. Peterson, M. Shearer, T. Witelski, R. Levy, Stability of Traveling Waves in Thin Liquid Films Driven By Gravity Surfactant, Proceedings 12th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Hyp 2008, to appear • A. Bertozzi, A. Münch, M. Shearer, K. Zumbrun, (2001) Stability of Compressive and Undercompressive Thin Film Travelling Waves, Euro. Jnl of Applied Mathematics, 12, 253-291 • A. Bertozzi, M. Brenner, (1997) Linear Stability and Transient Growth in Driven Contact Lines, Phys. Fluids, 9 (3), 530-53 • EXPERIMENT • Visualize the surfactant • (using insoluble fluorescent surfactant) • Examine the effect of the surfactant • on evolution of the height of the thin film • on a horizontal subsrate Green: Surfactant Molecules Red: Laser, used to visualize the height of the film