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 REFERENCES 1.E. Peterson, M. Shearer, T. Witelski, R. Levy, Stability of Traveling Waves in Thin Liquid Films Driven By Gravity Surfactant, Proceedings 12 th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Hyp 2008, to appear 2.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 3.A. Bertozzi, M. Brenner, (1997) Linear Stability and Transient Growth in Driven Contact Lines, Phys. Fluids, 9 (3), 530-53 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 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 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-D 1 One-Dimensional Regularized System: Perturb about traveling wave with Γ=0; linearized equations partly decouple Examine the Evans function and stability indicator function 2 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 ridge 3 Suggests linear instability 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-D 1 One-Dimensional Regularized System: Perturb about traveling wave with Γ=0; linearized equations partly decouple Examine the Evans function and stability indicator function 2 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 ridge 3 Suggests linear instability APPLICATIONS Surfactant Replacement Therapy Coating Flows Food Science APPLICATIONS Surfactant Replacement Therapy Coating Flows Food Science 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. 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. 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 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 EQUATIONS 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 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: 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: Perturbations placed on the inner steps of the traveling wave of the height profile Top two plots: height and surfactant profiles in variable ξ Bottom two plots: height and surfactant profiles in the variable x Traveling wave solution on an inclined plane h:blue, Γ:yellow Profile for the horizontal surface, h: blue, Γ: yellow erpeters@ncsu.edu Green: Surfactant Molecules Red: Laser, used to visualize the height of the film