Thin Fluid Films with SurfactantEllen 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 12th 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-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

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

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 surfactanton evolution of the height of the thin filmon a horizontal subsrate

EXPERIMENT

• Visualize the surfactant (using insoluble fluorescent surfactant)

• Examine the effect of the surfactanton evolution of the height of the thin filmon a horizontal subsrate

EQUATIONSEQUATIONS

h: height of thin filmΓ: 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 ξBottom 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 MoleculesRed: Laser, used to visualize the height of the film