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Simulating surfactant spreading: Influence of a physically motivated equation of state†

Published online by Cambridge University Press:  09 March 2017

DINA SINCLAIR ,
RACHEL LEVY  and
KAREN E. DANIELS
DINA SINCLAIR
Affiliation:
Mathematics Department, Harvey Mudd College, Claremont, CA, USA emails: [email protected], [email protected]
RACHEL LEVY
Affiliation:
Mathematics Department, Harvey Mudd College, Claremont, CA, USA emails: [email protected], [email protected]
KAREN E. DANIELS
Affiliation:
Department of Physics, North Carolina State University, Raleigh, NCUSA email: [email protected]
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Abstract

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In this paper, we present numerical simulations that demonstrate the effect of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension in surfactant-driven thin liquid films. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically motivated EoS; this provides a route to resolve some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and influences the dynamics of the simulations. Because an empirically motivated EoS has regions with distinct gradients, future mathematical models may be improved by considering more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational versus capillary forces has a larger influence on the dynamics than the other non-dimensional groups, but is nonetheless not a likely cause of discrepancy between simulations and experiments. Finally, we observe that the experimental approach using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the fluid meniscus itself does not significantly affect the dynamics.

Keywords

AMS classifications: 35Q35 PDEs in connection with fluid mechanics76A20 thin fluid films76M12 finite volume methods76B45 capillarity (surface tension)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 30 - 54
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

†

This work was funded by NSF grant DMS-FRG #096815 (RL and KED), Howard Hughes Medical Institute Undergraduate Science Education Program Award #52006301 (RL), and Research Corporation Cottrell Scholar Award #19788 (RL).

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