Travelling wave solutions in dilatant non-Newtonian thin films with second-order viscosity

Abstract

Given any positive speed, we prove the existence of a travelling wave solution for a viscous incompressible thin dilatant liquid film coated with an insoluble surfactant. Relying on first principles of fluid mechanics, lubrication theory leads to a coupled system of second-order partial differential equations for the film's height and the surfactant's concentration where a second-order approximation of the viscosity function is used. The travelling wave solution is $C^2$-regular and generates a heteroclinic orbit which connects two distinct equilibrium points of the associated plane dynamical system.

Keywords:

• Travelling wave
• heteroclinic orbit
• dilatant fluid
• thin film
• surfactant

2010 Mathematics Subject Classifications:

• 35C07
• 35K40
• 35K65
• 35Q35
• 76A20

Acknowledgments

The first author would like to express his sincere thanks to Institut für Angewandte Mathematik, Leibniz Universität Hannover for the support from October 2018 to September 2019 as a joint PhD student.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 See the discussion following Lemma  3.1.

2 Here and in the following we use the notation $f(H,0)$ and $g(H,0)$ for the limits $\underset{\mathrm{\Gamma }\to 0}{lim}f(H,\mathrm{\Gamma })$ and $\underset{\mathrm{\Gamma }\to 0}{lim}g(H,\mathrm{\Gamma }),$ respectively.

Additional information

Funding

This work is sponsored by China Scholarship Council and partially supported by National Natural Science Foundation of China under the grant number 11571381.