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Healing capillary films

Published online by Cambridge University Press:  16 January 2018

Zhong Zheng*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0EZ, UK
Marco A. Fontelos*
Affiliation:
Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, Madrid, 28049, Spain
Sangwoo Shin
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA
Michael C. Dallaston
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Dmitri Tseluiko
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
Serafim Kalliadasis
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Abstract

Consider the dynamics of a healing film driven by surface tension, that is, the inward spreading process of a liquid film to fill a hole. The film is modelled using the lubrication (or thin-film) approximation, which results in a fourth-order nonlinear partial differential equation. We obtain a self-similar solution describing the early-time relaxation of an initial step-function condition and a family of self-similar solutions governing the finite-time healing. The similarity exponent of this family of solutions is not determined purely from scaling arguments; instead, the scaling exponent is a function of the finite thickness of the prewetting film, which we determine numerically. Thus, the solutions that govern the finite-time healing are self-similar solutions of the second kind. Laboratory experiments and time-dependent computations of the partial differential equation are also performed. We compare the self-similar profiles and exponents, obtained by matching the estimated prewetting film thickness, with both measurements in experiments and time-dependent computations near the healing time, and we observe good agreement in each case.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

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Footnotes

Z. Zheng and M. A. Fontelos contributed equally to this work.

§

Present address: School of Computing, Electronics and Mathematics, and Flow Measurement and Fluid Mechanics Research Centre, Coventry University, Coventry CV1 5FB, UK.

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