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Universal self-similar attractor in the bending-driven levelling of thin viscous films

Pedersen, Christian; Salez, Thomas; Carlson, Andreas
Journal article; AcceptedVersion; Peer reviewed
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Pedersen2020.pdf (2.006Mb)
Year
2021
Permanent link
http://urn.nb.no/URN:NBN:no-94291

CRIStin
1963587

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Appears in the following Collection
  • Matematisk institutt [2705]
  • CRIStin høstingsarkiv [22560]
Original version
Proceedings of the Royal Society of London. Biological Sciences. 2021, 477 (2254), DOI: https://doi.org/10.1098/rspa.2021.0354
Abstract
We study theoretically and numerically the bending-driven levelling of thin viscous films within the lubrication approximation. We derive Green’s function of the linearized thin-film equation and further show that it represents a universal self-similar attractor at long times. As such, the rescaled perturbation of the film profile converges in time towards the rescaled Green’s function, for any summable initial perturbation profile. In addition, for stepped axisymmetric initial conditions, we demonstrate the existence of another, short-term and one-dimensional-like self-similar regime. We also characterize the convergence time towards the long-term universal attractor in terms of the relevant physical and geometrical parameters, and provide the local hydrodynamic fields and global elastic energy in the universal regime as functions of time. Finally, we extend our analysis to the nonlinear thin-film equation through numerical simulations.
 
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