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Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity
journal contribution
posted on 2016-12-16, 10:23 authored by Michael C. Dallaston, Dmitri TseluikoDmitri Tseluiko, Serafim KalliadasisConsider the dynamics of a thin film flowing down a heated substrate. The substrate
heating generates a temperature distribution on the free surface, which in turn induces
surface-tension gradients and corresponding thermocapillary stresses that affect the free
surface and therefore the fluid flow. We study here the effect of finite substrate thermal
diffusivity on the film dynamics. Linear stability analysis of the full Navier-Stokes and
heat transport equations indicates if the substrate diffusivity is sufficiently small, the film
becomes unstable at a finite wavelength and at a Reynolds number smaller than that
predicted in the long-wavelength limit. This property is captured in a reduced-order system
of equations derived using a weighted-residual integral-boundary-layer method. This
reduced-order model is also used to compute the bifurcation diagrams of solution branches
connecting the trivial flat film to travelingwaves including solitary pulses. The effect of finite
diffusivity is to separate a simultaneous Hopf-transcritical bifurcation into its individual
component bifurcations. The appropriate Hopf bifurcation then connects only to the solution
branch of negative-hump pulses, with wave speed less than the linear wave speed, while
the branch of positive-single-hump pulses merges with the branch of positive-two-hump
pulses at a supercritical Reynolds number. In the regime where finite-wavelength instability
occurs, there exists a Hopf-bifurcation pair connected by a branch of periodic solutions,
whose period cannot be increased indefinitely. Numerical simulation of the reduced-order
system shows the development of a train of coherent structures, each of which resembles
a stationary positive-hump pulse, and, in the regime of finite-wavelength instability,
wavelength selection and saturation to periodic traveling waves.
Funding
We acknowledge financial support from the Engineering and Physical Sciences Research Council of the UK through Grants No. EP/K008595/1 and No. EP/L020564/1. The work of D.T. was partly supported by EPSRC Grant No. EP/K041134/1.
History
School
- Science
Department
- Mathematical Sciences
Published in
Physical Review FluidsVolume
1Citation
DALLASTON, M., TSELUIKO, D. and KALLIADASIS, S., 2016. Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity. Physical Review Fluids, 1, 073903. DOI: 10.1103/PhysRevFluids.1.073903Publisher
© American Physical SocietyVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Publication date
2016-11-28Copyright date
2016ISSN
2469-990XPublisher version
Language
- en