Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity
Michael C. Dallaston
Dmitri Tseluiko
Serafim Kalliadasis
2134/23495
https://repository.lboro.ac.uk/articles/Dynamics_of_a_thin_film_flowing_down_a_heated_wall_with_finite_thermal_diffusivity/9385241
Consider 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.
2016-12-16 10:23:01
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