Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity

Abstract

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 that 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 reducedorder 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 traveling waves 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.

Robust low-dimensional modelling of falling liquid films subject to variable wall heating

“…In our analysis we have assumed throughout that the temperature at the wall is exactly specified. However, in physical implementation it is more likely that the boundary condition would involve either a specification of the heat flux or some linear combination of temperature and heat flux (conceivably coupled to another thermal problem within the wall such as considered by Dallaston, Tseluiko & Kalliadasis (2016)). Our modelling methodology can be easily extended to these more general boundary conditions (see appendix A).…”

Regular and complex singularities of the generalized thin film equation in two dimensions

“…Novel extensions in the present study compared with that work are the computation of asymmetric similarity solutions (which are stable in some parameter regimes and, therefore, of interest), and the use of continuation to compute periodic orbits in the scaled time and space coordinate system in which a similarity solution corresponds to a steady state. While Dallaston et al (2018) is an instance of numerical continuation being used to compute the stability of similarity solutions, the use of numerical continuation to compute the stability of steady states has previously been applied in thin film models in Thiele & Knobloch (2003 and to the Orr-Sommerfeld equations of interfacial hydrodynamic models in Dallaston, Tseluiko & Kalliadasis (2016). In addition, the tracing of periodic solution branches via continuation has been applied previously in Lin et al (2016), although in unscaled time and space.…”

Fluid Flow and Thin-Film Evolution near the Triple Line during Droplet Evaporation of Self-Rewetting Fluids

“…Atomic force microscopy in a tapping mode was successfully used for the characterization of the thin-film profile at about 1 nm away from the contact line, , but it is insufficient to further understand how the thin-film profile varies with time. In fact, the thin-film evolution is critical for understanding the thin-film dynamics. − …”