Abstract
Flow of a thin viscous film down a flat inclined plane becomes unstable to long-wave interfacial fluctuations when the Reynolds number based on the mean film thickness becomes larger than a critical value (this value decreases as the angle of inclination to the horizontal increases, and in particular becomes zero when the plate is vertical). Control of these interfacial instabilities is relevant to a wide range of industrial applications including coating processes and heat or mass transfer systems. This study considers the effect of blowing and suction through the substrate in order to construct from first principles physically realistic models that can be used for detailed passive and active control studies of direct relevance to possible experiments. Two different long-wave, thin-film equations are derived to describe this system; these include the imposed blowing/suction as well as inertia, surface tension, gravity and viscosity. The case of spatially periodic blowing and suction is considered in detail and the bifurcation structure of forced steady states is explored numerically to predict that steady states cease to exist for sufficiently large suction speeds since the film locally thins to zero thickness, giving way to dry patches on the substrate. The linear stability of the resulting non-uniform steady states is investigated for perturbations of arbitrary wavelength, and any instabilities are followed into the fully nonlinear regime using time-dependent computations. The case of small amplitude blowing/suction is studied analytically both for steady states and their stability. Finally, the transition between travelling waves and non-uniform steady states is explored as the amplitude of blowing and suction is increased.
Highlights
The flow of a viscous liquid film down an inclined plane under the action of gravity, inertia and surface tension, is a fundamental problem in fluid mechanics that has received considerable attention both theoretically and experimentally due to the richness of its dynamics and its wide technological applications, e.g. in coating processes and heat or mass transfer enhancement
The linear stability of a uniform film was first considered by Benjamin (1957) and Yih (1963), who used an Orr–Sommerfeld analysis to show that instability first appears at wavelengths that are large compared to the undisturbed film thickness, hs
Μ is the viscosity of the fluid; the flow becomes linearly unstable to long waves when R > Rc = (5/4) cot θ and we can see that the critical Reynolds number tends to zero as the plate becomes vertical
Summary
The flow of a viscous liquid film down an inclined plane under the action of gravity, inertia and surface tension, is a fundamental problem in fluid mechanics that has received considerable attention both theoretically and experimentally due to the richness of its dynamics and its wide technological applications, e.g. in coating processes and heat or mass transfer enhancement. There have been several theoretical and experimental studies of film flows down wavy inclined planes (typically with sinusoidal and step topographies), aiming to explore how topography affects stability and stability criteria such as the critical Reynolds number, how substrate heterogeneity interacts with nonlinear coherent structures and from a practical perspective, how topography induces flows that can be useful in heat or mass transfer by creating regions of recirculating fluid (see for example Tseluiko, Blyth & Papageorgiou (2013), and numerous references therein). Numerical methods The numerical calculations that we perform are of three types: computation of steady periodic solutions and their bifurcation structure, linear stability calculations of such steady states to perturbations of arbitrary wavelength and nonlinear time-evolution via initial value problems We conduct these calculations using the continuation software package AUTO-07P (Doedel et al 2009) and Matlab. We will generally present results for the weighted-residual model when the two models differ, but we note that a non-trivial bifurcation structure emerges even at zero Reynolds number, where the models are identical
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