Abstract
AbstractNew mechanisms are discovered regarding the effects of inertia in the transient Moffatt–Pukhnachov problem (J. Méc., vol. 187, 1977, pp. 651–673) on the evolution of the free surface of a viscous film coating the exterior of a rotating horizontal cylinder. Assuming two-dimensional evolution of the film thickness (i.e. neglecting variation in the axial direction), a multiple-timescale procedure is used to obtain explicitly parameterized high-order asymptotic approximations of solutions of the spatio-temporal evolution equation. Novel, hitherto-unexplained transitions from stability to instability are observed as inertia is increased. In particular, a critical Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_c$ is predicted at which occurs a supercritical pitchfork bifurcation in wave amplitude that is fully explained by the new asymptotic theory. For $\mathit{Re}<\mathit{Re}_c$, free-surface profiles converge algebraically-cum-exponentially to a steady state and, for $\mathit{Re}> \mathit{Re}_c$, stable temporally periodic solutions with leading-order amplitudes proportional to $(\mathit{Re}-\mathit{Re}_c)^{1/2}$ are found, i.e. in the régime in which previous related literature predicts exponentially divergent instability. For $\mathit{Re}=\mathit{Re}_c$, stable solutions are found that decay algebraically to a steady state. A model solution is proposed that not only captures qualitatively the interaction between fundamental and higher-order wave modes but also offers an explanation for the formation of the lobes observed in Moffatt’s original experiments. All asymptotic theory is convincingly corroborated by numerical integrations that are spectrally accurate in space and eighth/ninth-order accurate in time.
Highlights
The influences of gravitational and capillary effects on the stability of coating and rimming flows have been the subject of numerous asymptotic and numerical studies, the explicit influence of inertia has been less widely considered, e.g. as in Hosoi & Mahadevan (1999), Benilov & O’Brien (2005), Noakes, King & Riley (2006), Kelmanson (2009b), Pougatch & Frigaard (2011) and Benilov & Lapin (2013)
Whilst the present results — and those in a plethora of related theoretical studies — are derived from analyses of two-dimensional coating/rimming models that neglect axial effects, it is well-known that the experiments of, e.g., Moffatt (1977) and Hynes (1978) reveal an onset of instability that results in the development of three-dimensional axially-and-azimuthally isolated “lobe-like” profiles
Many qualitative and quantitative results have been found relating to the temporal dynamics of coating flow, and the present study is conducted in the spirit of these papers, of which Kelmanson (2009b) forms the primary motivation
Summary
The influences of gravitational and capillary effects on the stability of coating and rimming flows have been the subject of numerous asymptotic and numerical studies, the explicit influence of inertia has been less widely considered, e.g. as in Hosoi & Mahadevan (1999), Benilov & O’Brien (2005), Noakes, King & Riley (2006), Kelmanson (2009b), Pougatch & Frigaard (2011) and Benilov & Lapin (2013). The use of algebraic manipulators has enabled the development and implementation of increasingly powerful automated multiple-timescale asymptotic methods (Hinch & Kelmanson (2003), Hinch, Kelmanson & Metcalfe (2004), Kelmanson (2009a, 2009b), Groh & Kelmanson (2009, 2012)), in the last of which all results are validated against spectrally accurate transient numerical methods‡ Through such asymptotic studies, many qualitative and quantitative results have been found relating to the temporal dynamics of coating flow, and the present study is conducted in the spirit of these papers, of which Kelmanson (2009b) forms the primary motivation. By contrast, using a spectrally accurate computational technique developed and validated in Groh & Kelmanson (2009, 2012), new numerical integrations (discussed in §2.2) of the thin-film evolution equation derived in Kelmanson (2009b) presently reveal the existence of cyclic solutions for a range Re > Rec, in which both the linearized stability analysis and the two-timescale asymptotics predict full-blown instability. A model form of solution is proposed which captures well the qualitative behaviour of the numerics and proposes (for future work) how asymptotic methods might be developed to model the higher-mode interactions in a more accurate quantitative fashion
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