Ice formation within a thin film flowing over a flat plate

Moore, M. R.; Mughal, M. S.; Papageorgiou, D. T.

Abstract

We present a model for ice formation in a thin, viscous liquid film driven by a Blasius boundary layer after heating is switched off along part of the flat plate. The flow is assumed to initially be in the Nelson et al. (J. Fluid Mech., vol. 284, 1995, pp. 159–169) steady-state configuration with a constant flux of liquid supplied at the tip of the plate, so that the film thickness grows like 𝑥1/4 in distance along the plate. Plate cooling is applied downstream of a point, 𝐿𝑥0 , an 𝑂(𝐿) -distance from the tip of the plate, where 𝐿 is much larger than the film thickness. The cooling is assumed to be slow enough that the flow is quasi-steady. We present a thorough asymptotic derivation of the governing equations from the incompressible Navier–Stokes equations in each fluid and the corresponding Stefan problem for ice growth. The problem breaks down into two temporal regimes corresponding to the relative size of the temperature difference across the ice, which are analysed in detail asymptotically and numerically. In each regime, two distinct spatial regions arise, an outer region of the length scale of the plate, and an inner region close to 𝑥0 in which the film and air are driven over the growing ice layer. Moreover, in the early time regime, there is an additional intermediate region in which the air–water interface propagates a slope discontinuity downstream due to the sudden onset of the ice at the switch-off point. For each regime, we present ice profiles and growth rates, and show that for large times, the film is predicted to rupture in the outer region when the slope discontinuity becomes sufficiently enhanced.

Citation

Moore, M. R., Mughal, M. S., & Papageorgiou, D. T. (2017). Ice formation within a thin film flowing over a flat plate. Journal of Fluid Mechanics, 817, 455-489. https://doi.org/10.1017/jfm.2017.100

• Journal Article Type: Article
• Acceptance Date: Feb 10, 2017
• Online Publication Date: Mar 22, 2017
• Publication Date: Apr 25, 2017
• Deposit Date: Nov 17, 2021
• Publicly Available Date: Feb 23, 2022
• Journal: Journal of Fluid Mechanics
• Print ISSN: 0022-1120
• Electronic ISSN: 1469-7645
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 817
• Pages: 455-489
• DOI: https://doi.org/10.1017/jfm.2017.100
• Keywords: Mechanical Engineering; Mechanics of Materials; Condensed Matter Physics
• Public URL: https://hull-repository.worktribe.com/output/3883149