Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T04:19:36.788Z Has data issue: false hasContentIssue false
  • English
  • Français

An asymptotic approach to the crenulation instability

Published online by Cambridge University Press:  09 August 2017

Carlo Camporeale Open the ORCID record for Carlo Camporeale [Opens in a new window]
Carlo Camporeale*
Affiliation:
Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Torino 10129, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A novel linear stability analysis for the pattern formation of ripple-like corrugations, over the surface of stalactites due to calcite-laden falling films, is presented by using a modified version of the gradient expansion technique. A fully analytical theory combining thin film hydrodynamics, Kármán–Pohlhausen solution of the advection–diffusion equation for calcium concentration and an evolution equation for the shape of the liquid–solid interface is developed. Analytical formulas for the selected wavelength and the corresponding phase velocity are provided, which match previous numerical solutions. The obtained easy-to-use mathematical results have a potential for developing a novel set of paleo-reconstruction proxies based on the link between morphological patterns and paleo-flows.

JFM classification

Instability: Absolute/convective instability Phase change: Morphological instability Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 636 - 652
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55. Department of Commerce, United States of America.Google Scholar
Camporeale, C. 2015 Hydrodynamically locked morphogenesis in karst and ice flutings. J. Fluid Mech. 778, 89–119.CrossRefGoogle Scholar
Camporeale, C. & Ridolfi, L. 2012a Hydrodynamic-driven stability analysis of morphological patterns on stalactites and implications for cave paleoflow reconstructions. Phys. Rev. Lett. 108, 238501.CrossRefGoogle ScholarPubMed
Camporeale, C. & Ridolfi, L. 2012b Ice ripple formation at large Reynolds numbers. J. Fluid Mech. 694, 225–251.CrossRefGoogle Scholar
Chan, P. Y. & Goldenfeld, N. 2007 Steady states and linear stability analysis of precipitation pattern formation at geothermal hot springs. Phys. Rev. E 76 (4, part 2), 046104.Google ScholarPubMed
Chen, A. S.-H. & Morris, S. W. 2013 On the origin and evolution of icicle ripples. New J. Phys. 15, 103012.Google Scholar
Dreybrodt, W. 1988 Processes in Karst Systems. Springer.CrossRefGoogle Scholar
Fairchild, I. J., Smith, C. L., Baker, A., Fuller, L., Spotl, C., Mattey, D., McDermott, F. & Edinburgh Ion Microprobe Facility 2006 Modification and preservation of environmental signals in speleothems. Earth-Sci. Rev. 75, 105–153.CrossRefGoogle Scholar
Genty, D., Blamart, D., Ouahdi, R., Gilmour, M., Baker, A., Jouzel, J. & Van-Exter, S. 2003 Precise dating of Dansgaard–Oeschger climate oscillations in western Europe from stalagmite data. Nature 421 (6925), 833–837.CrossRefGoogle ScholarPubMed
Hutter, K. 1983 Theoretical Glaciology. Kluwer.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films, Springer Series on Applied Mathematical Sciences. Springer.CrossRefGoogle Scholar
Kaufmann, G. & Dreybrodt, W. 2007 Calcite dissolution kinetics in the system CaCO3–H2O–CO2 at high undersaturation. Geochim. Cosmochim. Acta 71 (6), 1398–1410.CrossRefGoogle Scholar
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. Lond. A 466 (2115), 659–694.Google Scholar
Plummer, L. N., Wigley, T. M. L. & Parkhurst, D. L. 1978 Kinetics of calcite dissolution in CO2 water system in at 5 °C–60 °C and 10.0–1.0 atm CO2 . Am. J. Sci. 278 (2), 179–216.Google Scholar
Pradas, M., Pavliotis, G. A., Kalliadasis, S., Papageorgiou, D. T. & Tseluiko, D. 2012 Additive noise effects in active nonlinear spatially extended systems. Eur. J. Appl. Maths 23 (5), 56317591.Google Scholar
Romanov, D., Kaufmann, G. & Dreybrodt, W. 2008 Modeling stalagmite growth by first principles of chemistry and physics of calcite precipitation. Geochim. Cosmochim. Acta 72 (2), 423–437.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, 1st edn. Springer.CrossRefGoogle Scholar
Short, M. B., Baygents, J. C., Beck, J. W., Stone, D. A., Toomey, R. S. & Goldstein, R. E. 2005a Stalactite growth as a free-boundary problem: a geometric law and its platonic ideal. Phys. Rev. Lett. 94 (1), 018501.CrossRefGoogle ScholarPubMed
Short, M. B., Baygents, J. C. & Goldstein, R. E. 2005b Stalactite growth as a free-boundary problem. Phys. Fluids 17 (8), 083101.CrossRefGoogle Scholar
Tseluiko, D. & Blyth, M. G. 2009 Effect of inertia on electrified film flow over a wavy wall. J. Engng Maths 65 (3), 229–242.CrossRefGoogle Scholar
Tseluiko, D., Blyth, M. G. & Papageorgiou, D. T. 2013 Stability of film flow over inclined topography based on a long-wave nonlinear model. J. Fluid Mech. 729, 63817671.CrossRefGoogle Scholar
Ueno, K., Farzaneh, M., Yamaguchi, S. & Tsuji, H. 2010 Numerical and experimental verification of a theoretical model of ripple formation in ice growth under supercooled water film flow. Fluid Dyn. Res. 42 (2), 025508.CrossRefGoogle Scholar
Vesipa, R., Camporeale, C. & Ridolfi, L. 2015 Thin-film induced morphological instabilities over calcite surfaces. Proc. R. Soc. Lond. A 471 (2176), 20150031.Google ScholarPubMed