Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T08:49:51.188Z Has data issue: false hasContentIssue false
  • English
  • Français

SURFACE TEMPERATURE RECONSTRUCTION BASED ON THE THERMOCAPILLARY EFFECT

Part of: Two-phase and multiphase flows

Published online by Cambridge University Press:  19 August 2011

M. SELLIER  and
S. PANDA
M. SELLIER*
Affiliation:
Department of Mechanical Engineering, The University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand (email: [email protected])
S. PANDA
Affiliation:
Department of Mathematics, National Institute of Technology, Calicut NIT Post. 673601, Kerala, India (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A thin liquid film subject to a temperature gradient is known to deform under the action of thermocapillary stresses which induce convective cells. The free surface deformation can be thought of as the signature of the imposed temperature gradient, and this study investigates the inverse problem of trying to reconstruct the temperature field from known free surface variations. The present work builds on the analysis of Tan et al. [“Steady thermocapillary flows of thin liquid layers I. Theory”, Phys. Fluids A2 (1990) 313–321, doi:10.1063/1.857781] which provides a long-wave evolution equation for the fluid film thickness variation on nonuniformly heated substrates and proposes a solution strategy for the planar flow version of this inverse problem. The present analysis reveals a particular case for which there exists an explicit, closed-form solution expressing the local substrate temperature in terms of the local film thickness and its spatial derivatives. With some simplifications, this analysis also shows that this solution applies to three-dimensional flows. The temperature reconstruction strategies are successfully tested against “artificial” experimental data (obtained by solving the direct problem for known temperature profiles) and actual experimental data.

Keywords

thin filmlubrication approximationfinite volume methodinverse problem

MSC classification

Secondary: 76T10: Liquid-gas two-phase flows, bubbly flows
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 2 , October 2010 , pp. 146 - 159
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Burelbach, J. P., Bankoff, S. G. and Davis, S. H., “Steady thermocapillary flows of thin liquid layers. II. Experiment”, Phys. Fluids A 2 (1990) 322–333, doi:10.1063/1.857782.CrossRefGoogle Scholar
[2]Davis, S. H., “Thermocapillary instabilities”, Annu. Rev. Fluid Mech. 19 (1987) 403–435,doi:10.1146/annurev.fluid.19.1.403.CrossRefGoogle Scholar
[3]Decré, M. M. J. and Baret, J. C., “Gravity-driven flows of viscous liquids over two-dimensional topographies”, J. Fluid. Mech. 487 (2003) 147–166, doi:10.1017/S0022112003004774.CrossRefGoogle Scholar
[4]Ha, Y., Kim, Y.-J. and Myers, T. G., “On the numerical solution of a driven thin film equation”, J. Comput. Phys. 227 (2008) 7246–7263, doi:10.1016/j.jcp.2008.04.007.CrossRefGoogle Scholar
[5]Kalliadasis, S., Kiyashko, A. and Demsekhin, E. A., “Marangoni instability of a thin liquid film heated from below by a local heat source”, J. Fluid Mech. 475 (2003) 377–408, doi:10.1017/S0022112002003014.CrossRefGoogle Scholar
[6]Koehler, T. P., Experimental and numerical investigation of thermocapillary effects in thin liquid layers, Ph.D. Thesis, Georgia Institute of Technology, 2007.Google Scholar
[7]Liu, T., Campbell, B. T., Burns, S. P. and Sullivan, J. P., “Temperature- and pressure-sensitive luminescent paints in aerodynamics”, Appl. Mech. Rev. 50 (1997) 227–246, doi:10.1115/1.3101703.CrossRefGoogle Scholar
[8]Nozhat, W. M., “Measurement of liquid-film thickness by laser interferometry”, Appl. Optics 36 (1997) 7864–7869, doi:10.1364/AO.36.007864.CrossRefGoogle ScholarPubMed
[9]Ruyer-Quil, C., Scheid, B., Kalliadasis, S., Velarde, M. G. and Zeytounian, R. Kh., “Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation”, J. Fluid Mech. 538 (2005) 199–222, doi:10.1017/S0022112005005422.CrossRefGoogle Scholar
[10]Schatz, M. F. and Neitzel, G. P., “Experiments on thermocapillary instabilities”, Annu. Rev. Fluid Mech. 33 (2001) 93–127, doi:10.1146/annurev.fluid.33.1.93.CrossRefGoogle Scholar
[11]Scheid, B., Oron, A., Colinet, P., Thiele, U. and Legros, J. C., “Nonlinear evolution of nonuniformly heated falling liquid films”, Phys. Fluids 14 (2002) 4130–4151, doi:10.1063/1.1515270.CrossRefGoogle Scholar
[12]Sellier, M., “Substrate design or reconstruction from free surface data for thin film flows”, Phys. Fluids 20 (2008) 062106, (4 pages), doi:10.1063/1.2939404.CrossRefGoogle Scholar
[13]Settles, G. S., Schlieren and shadowgraph techniques: visualizing phenomena in transparent media (Experimental fluid mechanics) (Springer, New York, 2001).CrossRefGoogle Scholar
[14]Tan, M. J., Bankoff, S. G. and Davis, S. H., “Steady thermocapillary flows of thin liquid layers. I. Theory”, Phys. Fluids A 2 (1990) 313–321, doi:10.1063/1.857781.CrossRefGoogle Scholar
[15]Versteeg, H. K. and Malalasekera, W., An introduction to computational fluid dynamics: the finite volume method (Pearson Education, Harlow, UK, 1995).Google Scholar