×

Healing of thermocapillary film rupture by viscous heating. (English) Zbl 1430.76047

Summary: Thin liquid films sitting on a heated solid substrate and surrounded by a colder ambient gas phase are strongly affected by surface-shear stresses induced by surface tension and temperature gradients, as well as by viscous and capillary forces. The temperature dependence of surface tension may lead to thinning of liquid-film depressions promoting instability which takes place when a critical temperature difference \(\Delta\vartheta_{cr}\) between the substrate and the ambient gas phase is exceeded. In this article we show theoretically that viscous heating, previously neglected in related literature, may delay or suppress the thermocapillary instability and leads to film healing. The viscous heating effect, by inhibiting heat transfer, prevents the system from reaching the critical value \(\Delta\vartheta_{cr}\) required to bring about instability. As a consequence, the system remains within the stability region, suppressing film rupture. The presence of the viscous heating effect leads to a persistent circulating motion of two counter-rotating vortices lying diametrically opposite to a depression of the liquid-gas interface reducing the wavelength of disturbances to one half of its initial value. This effect has yet to be observed in experiment.

MSC:

76A20 Thin fluid films
76E30 Nonlinear effects in hydrodynamic stability

Keywords:

thin films
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aero, E. L.; Bulygin, A. N.; Kuvshinskii, E. V., Asymmetric hydromechanics, Z. Angew. Math. Mech., 29, 2, 333-346, (1965) · Zbl 0141.42603
[2] Benney, D. J., Long waves on liquid films, J. Math. Phys., 45, 2, 150-155, (1966) · Zbl 0148.23003
[3] Braun, R. J., Dynamics of the tear film, Annu. Rev. Fluid Mech., 44, 267-297, (2012) · Zbl 1357.76110
[4] Cordero, M. L.; Burnham, D. R.; Baroud, C. N.; Mcgloin, D., Thermocapillary manipulation of droplets using holographic beam shaping: Microfluidic pin ball, Appl. Phys. Lett., 93, 3, (2008)
[5] Cross, M. C.; Hohenberg, P. C., Pattern formation outside of equilibrium, Rev. Mod. Phys., 65, 3, 851-1112, (1993) · Zbl 1371.37001
[6] Davis, M. J.; Gratton, M. B.; Davis, S. H., Suppressing van der Waals driven rupture through shear, J. Fluid Mech., 661, 522-539, (2010) · Zbl 1205.76115
[7] Davis, S. H., Thermocapillary instabilities, Annu. Rev. Fluid Mech., 19, 1, 403-435, (1987) · Zbl 0679.76052
[8] Davis, S. H.2002Interfacial fluid dynamics. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 1-51. Cambridge University Press. · Zbl 1012.76021
[9] Gebhart, B., Effects of viscous dissipation in natural convection, J. Fluid Mech., 14, 2, 225-232, (1962) · Zbl 0106.39903
[10] Halpern, D.; Grotberg, J. B., Fluid-elastic instabilities of liquid-lined flexible tubes, J. Fluid Mech., 244, 615-632, (1992) · Zbl 0775.76054
[11] Huang, H.; Delikanli, S.; Zeng, H.; Ferkey, D. M.; Pralle, A., Remote control of ion channels and neurons through magnetic-field heating of nanoparticles, Nanotechnology, 5, 8, 602-606, (2010)
[12] Johns, L. E.; Narayanan, R., Frictional heating in plane Couette flow, Proc. R. Soc. Lond. A, 453, 1963, 1653-1670, (1997) · Zbl 0886.76020
[13] Joseph, D. D., Stability of frictionally-heated flow, Phys. Fluids, 8, 12, 2195-2200, (1965)
[14] Kataoka, D. E.; Troian, S. M., Stabilizing the advancing front of thermally driven climbing films, J. Colloid Interface Sci., 203, 2, 335-344, (1998)
[15] Kataoka, D. E.; Troian, S. M., Patterning liquid flow on the microscopic scale, Nature, 402, 6763, 794-797, (1999)
[16] Kerchman, V. I.; Frenkel, A. L., Interactions of coherent structures in a film flow: simulations of a highly nonlinear evolution equation, Theor. Comput. Fluid Dyn., 6, 4, 235-254, (1994) · Zbl 0820.76042
[17] Kirkinis, E., Magnetic torque-induced suppression of van-der-Waals-driven thin liquid film rupture, J. Fluid Mech., 813, 991-1006, (2017) · Zbl 1383.76037
[18] Kirkinis, E.; Davis, S. H., Stabilization mechanisms in the evolution of thin liquid-films, Proc. R. Soc. Lond. A, 471, (2015) · Zbl 1371.76013
[19] Landau, L. D.; Lifshitz, E. M., Electrodynamics of Continuous Media, (1960), Pergamon Press · Zbl 0122.45002
[20] Landau, L. D. & Lifshitz, E. M.1987Fluid Mechanics, , vol. 6. Pergamon Press. · Zbl 0081.22207
[21] Maggi, C.; Saglimbeni, F.; Dipalo, M.; De Angelis, F.; Di Leonardo, R., Micromotors with asymmetric shape that efficiently convert light into work by thermocapillary effects, Nat. Commun., 6, 7855, (2015)
[22] Oron, A.; Davis, S. H.; Bankoff, S. G., Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69, 3, 931-980, (1997)
[23] Polo-Corrales, L.; Rinaldi, C., Monitoring iron oxide nanoparticle surface temperature in an alternating magnetic field using thermoresponsive fluorescent polymers, J. Appl. Phys., 111, 7, 07B334, (2012)
[24] Rinaldi, C.2002 Continuum modeling of polarizable systems. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.
[25] Shampine, L. F.; Reichelt, M. W., The matlab ode suite, SIAM J. Sci. Comput., 18, 1, 1-22, (1997) · Zbl 0868.65040
[26] Stewart, P. S.; Davis, S. H., Self-similar coalescence of clean foams, J. Fluid Mech., 722, 645-664, (2013) · Zbl 1287.76220
[27] Subrahmaniam, N.; Johns, L. E.; Narayanan, R., Stability of frictional heating in plane Couette flow at fixed power input, Proc. R. Soc. Lond. A, 458, 2027, 2561-2569, (2002) · Zbl 1015.76023
[28] Turcotte, D. L.; Hsui, A. T.; Torrance, K. E.; Schubert, G., Influence of viscous dissipation on Bénard convection, J. Fluid Mech., 64, 2, 369-374, (1974) · Zbl 0288.76043
[29] Vanhook, S. J.; Schatz, M. F.; Swift, J. B.; Mccormick, W. D.; Swinney, H. L., Long-wavelength surface-tension-driven Bénard convection: experiment and theory, J. Fluid Mech., 345, 45-78, (1997) · Zbl 0927.76036
[30] Vo, T. Q.; Kim, B.-H., Transport phenomena of water in molecular fluidic channels, Sci. Rep., 6, 33881, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.