Abstract
The steady laminar two-dimensional thermocapillary convection of two immiscible liquid layers in a thin annular pool with one free surface, one liquid/liquid interface subjected to a radial temperature gradient was investigated using asymptotical analysis. The pool is heated from the inner cylindrical wall and cooled at the outer wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the lower layer thickness to the gap width, trends to zero. The numerical experiments are also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. The results indicate that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.
Similar content being viewed by others
References
Cormack, D.E., Leal, L.G., Imberger, J.: Natural convection in a shallow cavity with differentially heated end walls. Part 1. Asymptotic theory. J. Fluid Mech. 65(2), 209–229 (1974)
Doi, T., Koster, J.N.: Thermocapillary convection in two immiscible liquid layers with free surface. Phys. Fluids A5(8), 1914–1927 (1993)
Fontaine, J.P., Sani, R.L.: Flow and transport in a multilayered fluid system-I. Influence of 1- or μ-g environment. Int. J. Heat Mass Transfer 39(13), 2751–2770 (1996a)
Fontaine, J.P., Sani, R.L.: Flow and transport in a multilayered fluid system-II. Influence of thermal boundary conditions and encapsulant. Int. J. Heat Mass Transfer 39(13), 2846–2852 (1996b)
Gupta, N.R., Haj-Hariri, H., Borhanc, A.: Thermocapillary flow in double-layer fluid structures: an effective single-layer model. J. Colloid Interface Sci. 293(1), 158–171 (2006)
Gupta, N.R., Haj-Hariri, H., Borhanc, A.: Thermocapillary convection in double-layer fluid structures within a two-dimensional open cavity. J. Colloid Interface Sci. 315(1), 237–247 (2007)
Jurisch, M., Borner, F., Bunger, T., Eichler, S., Flade, T., Kretzer, U., Kohler, A., Stenzenberger, J., Weinert, B.: LEC- and VGF-growth of SI GaAs single crystals - recent developments and current issues. J. Cryst. Growth 275(1–2), 283–291 (2005)
Leppinen, D.M.: Natural convection in a shallow cylindrical annuli. Int. J. Heat Mass Transfer 45(14), 2967–2981 (2002)
Li, Y.R., Zhang, W.J., Wang, S.C.: Two-dimensional numerical simulation of thermocapillary convection in annular two-layer system. Microgravity Sci. Technol. 20(3–4), 313–317 (2008a)
Li, Y.R., Zhao, X.X., Wu, S.Y., Peng, L.: Asymptotic solution of thermocapillary convection in a thin annular pool of silicon melt. Phys. Fluids 20(8), 082107 (2008b)
Liu, Q.S., Chen, G., Roux, B.: Thermogravitational and thermocapillary convection in a cavity containing two superposed immiscible liquid layers. Int. J. Heat Mass Transfer 36(1), 101–117 (1993)
Liu, Q.S., Roux, B., Velarde, M.G.: Thermocapillary convection in two-layers system. Int. J. Heat Mass Transfer 41(11), 1499–1511 (1998)
Madruga, S., Perez-Garcia, C., Lebon, G.: Convective instabilities in two superposed horizontal liquid layers heated laterally. Phys. Rev. E 68(4), 041607 (2003)
Madruga, S., Perez-Garcia, C., Lebon, G.: Instabilities in two-liquid layers subject to a horizontal temperature gradient. Theor. Comput. Fluid Dyn. 18(2–4), 277–284 (2004)
Merker, G.P., Leal, L.G.: Natural convection in a shallow annular cavity. Int. J. Heat Mass Transfer 23(5), 677–686 (1980)
Metz, E.P.A., Miller, R.C., Mazelsky, R.: A technique for pulling single crystals of volatile materials. J. Appl. Phys. 33(6), 2016–2017 (1962)
Morton, J.L., Ma, N., Bliss, D.F., Bryant, G.G.: Magnetic field effects during liquid-encapsulated Czochralski growth of doped photonic semiconductor crystals. J. Cryst. Growth 250(1–2), 174–182 (2003)
Nepomnyashchy, A.A., Simanovskii, I.B.: Convective flows in a two-layer system with a temperature gradient along the interface. Phys. Fluids 18(3), 032105 (2006)
Nepomnyashchy, A.A., Simanovskii, I.B.: Nonlinear stability of buoyancy-thermocapillary flows in two-layer systems with a temperature gradient along the interface. J. Phys.: Conference Series 64, 012017 (2007)
Nepomnyashchy, A.A., Simanovskii, I.B., Braverman, L.M.: Spiral thermocapillary flows in two-layer systems. Eur. J. Mech. B Fluids 18(6), 1005–1026 (1999)
Sen, A.K., Davis, S.H.: Steady thermocapillary flows in two-dimensional slots. J. Fluid Mech. 121, 163–186 (1982)
Villers, D., Platten, J.K.: Thermal convection in superposed immiscible liquid layers. Appl. Sci. Res. 45(2), 145–152 (1988)
Villers, D., Platten, J.K.: Influence of interfacial tension gradients on thermal convection in two superposed immiscible liquid layers. Appl. Sci. Res. 47(2), 177–202 (1990)
Walker, J.S., Henry, D.H., Hadid, B.: Magnetic stabilization of the buoyant convection in the liquid-encapsulated Czochralski process. J. Cryst. Growth 243(1), 108–116 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, YR., Wang, SC., Wu, SY. et al. Asymptotic Solution of Thermocapillary Convection in a Differentially Heated Thin Annular Two-Layer Pool. Microgravity Sci. Technol. 22, 193–203 (2010). https://doi.org/10.1007/s12217-009-9174-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12217-009-9174-0