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On unsteady free convection in vertical slots due to prescribed fluxes of heat or mass at the vertical walls

  • Fritz H. Bark (a1), Farid Alavyoon (a1) and Anders A. Dahlkild (a1) (a2)


Unsteady convection of an initially homogeneous fluid in a vertical slot is investigated theoretically in the limit of large Rayleigh and Prandtl/Schmidt numbers. The motion is driven by prescribed fluxes of heat or mass at the vertical walls of the slot. The ‘heat-up’ problem is considered, i.e. the fluxes are specified to change instantaneously from zero to finite constant values. Perturbation methods are used to compute approximate solutions for the initial period and for the slow approach to the asymptotic state. Numerical solutions of the full problem are also given. It is shown that a significant stratification is set up after short time and that the system thereafter evolves as a strongly stratified fluid on a timescale that is proportional to $Ra^{\frac{2}{9}}$. During the latter part of the process, linear buoyancy layers of thickness $\sim Ra^{\frac{2}{9}}$ appear on the vertical walls. On the horizontal walls, there are nonlinear boundary layers of thickness $\sim Ra^{-\frac{1}{9}}$, whose structure is akin to that of a Stewartson E¼ layer. The theoretical predictions are found to be in good agreement with experimental results.



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Awakura, Y., Ebata, A. & Kondo, Y. 1979 J. Electrochem. Soc. 126, 23.
Batchelor, G. K. 1954 Q. J. Appl. Maths 12, 209.
Bejan, A. 1979 J. Fluid Mech. 71, 729.
Bennets, D. A. & Hocking, L. M. 1973 Proc. R. Soc. Lond. A 333, 469.
Carey, V. P. 1983 Intl J. Heat Mass Transfer 26, 911.
Carey, V. P. 1984 Intl J. Heat Mass Transfer 27, 419.
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.
Daniels, P. G. 1985 Intl J. Heat Mass Transfer 28, 2071.
Daniels, P. G. 1987a J. Fluid Mech. 176, 419.
Daniels, P. G. 1987b Q. J. Appl. Maths 40, 257.
Eckert, E. R. & Carlson, W. O. 1961 Intl J. Heat Mass Transfer 2, 206.
Eklund, A., Alavyoon, F., Simonsson, D., Karlsson, R. I. & Bark, F. H. 1991 Electrochimica Acta 36, 1345.
Elder, J. W. 1965 J. Fluid Mech. 23, 77.
Elder, J. W. 1966 J. Fluid Mech. 24, 823.
Fusegi, T., Hyun, J. M. & Kuwahara, K. 1991 Intl J. Heat and Mass Transfer (to appear).
Gebhart, B., Jaluria, Y., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy Induced Flow and Transport. Springer.
Gill, A. E. 1966 J. Fluid Mech. 26, 515.
Gu, H., Nguyen, T. V. & White, R. E. 1987 J. Electrochem. Soc. 134, 2953.
Hine, F. 1985 Electrode Processes and Electrochemical Engineering. Plenum.
Hyun, J. M. 1984 Intl J. Heat Mass Transfer 27, 1986.
Hyun, J. M. 1985a J. Phys. Soc. Japan 54, 942.
Hyun, J. M. 1985b Intl J. Heat Mass Transfer 29, 499.
Ivey, G. N. 1984 J. Fluid Mech. 144, 389.
Jischke, M. C. & Doty, R. T. 1975 J. Fluid Mech. 71, 729.
Johansson, C. 1987 A new method for computing unsteady two-dimensional convective-diffusive mass transport. Licentiate thesis, Internal Rep. 87–1, Dep. of Comp. Sci., University of Uppsala.
Karlsson, R. I., Alavyoon, F. & Eklund, A. 1990 Laser Anemometry - Proc. 3rd Intl Conf., p. 329. Springer.
Kimura, S. & Bejan, A. 1984 Trans. ASME C: J. Heat Transfer 106, 99
Kuiken, H. K. 1978 J. Engng Maths 2, 353.
Le FeAvre, E. J. 1957 In Proc. IXth IUTAM Congress, Brussels, p. 168.
Newman, J. 1973 Electrochemical Systems. Prentice.
Ostrach, S. 1963 NACA Rep. 1111.
Patterson, J. 1983 Trans. ASME 106, 104.
Patterson, J. & Imberger, J. 1980 J. Fluid Mech. 100, 65.
Prandtl, L. 1952 Essentials of Fluid Dynamics. Blackie & Son.
Rahm, L. 1985 Mathematical Modelling 6, 19.
Rahm, L. & Walin, G. 1979a J. Fluid Mech. 90, 315.
Rahm, L. & Walin, G. 1979b Astrophysics. Fluid Dyn. 13, 51.
Sakurai, T. & Matsuda, T. 1972 J. Fluid Mech. 54, 417.
Schladow, S. G. 1990 J. Fluid Mech. 213, 589.
Smith, S. H. 1981 SIAM J. Appl. Maths 40, 78.
Vahl Davis, G. de 1968 Intl J. Heat Mass Transfer 11, 1675.
Veronis, G. 1967a Tellus 19, 326.
Veronis, G. 1967b Tellus 19, 620.
Veronis, G. 1970 Ann. Rev. Fluid Mech. 2, 37.
Walin, G. 1971 J. Fluid Mech. 48, 647.
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