Convective heat transport in a fluid layer of infinite Prandtl number: upper bounds for the case of rigid lower boundary and stress-free upper boundary

N. K. Vitanov

doi:10.1007/s100510051136

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2023 Impact factor 1.6

Condensed Matter and Complex Systems

Eur. Phys. J. B 15, 349-355 (2000)
https://doi.org/10.1007/s100510051136

Convective heat transport in a fluid layer of infinite Prandtl number: upper bounds for the case of rigid lower boundary and stress-free upper boundary

N. K. Vitanov

Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany

Received: 29 September 1999
Published online: 15 May 2000

Abstract

We present the theory of the multi-α-solutions of the variational problem for the upper bounds on the convective heat transport in a heated from below horizontal fluid layer with rigid lower boundary and stress-free upper boundary. A sequence of upper bounds on the convective heat transport is obtained. The highest bound ${\rm Nu} = 1+(1/6)R^{1/3}$ is between the bounds ${\rm Nu}=1+0.152R^{1/3}$ for the case of a fluid layer with two rigid boundaries and ${\rm Nu}=1+0.3254 R^{1/3}$ for the case of a fluid layer with two stress-free boundaries. As an additional result of the presented theory we obtain small corrections of the boundary layer thicknesses of the optimum fields for the case of fluid layer with two rigid boundaries. These corrections lead to systematically lower upper bounds on the convective heat transport in comparison to the bounds obtained in [5].