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Nonlinear convection in a porous layer with finite conducting boundaries. (English) Zbl 0559.76090

The problem of finite-amplitude thermal convection in a porous layer with finite conducting boundaries is investigated. The nonlinear problem of three-dimensional convection is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. Square-flow-pattern convection is found to be preferred in a bounded region \(\Gamma\) in the \((\gamma_ b,\gamma_ t)\)-space, where \(\gamma_ b\) and \(\gamma_ t\) are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid. Two-dimensional rolls are found to be the preferred pattern outside \(\Gamma\). The qualitative features of the convection problem appear to be essentially symmetric with respect to \(\gamma_ b\) and \(\gamma_ t\). The dependence of the heat transported by convection on \(\gamma_ b\) and \(\gamma_ t\) is computed for the various solutions analysed in the paper.

MSC:

76S05 Flows in porous media; filtration; seepage
76R10 Free convection
80A20 Heat and mass transfer, heat flow (MSC2010)
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References:

[1] Lapwood, Proc. Camb. Phil. Soc. 44 pp 508– (1948)
[3] DOI: 10.1017/S0022112067001661 · Zbl 0159.28202
[4] DOI: 10.1063/1.862066 · Zbl 0375.76075
[7] DOI: 10.1017/S0022112074001996 · Zbl 0289.76054
[8] DOI: 10.1017/S0022112078001664
[10] DOI: 10.1017/S0022112065001271 · Zbl 0134.21801
[11] DOI: 10.1017/S002211207200059X · Zbl 0252.76066
[12] DOI: 10.1038/288442a0
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