## 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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