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Stationary instability of the convective flow between differentially heated vertical planes. (English) Zbl 0673.76045
Summary: An asymptotic theory describes the stationary instability of convective flow between differentially heated vertical planes at large Prandtl numbers. The theory is concerned with the structure for $$A\gg 1$$, where A is a Rayleigh number based on the horizontal temperature difference and the distance between the planes. As such it is relevant to the instability of flow in a vertical slot of aspect ratio $$h\gg 1$$ where the convective regime corresponds to order-one values of a non-dimensional parameter $$\gamma$$ which partly depends on the vertical temperature gradient generated in the slot and can be approximated by $$\gamma^ 4=A/8h$$. Instability is shown to set in at a critical value of $$\gamma$$ that compares well with experimental observation. The lower branch of the neutral curve conforms to a boundary-layer type approximation while the upper branch has a critical-layer structure midway between the planes which becomes fully developed as the first reversal of the vertical velocity of the base flow is encountered near the centreline.

##### MSC:
 76E15 Absolute and convective instability and stability in hydrodynamic stability 80A20 Heat and mass transfer, heat flow (MSC2010) 76M99 Basic methods in fluid mechanics
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##### References:
 [1] Daniels, Proc. R. Soc. Lond. A401 pp 145– (1985) · Zbl 0575.76084 · doi:10.1098/rspa.1985.0092 [2] DOI: 10.1016/0017-9310(85)90100-0 · doi:10.1016/0017-9310(85)90100-0 [3] DOI: 10.1017/S0022112078000452 · doi:10.1017/S0022112078000452 [4] Batchelor, Q. Appl. Maths 12 pp 209– (1954) [5] DOI: 10.1063/1.1694263 · Zbl 0307.76023 · doi:10.1063/1.1694263 [6] DOI: 10.1017/S0022112069001467 · Zbl 0167.25704 · doi:10.1017/S0022112069001467 [7] DOI: 10.1017/S0022112081000827 · doi:10.1017/S0022112081000827 [8] DOI: 10.1175/1520-0469(1966)023 2.0.CO;2 · doi:10.1175/1520-0469(1966)023 · doi:2.0.CO;2 [9] DOI: 10.1017/S0022112078000427 · doi:10.1017/S0022112078000427 [10] DOI: 10.1016/0017-9310(73)90161-0 · doi:10.1016/0017-9310(73)90161-0 [11] DOI: 10.1017/S0022112069001431 · Zbl 0165.57901 · doi:10.1017/S0022112069001431 [12] DOI: 10.1017/S0022112065001246 · doi:10.1017/S0022112065001246 [13] DOI: 10.1017/S0022112087000740 · Zbl 0612.76091 · doi:10.1017/S0022112087000740
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