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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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