Amplitude equations for non linear convection in high vertical channels

Abstract

The problem of weakly non linear convection in vertical (planar or cylindrical) layers of fluid is not easily tractable by means of analytical methods. For instance the perturbation expansion in powers of the amplitude of the instability (or C-expan sion) does not work. Then, using the disparity between the small horizontal scale, R, and the large vertical scale, h, we have developed an expansion scheme in powers of λ 1 = 2R/h. The governing equations are solved at each order in λ⁻¹ and the first solvability condition which is not trivially satisfied yields the evolution equation for the slowly varying function A(Z,T). Three typical equations are obtained owing to the symmetry of the horizontal mode of flow.