Double-component convection due to different boundary conditions in an infinite slot diversely oriented to the gravity

摘要

Onset of small-amplitude oscillatory and both small- and finite-amplitude steady double-component convection arising due to component different boundary conditions in an infinite slot is studied for various slot orientations to the gravity. The main focus is on two compensating background gradients of I tie components. The physical mechanisms underlying steady and oscillatory convection are analyzed from the perspective of a universally consistent understanding of the effects of different boundary conditions. In a horizontal slot with inviscid fluid addressed by Welander [P. Welander, Tellus Ser. A 41 (1989) 66], oscillatory convection sets in with the most unstable wave number and oscillation frequency being zero. Exact expressions for the critical fixed-value background gradient and the respective group velocity at zero wave number are derived from the long-wavelength expansion both for the horizontal slot with independently varying background gradients and for the inclined slot with the compensating gradients. In the horizontal slot with viscous fluid, the dissipation of along-slot perturbation-cell motion reduces efficiency of the oscillatory instability feedback and thus prevents the most unstable wavelength from being infinite. Based on this interpretation, the oscillatory instability of a three-dimensional (313) nature is predicted for an interval of long two-dimensional (2D) wavelengths in an inclined slot, and such 3D instability is indeed shown to arise. Related general conditions for three-dimensionality of most unstable disturbances are also formulated. As the slot orientation changes from the horizontal by angle 0 ( >= pi/2), the oscillatory 2D marginal-stability boundaries in inviscid and viscous fluid are expected to eventually transform into respective steady ones. Oscillatory instability in the vertical slot with viscous fluid, first reported by Tsitverblit [N. Tsitverblit, Phys. Rev. E 62 (2000) R7591], is of a quasi-steady nature. Its (new) mechanism is identified. It is underlain by differential gradient diffusion. As the horizontal slot at theta = pi, addressed by Tsitverblit [N. Tsitverblit, Phys. Fluids 9 (1997) 2458], changes its orientation to vertical, the wave number interval of linear steady instability shrinks to the vicinity of the most unstable zero wave number and vanishes. Consistently with the basic nature of finite-amplitude steady convection being the same in the horizontal and vertical slots, the respective convective flows are continuously transformed into each other. The dissimilarity between the nature of finite-amplitude steady convective flows in the horizontal slot with theta = 0, revealed by Tsitverblit [N. Tsitverblit, Phys. Lett. A 329 (2004) 445], and that in the vertical slot is shown to eventually give rise to a region of hysteresis in theta is an element of (0,pi/2). (c) 2006 Elsevier Inc. All rights reserved.