Instabilities of longitudinal rolls in the presence of Poiseuille flow

Abstract

Longitudinal rolls represent the preferred form of convection in a horizontal fluid layer heated from below in the presence of parallel shear flows for sufficiently low Reynolds numbers and for a finite range of the Rayleigh number above the critical value Rac. In this paper properties of the longitudinal rolls and their stability with respect to three-dimensional disturbances are investigated in the case of Poiseuille flow. While the convective heat transport is independent of the Reynolds number, the mass flux through the channel at a given Reynolds number decreases with increasing Rayleigh number. A wavy instability is found to set in at a finite Reynolds number and relatively low Rayleigh numbers, depending on the Prandtl number P. In particular, the stability region for longitudinal rolls is analysed for P = 0.025, 0.1, 0.71, 2.5, and 7. For sufficiently small Reynolds number the oscillatory, the skewed varicose or the knot instability can precede the wavy instability. For P = 7 the wavy instability is preceded by a modified knot instability throughout the Reynolds-number range that has been investigated. In spite of the difference hi symmetry, the results for Poiseuille flow resemble those obtained earlier hi the case of plane Couette flow.