K. Fujimura and R. E. Kelly : Mixed mode convection in an inclined slot, J. Fluid Mech. 246 (1993) 545-568.
Abstract:
Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263897. For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls R_c^L is somewhat less than that for transverse stationary rolls, R_c^S, and for transverse travelling waves, R_c^T, longitudinal rolls occur first and then remain stable as R is increased beyond R_c^S or R_c^T; no mixed mode state occurs. In contrast, if R_c^S or R_c^T is slightly below R_c^L, pure transverse modes exist for only a relatively small rang of R beyond R_c^S or R_c^T. Thereafter, a three-dimensional mixed mode state occurs well before R_c^L is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches R_c^L, the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than R_c^L. Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge a stable supercritical state.
Related papers