Mixed mode convection in an inclined slot

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 0.263 897, For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls R^L_C is somewhat less than that for transverse stationary rolls, R^S_C, and for transverse travelling waves, R^T_C, longitudinal rolls occur first and then remain stable as R is increased beyond R^S_C or R^T_C; no mixed mode state occurs. In contrast, if R^S_c or R^T_c is slightly below R^L_C, pure transverse modes exist for only a relatively small range of R beyond R^S_C or R^T_C. Thereafter, a three-dimensional mixed mode state occurs well before R^L_C is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches R^L_C the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than R^L_C. 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 as the stable supercritical state.