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 RLC is somewhat less than that for transverse stationary rolls, RSC, and for transverse travelling waves, RTC, longitudinal rolls occur first and then remain stable as R is increased beyond RSC or RTC; no mixed mode state occurs. In contrast, if RSc or RTc is slightly below RLC, pure transverse modes exist for only a relatively small range of R beyond RSC or RTC. Thereafter, a three-dimensional mixed mode state occurs well before RLC is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches RLC the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than RLC. 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.