Linear stability of Rayleigh-Bénard-Poiseuille flow of water near 4°C in a channel bounded by slip walls

The onset of mixed convection of cold water in a horizontal channel is studied using linear stability analysis . The two walls of the channel are modeled using slip conditions and are maintained at different constant temperatures. A model with a parabolic relationship is used to predict the variation of density with temperature in the range around its maximum density. The finite difference method is used to solve numerically the linearized equations, and the critical eigenvalue is obtained using the iterative Newton-Raphson method. The effects of the dimensionless temperature ratio γ and the slip parameter Ω on the onset of thermal instabilities are studied. The obtained results showed that these two parameters induced modifications in the critical Rayleigh number and the corresponding critical wavenumber. The results also indicated that beyond a certain value of dimensionless temperature ratio the slip parameter of the upper wall no longer influences the critical threshold for the onset of mixed convection .