Governing equations: Under appropriate nondimensionalization, the governing equations are \[ \nabla \cdot \mathbf{u} = 0 \] \[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \text{Pr} \nabla^2 \mathbf{u} + \text{Ra} \, \text{Pr} \, \theta \, \mathbf{e}_y \] \[ \frac{\partial \theta}{\partial t} + (\mathbf{u} \cdot \nabla)\theta = \nabla^2 \theta \] where \(Pr\) is the Prandtl number and \(Ra\) the Rayleigh number

Assume that the flow is two-dimensional, i.e. \[ \mathbf{u} = (v_x, v_y)^T \]

Fixed point: the conducting state

• The fixed point is solution to \[ \frac{d^2 \Theta}{dy^2} = 0 \] with appropriate boundary conditions
• The nondimensional temperature profile \(\Theta(y)\) is given by \[ \Theta(y) = 1 - y \]
• It corresponds to a pure conduction state (i.e. \(\mathbf{u} = 0\))