Stability analysis with heat transfer. Set up the equations for steady state for the Bnard problem. Now

Stability analysis with heat transfer. Set up the equations for steady state for the Bénard problem.

Now perturb the temperature and use an energy equation to derive an equation for the temperature perturbation. This in turn needs an equation for the velocity profile. Derive this equation.

Thus we have two perturbation variables to handle here. How would you proceed further?