In this example, we want to see which are the effects of an horizontal gradient of temperature.Theory
In our case, the density is dependent on the temperature, so the isochore lines are vertical. Moreover, the isobar lines are horizontal, due to the hydrostatic configuration. Thus, the theory explains us that this configuration is not stable. Indeed, as shown on the draw, the gravity center (G) of a fluid element is not the same that the center (P) due to the pressure forces, so there is apparition of a momentum which creates a local rotation.Simulation
The configuration of the simulation is the following :
 Initial conditions. the fluid velocity is equal to zero everywhere,
. the temperature is given by T(x) = 301.51.5 cos(50 Pi x), that is to say a difference of 3° C between the left and right sides and a temperature equal for all points placed on a vertical line.
 Boundary conditionsthe temperature on the both walls is equal to the initial temperature
The results obtained are the following :


We notice that the fluid is rolling in the whole domain which is given
to it. This allows a heat transfer by convection. This one is stopped in
the first top half of the domain when there is a steadystate flow.