? Simple CFD problem

Davy

Recently, I met a simple flow problem which make us confuse. The problem is as following:

______________

I I

I a I

I_____________I

I I

I b I

I_____________I

In the rectangle vessel, Work fluid (a) is slightly heavier than work fluid (b) ( about one percent and neglect the surface tension ). According to our experience, the bouyancy flow will take place. However, I found there is at all no motion when the stream function-vorticity method to calculate this problem. If we used bossinesq approximation to deal with bouyancy force. If you check the vorticity equation, you will find that the item of the bouyancy force doesn't effect, thus it causes this result. I'd like to know how to accurately define the bouyancy force in this case. Thanks for your help in advance,

Davy