Inflation on common Face using cutcell mesh
 

I am trying to simulate a multifluid natural convection problem. Simple way to understand the problem is its 2 fluids put in three compartments as shown in the attached picture. Now I need to have very fine all around the walls including the common walls between two seperating fluids i.e two way inflation on common face of 2 bodies. I couldn't generate this inflation (particularly on common face) using cut cell mesh. Do you have any idea, why inflation is not generated on common face?
My geometry in very complicated (not as shown in picture) so can't use any other method to generate hex dominant mesh so prefering cut cell.

Thanks.