How could the continuity equation be enabled if the fluid density variation in momentum equation permit the buoyancy induced flows?

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Filippo Maria Denaro

I am not sure about your question... buoyancy-driven flow is general... for example, for small temperature variation you can insert the Bousinnesq assumption (a Taylor series around the base state) in the incompressible flow model and the continuity equation reduces to the divergence-free constraint on the velocity field. Otherwise, a full compressible flow model is required and the continuity is the classical transport equation for density.

Mladen Bošnjaković

Also I am not sure about your question ...

Continuity equation have to be fullfiled in default. Also Navier-stokes equitation.

Maybe following article can help.

http://m.ijsimm.com/Full_Papers/Fulltext2015/text14-2_238-249.pdf