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In this study, the conjugate natural convection in a square cavity filled with $$\hbox {Al}_2\hbox {O}_3$$ Al 2 O 3 –water nanofluid with an inner conducting solid block is studied numerically using nonhomogeneous Buongiorno’s two-phase model. The left wall of the cavity is partially heated and the remaining parts of the wall are adiabatic, while the right wall is fully cooled. The top...