Nanofluid based heat transfer approaches have a tremendous prospect to develop novel cost-effective cooling technologies. In response to this potential development, a problem of unsteady copper oxide-water nanofluid flow and natural convective heat transfer within a quadrilateral vessel with uniform heating of bottom wall using modified Buongiorno model are investigated. The sloping wall of the vessel is maintained at constant low temperature and the uniform thermal condition on the bottom heated wall is considered, whereas the upper horizontal wall is regarded as adiabatic. The governing equations along with boundary conditions are solved using the Galerkin finite element method. Partial differential equation solver COMSOL Multiphysics with Matlab interface is used in the simulation. The results of the present problem of a certain situation as a special case have been verified by the previously published standard numerical investigations. The flow, thermal and concentration fields, local and average Nusselt number for various pertinent parameters entered into the problem have been analyzed. The time evolutions for a steady-state solution are also examined. The results show that the adjustment factor with the optimal nanoparticle volume fraction and the thermal Rayleigh number controls the optimal heat transfer. The trapezoidal vessel having higher sloping angles with the vertical axis exhibits higher heat transfer. Heat transfer decreases rapidly in 1–10 nm size nanoparticles for a nanofluid solution.