  The present study deals with the finite element discretization of nanofluid
convective transport in an enclosure with variable properties. We study the
Buongiorno model, which couples the Navier-Stokes equations for the base fluid,
an advective-diffusion equation for the heat transfer, and an advection
dominated nanoparticle fraction concentration subject to thermophoresis and
Brownian motion forces. We develop an iterative numerical scheme that combines
Newton's method (dedicated to the resolution of the momentum and energy
equations) with the transport equation that governs the nanoparticles
concentration in the enclosure. We show that Stream Upwind Petrov-Galerkin
regularization approach is required to solve properly the ill-posed Buongiorno
transport model being tackled as a variational problem under mean value
constraint. Non-trivial numerical computations are reported to show the
effectiveness of our proposed numerical approach in its ability to provide
reasonably good agreement with the experimental results available in the
literature. The numerical experiments demonstrate that by accounting for only
the thermophoresis and Brownian motion forces in the concentration transport
equation, the model is not able to reproduce the heat transfer impairment due
to the presence of suspended nanoparticles in the base fluid. It reveals,
however, the significant role that these two terms play in the vicinity of the
hot and cold walls.
