@ARTICLE{Riahi2022,
author={Riahi, M.K. and Ali, M. and Addad, Y. and Abu-Nada, E.},
title={Combined Newton-Raphson and Streamlines-Upwind Petrov-Galerkin iterations for nanoparticles transport in buoyancy-driven flow},
journal={Journal of Engineering Mathematics},
year={2022},
volume={132},
number={1},
doi={10.1007/s10665-021-10205-4},
art_number={22},
url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85123031898&doi=10.1007%2fs10665-021-10205-4&partnerID=40&md5=47681d80babdff9ff79a672932990c0f},
affiliation={Department of Applied Mathematics, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates; Emirates Nuclear Technology Center, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates; Department of Nuclear Engineering, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates; Department of Mechanical Engineering, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates},
abstract={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 NavierÐStokes 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 NewtonÕs 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 the Stream-Upwind PetrovÐGalerkin regularization approach is required to solve properly the transport equation in BuongiornoÕs model, in the Finite Element framework. Indeed, we formulate this ill-posed equation as a variational problem under mean value constraint. Numerical analysis and 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. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.},
author_keywords={Advection-dominated equation;  Finite element method;  Nanofluid;  Nanofluid heat transfer;  NavierÐStokes equations;  NewtonÐRaphson method;  Stream-Upwind PetrovÐGalerkin},
document_type={Article}
}

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We study the Buongiorno model, which couples the NavierÐStokes 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 NewtonÕs 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 the Stream-Upwind PetrovÐGalerkin regularization approach is required to solve properly the transport equation in BuongiornoÕs model, in the Finite Element framework. Indeed, we formulate this ill-posed equation as a variational problem under mean value constraint. 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