Original Research Paper Numerical investigation of natural convection of Al 2 O 3 -water nanoﬂuid in a wavy cavity with conductive inner block using Buongiorno’s two-phase model I. Hashim a , A.I. Alsabery a,b,⇑ , M.A. Sheremet c , A.J. Chamkha d,e a School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia b Refrigeration & Air-conditioning Technical Engineering Department, College of Technical Engineering, The Islamic University, Najaf, Iraq c Department of Theoretical Mechanics, Tomsk State University, 634050 Tomsk, Russia d Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia e RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021, Ras Al Khaimah, United Arab Emirates article info Article history: Received 18 August 2018 Received in revised form 19 October 2018 Accepted 19 November 2018 Available online 28 November 2018 Keywords: Natural convection Thermophoresis and Brownian Wavy cavity Conductive inner block Buongiorno model abstract By employing the ﬁnite element method, thermophoresis and Brownian diffusion are studied numeri- cally relating to the natural convection in a wavy cavity that is ﬁlled with an Al 2 O 3 -water nanoﬂuid pos- sessing a central heat-conducting solid block that is inﬂuenced by the local heater located on the bottom wall. An isothermal condition is established in the two wavy vertical walls, while adiabatic condition is for the top horizontal wall. Partial heating is applied to the bottom of the horizontal wall, while the remaining part remains in the adiabatic condition. Empirical correlations are employed for the thermal conductivity and dynamic viscosity of the nanoﬂuid. The number of oscillations (1 6 N  4), Rayleigh number (10 3 6 Ra  10 6 ), nanoparticles volume fraction (0 6 /  0:04) and dimensionless length of the bottom heater (0:2 6 H 6 0:8) govern the parameters in this study. The grid independency test, as well as experimental and numerical data from other published works, was employed to validate the developed computational code comprehensively. Based on the obtained results, it was found that the heat transfer inside the cavity is enhanced by introducing nanoparticles as well as a selection of optimal number of oscillations. Ó 2018 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. 1. Introduction In various engineering applications, including heat transfer improvement in heat exchangers and solar collectors, cooling of electronic devices, mass and heat transfers occurring in chemical reactors, heat sink, and many others, convective heat transfer is considered a key phenomenon in wavy channels and enclosures [1–4]. While an effective approach to enhance heat transfer would be to alter the cavity walls to grow the heat transfer surface, these days, for the same purpose, the heat transfer medium can be chan- ged by employing nanoparticles of metal oxides or metals. It has been shown that the thermal conductivity could be increased and also improves heat transfer by introducing low concentrations of metal nanoparticles [5–9]. There have been numerous experi- mental and theoretical studies focusing on the considered topic [10–25]. The problem of natural convection in a cavity ﬁlled with nanoﬂuid and having double wavy walls with different phase devi- ations was numerically evaluated by Tang et al. [10]. They explored that the increasing of the nanoﬂuid volume fraction led to a signif- icant enhancement on the surface heat transfer coefﬁcient and the temperature distribution inside the considered cavity trued to be more homogenous with the large volume fraction. By employing the single-phase nanoﬂuid model with Maxwell and Brinkman cor- relations, Mustafa et al. [11] investigated how nanoparticles (silver and alumina) interacted with an external applied magnetic ﬁeld with the presence of internal heat generation along the vertical rough surface. They described that the local Nusselt number and the local skin friction coefﬁcient have been shown to be a decreas- ing function for the wavy surface’s amplitude. The improvement of laminar forced convection cooling was experimentally evaluated by Khoshvaght-Aliabadi et al. [12] in a wavy heat sink by employ- ing two passive techniques, which involved Al 2 O 3 -water nanoﬂuid and rectangular ribs. Using ribs in the wavy heat sinks was found to be signiﬁcantly helpful to improve the heat transfer because of https://doi.org/10.1016/j.apt.2018.11.017 0921-8831/Ó 2018 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. ⇑ Corresponding author at: School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia. E-mail address: alsabery_a@ukm.edu.my (A.I. Alsabery). Advanced Powder Technology 30 (2019) 399–414 Contents lists available at ScienceDirect Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt