International Journal of Thermal Sciences 147 (2020) 106116 1290-0729/© 2019 Published by Elsevier Masson SAS. Effect of backward facing step shape on 3D mixed convection of Bingham fluid Fetta Danane a, b, * , Ahlem Boudiaf a , Omar Mahfoud c , Seif-Eddine Ouyahia a, d , Nabila Labsi a , Youb Khaled Benkahla a a Laboratory of Transport Phenomena, Faculty of Mechanical and Process Engineering, USTHB, B.P. 32, El-Alia Bab-Ezzouar, 16111, Algiers, Algeria b Centre de D�eveloppement des Energies Renouvelables, CDER, 16340, Algiers, Algeria c Faculty of Mechanical and Process Engineering, USTHB, B.P. 32, El-Alia Bab-Ezzouar, 16111, Algiers, Algeria d Sonatrach – Central Direction of Research and Development, Avenue of November 1st, Boumerdes 35000, Algeria A R T I C L E INFO Keywords: Backward facing step Bingham fluid Mixed convection Step shape ABSTRACT The present paper examines the viscous laminar mixed convection flow of a Bingham viscoplastic fluid in order to enhance the thermal performance and thus reduce the pressure drop. To achieve this, three-dimensional numerical simulations for vertical, inclined, and round backward facing step inside a horizontal rectangular duct are established. The effects of variation in buoyancy, viscous and inertia forces variations as well as the step shape, on thermal and hydrodynamic behaviours of the fluid are investigated. The structure of the flow is significantly influenced by the variation of Bingham and Richardson numbers and the backward facing steps shape. It is found that the recirculation zones, which are the main source of thermal and hydrodynamic losses, are intensified with the increase in buoyancy forces represented by the Richardson number. Moreover, the recirculation zones are attenuated as the viscous forces predominate. It is also noticed that the recirculation zones decrease or disappear as the geometry is modified by reducing the angle or the shape of the step. The change of the step shape reduces significantly the pressure drops of about 11.77% and 58.83% for the inclined and rounded backward facing steps; respectively, compared to the vertical backward facing step. However, the pressure coefficient decreases along the duct and increases in presence of a recirculation zone. Also, the highest value of the total Nusselt number is obtained for a round backward facing step in mixed convection. Otherwise, an increase in the Bingham number extends the plug structure in the three geometries. 1. Introduction Backward-Facing Step (BFS) is one of representative models, estab- lished as a benchmark configuration for separated flow studies in fluid mechanics. It constitute a solution to various real industrial problems. Although theoretical, could find applications in several industrial de- vices met in engineering daily life. Some of them are industries engine processes, separation flow behind vehicles, inlet tunnel flow of engine, inside a condenser/combustor exchangers or electronic chips. However, the viscoplastic non-Newtonian fluid flow through backward facing step (BFS), as considered in our study, is met especially in extrusions (e.g. extrusion of pasta, hot rolling, and extrusion of plastic) and in food transformation process industries. Thus, many numerical, experimental and analytical studies, focused on these phenomena, and could be find in Refs. [1–4]. Many researches have been carried out to study the pressure drop problems issues in order to ensure an efficient heat transfer of flowing fluids inside ducts with singularities, presented by a sudden enlargement [5–9]. The decrease in pressure drop is directly related to the reduction of the abrupt change of the duct cross-section while the fluids flow, facilitating flow separation phenomenon. The latter, which is of great importance, has been further studied in several significant ways. How- ever, pressure drop phenomena is still an active area of many researches. Recently, Chen et al. [10] provided an overview of the working principals of Backward-Facing Step and a review for the advancement of such technologies from the most recent publications. This paper, sum- marize most of solutions to various real industrial problems. The authors * Corresponding author. Laboratory of Transport Phenomena, Faculty of Mechanical and Process Engineering, USTHB, B.P. 32, El-Alia Bab-Ezzouar, 16111, Algiers, Algeria. Tel.: þ213 554 155 040. E-mail address: f.danane@cder.dz (F. Danane). Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts https://doi.org/10.1016/j.ijthermalsci.2019.106116 Received 26 May 2019; Received in revised form 5 September 2019; Accepted 23 September 2019