Numerical Study of Natural Convection for Generalized Second-Grade Fluids Confined in a Square Cavity Subjected to Horizontal Heat Flux Hamza Daghab 1 , Mourad Kaddiri 1* , Said Raghay 2 , Mohamed Lamsaadi 3 , Hassan El Harfi 3 1 Industrial Engineering Laboratory, Sultan Moulay Slimane University, B.P. 523, Béni-Mellal 23000, Morocco 2 Laboratory of Applied Mathematics and Computing, Cadi Ayyad University, B.P. 549, Marrakech 40000, Morocco 3 Polydisciplinary Faculty, Sultan Moulay Slimane University, B.P. 592, Béni-Mellal 23000, Morocco Corresponding Author Email: mouradkadiri@usms.ma https://doi.org/10.18280/ijht.390203 ABSTRACT Received: 24 August 2020 Accepted: 17 December 2020 Two-dimensional steady laminar natural convection of a viscoelastic fluid represented by generalized second-grade fluid model in a square enclosure is studied. The cavity is submitted at its vertical sides to a uniform density of heat flux while the horizontal walls are insulated, without slipping conditions at all the solid boundaries. The governing conservation and constitutive equations with the corresponding boundary conditions are solved by finite volume method in a collocated grid system. The contributions of shear rate dependent and elastic characteristics of the viscoelastic fluid are investigated on momentum and heat transport. The effects of elastic number (E) in the range 0 - 1 on heat transfer and fluid motion are interpreted for a power-law index (n) in the range 1.4 - 0.6 and nominal values of Rayleigh number (Ra) range of 10 3 to 10 5 . Keywords: finite volume, generalized second-grade model, natural convection, numerical study, square cavity, viscoelastic fluids 1. INTRODUCTION The study of viscoelastic flows is very important due to their wide range of applications in many areas such as geophysics, biomechanical engineering, chemical and petroleum industries. The viscoelastic fluids include both viscous and elastic effects, making their mathematical modeling complicated than the Newtonian fluids one. To describe the behavior of viscoelastic fluids different models have been used. Amongst these models, Rivlin-Ericksen fluids or differential type fluids proposed by Rivlin and Ericksen [1], have received special attention. As a particular case of Rivlin-Ericksen fluids we find second grade fluids, relevant issues concerning these fluids have been discussed in detail by Dunn and Fosdick [2], Fosdick and Rajagopal [3], and Fosdick and Rajagopal [4], respectively. It is clear that second-grade fluids show only the normal stress effects. However, many realistic fluids exhibit combined effects of normal stress and shear-thinning/shear-thickening behavior. In such cases, generalized second-grade fluids, proposed by Man and Sun [5], are suitable. Because of this motivation, the model considered in the present study is a generalized second-grade fluid. Thermal transfer of viscoelastic fluids has attained tremendous number of studies because of its presence in many areas. The previous investigations have analytically and numerically analyzed this phenomenon in order to interpret and show the effects of elasticity on the flow and heat transfer characteristics related to the viscoelastic flow in different geometries. In the precedent studies, it has been found that the elasticity affects heat transfer through the change of Nusselt number. Shenoy and Mashelkar [6] have analyzed natural convection of viscoelastic fluid by using the approximate integral method. They have noted that the effect of elasticity on Nusselt number depends on the value of Weissenberg number. Also, a set of studies [7-10] have been done for viscoelastic second-grade fluids in different geometries. It has been noticed that the elasticity is considered as a resistance force, which trends to depreciate heat transfer within the flow and decelerate the fluid motion. Natural convection flows arise from density variations with temperature or concentration within a non-isothermal-fluid under the influence of gravity. This is frequently encountered in many industrial and practical engineering areas that include heat exchangers, nuclear reactors, geothermal systems, metallurgical processes, crystal growth and others. Buoyancy- driven convection for non-Newtonian fluids in enclosures has been broadly studied. Such one-dimensional flows often allow getting an analytical solution of the governing conservation and constitutive equations. In this context, Lamsaadi et al. [11] have analytically and numerically studied natural convection of power law fluids in a shallow cavity subjected to a uniform density of heat flux at its horizontal walls. The parallel flow approximation has been used to simplify the governing non- linear differential equations. They have demonstrated that the flow characteristics are sensitive to the power law index but not to the Prandtl number, when this later is sufficiently large. Another model of non-Newtonian fluids has been used in the study of Allaoui and Vasseur [12], which is the Carreau- Yasuda non-Newtonian fluid. They have deduced a semi- analytical solution for natural convection of a Carreau-Yasuda fluid in a vertical enclosure heated from the side walls, founding pseudo-rheological behavior generates a significant modification in heat transfer characteristics. Whereas 2-D flows require a numerical study to solve the non-linear governing equations. Natural convection of power law fluids inside square enclosures has been numerically studied by Turan et al. [13-17] for various boundary conditions. It has been noted that Nusselt number and the flow intensity are increasing functions with rising Rayleigh number (Ra) and decreasing power law index (n), and heat transfer takes place International Journal of Heat and Technology Vol. 39, No. 2, April, 2021, pp. 345-354 Journal homepage: http://iieta.org/journals/ijht 345