FEM solution to natural convection flow of a micropolar nanofluid in the presence of a magnetic field

Abstract

The two-dimensional, laminar, unsteady natural convection flow in a square enclosure filled with aluminum oxide (\(\hbox {Al}_{2} \hbox {O}_{3}\))–water nanofluid under the influence of a magnetic field, is considered numerically. The nanofluid is considered as Newtonian and incompressible, the nanoparticles and water are assumed to be in thermal equilibrium. The mathematical modelling results in a coupled nonlinear system of partial differential equations. The equations are solved using finite element method (FEM) in space, whereas, the implicit backward difference scheme is used in time direction. The results are obtained for Rayleigh (Ra), Hartmann (Ha) numbers, and nanoparticles volume fractions (\(\phi\)), in the ranges of \(10^3 \le Ra \le 10^7\), \(0\le Ha \le 500\) and \(0 \le \phi \le 0.2\), respectively. The streamlines and microrotation contours are observed to show similar behaviors with altering magnitudes. For low Ra values, when \(Ha=0\), symmetric vortices near the walls and a central vortex in opposite direction are observed in vorticity. As Ra increases, the central vortex splits into two due to the circulation in the effect of the buoyant flow. Boundary layer formation is observed when Ha increases for almost all Rayleigh numbers in both streamlines and vorticity. The isotherms have horizontal profiles for high Ra values owing to convective dominance over conduction. As Ha is increased, the convection effect is reduced, and isotherms tend to have vertical profiles. This study presents the first FEM application for solving highly nonlinear PDEs defining micropolar nanofluid flow especially for large values of Rayleigh and Hartmann numbers.