Studying the effects of a longitudinal magnetic field and discrete isoflux heat source size on natural convection inside a tilted sinusoidal corrugated enclosure

In the present work, the effects of the longitudinal magnetic field and the heat source size on natural convection heat transfer through a tilted sinusoidal corrugated enclosure for different values of enclosure inclination angles are analyzed and solved numerically by using the finite volume technique based on body fitted control volumes with a collected variable arrangement. A constant heat flux source is discretely embedded at the central part of the bottom wall whereas the remaining parts of the bottom wall and the upper wall are assumed adiabatic, and two vertical sinusoidal corrugated walls are maintained at a constant low temperature. The range of the variable parameters considered in the present analysis is as follows: the enclosure inclination angle is varied from 0° to 135°, the ratio of the size of the heating element to enclosure width varied from 20 to 80% of enclosure reference length, Hartmann number is varied from 0 to 100, and Rayleigh number varied from 103 to 106. Liquid gallium with constant Prandtl number (0.02) is used as a working fluid with constant properties except the density. The obtained results indicated that streamlines are affected strongly by the magnetic field especially for small values of inclination angle (Φ=0°Φ=0°) and Rayleigh number (Ra=103–106). The magnetic field effect decreases with an increase in the enclosure inclination angle (Φ>0°Φ>0°) especially for large values of Rayleigh number. The increase in Hartmann number will cause the temperature lines to become symmetrical in shape for large values of Rayleigh number (Ra=105–106). The results also explain that the temperature lines are very little affected by the inclination angle especially for small values of (ε=0.4ε=0.4) and (Ra=104), but this effect will increase especially for (ε=0.8ε=0.8) and (Ra=106). The Nusselt number increases first with an increase in inclination angle (0°≤Φ≤45°)(0°≤Φ≤45°), then is slightly affected for (45°<Φ≤90°)(45°<Φ≤90°), and finally decreases for (90°<Φ≤135°)(90°<Φ≤135°). An empirical correlation is developed by using Nusselt number versus Hartmann and Rayleigh numbers, and enclosure inclination angle. The increase in Hartmann number and the ratio of heating element to enclosure width will decrease the Nusselt number. Furthermore, four mathematical correlations are extracted from the results and presented, which can be used to accurately predict the average Nusselt number in terms of enclosure inclination angle, Hartmann, and Rayleigh numbers.