geometry and coordinates system. 

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... schematic drawing of the geometry and the Cartesian coordinate system employed in solving the problem is shown in Fig. ...

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... before the same behave will be shown in Fig. 21 for the variation of the average Nusselt number with Mn for different values of the radius ratio Rr, that is increasing Mn cause to decrease ...

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... 14-16 show the variation of the average Nusselt number with Mn for different values of Ra * , it is clear that for low values of Ra * , the effect of Mn is very slight but for Ra * = 5,000 and 10 4 , Nu decrease with Mn then nearly be constant when Mn = 60 and on and increasing the length of fins cause to decrease Nu as shown in Figs. 17 and 18. At high Ra * Nusselt number was high owing to the effect of convection mode of heat transfer but at low Ra * the mode of heat transfer is conduction and its value increase with the increase of Ra * . The variation of the average Nusselt number with Ra * for different values of the radius ratio Rr is shown in Fig. 20, which indicate that decreasing Rr (i.e., increasing the outer radius cause to decrease Nu for low values of Ra * but increasing Ra * show the inverse ...

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... horizontal vertical Mn=60, horizontal vertical Mn=100,horizontal vertical Fig. 19 The variation of average Nu with magnetohydrodynamic for Ra = 5,000 and radius ratio = 0.435. ...

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... dimensionless temperature distribution within the enclosure is presented in a contour map form. One section was selected in the (Z-R) plane along the length of the annulus, and the other in the (R-) plane in a manner allowed studying the temperature distribution within each plane. Fig. 3 shows the symmetry and uniformity of the temperature distribution for horizontal annulus when there is magnetohydrodynamic effect (Mn = 40) and increasing Mn cause the values of streamlines to drop and the temperature to increase as shown in Fig. 4. Increase Ra * and/or decrease the radius ratio results in a thicker cold layer near the bottom wall and a high temperature field near the top wall. More heat is transported upward, and a large difference of temperature is observed between the upper and lower parts of the annulus as shown in Fig. 5. It was observed that as Rr increase, isotherms shift towards the outer (cold) cylinder and the streamlines values dropped while the waviness will be clear due to the existence of the fins for Mn = 0 as shown in Figs. 6-8. The same behavior was observed for vertical cylinder when Rr increases in Figs. 9 and 10 and since this research was achieved for a steady state laminar region, thus in the warm region at the top end of the vertical cylinder, the isotherms come to be as a concentric circles located at the center of the annulus and distributed between the hot and cold cylinders. From the foregoing it is clear that the streamlines value was dropped with the increasing of Mn at constant value of Ra and Rr, because the force which created from MHD acting in the opposite direction of the force that caused the motion of the fluid. The growths of the boundary layers on the vertical wall are also observed to be affected by variation of Rr. The decrease in Rr is seen to reduce the rate of boundary layer growth on the hot wall. This behavior is reversed on the cold wall. The net result is a shift of the core toward the top edge of the cooled wall. This shift is further strengthening if there is an increase in Ra * . is a clear decrease in the velocity, While increasing Ra * from 10 2 to 10 4 increase the velocity slightly when there is the effect of Mn as shown in Fig. 12. Decreasing Rr (increasing the outer radius) cause the fluid to cooled before reaching the outer cylinder and its velocity will be reduced as shown in Fig. 13. Also, for horizontal annulus, the velocity in the upper region is much higher as compared to that in the lower ...

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... dimensionless temperature distribution within the enclosure is presented in a contour map form. One section was selected in the (Z-R) plane along the length of the annulus, and the other in the (R-) plane in a manner allowed studying the temperature distribution within each plane. Fig. 3 shows the symmetry and uniformity of the temperature distribution for horizontal annulus when there is magnetohydrodynamic effect (Mn = 40) and increasing Mn cause the values of streamlines to drop and the temperature to increase as shown in Fig. 4. Increase Ra * and/or decrease the radius ratio results in a thicker cold layer near the bottom wall and a high temperature field near the top wall. More heat is transported upward, and a large difference of temperature is observed between the upper and lower parts of the annulus as shown in Fig. 5. It was observed that as Rr increase, isotherms shift towards the outer (cold) cylinder and the streamlines values dropped while the waviness will be clear due to the existence of the fins for Mn = 0 as shown in Figs. 6-8. The same behavior was observed for vertical cylinder when Rr increases in Figs. 9 and 10 and since this research was achieved for a steady state laminar region, thus in the warm region at the top end of the vertical cylinder, the isotherms come to be as a concentric circles located at the center of the annulus and distributed between the hot and cold cylinders. From the foregoing it is clear that the streamlines value was dropped with the increasing of Mn at constant value of Ra and Rr, because the force which created from MHD acting in the opposite direction of the force that caused the motion of the fluid. The growths of the boundary layers on the vertical wall are also observed to be affected by variation of Rr. The decrease in Rr is seen to reduce the rate of boundary layer growth on the hot wall. This behavior is reversed on the cold wall. The net result is a shift of the core toward the top edge of the cooled wall. This shift is further strengthening if there is an increase in Ra * . is a clear decrease in the velocity, While increasing Ra * from 10 2 to 10 4 increase the velocity slightly when there is the effect of Mn as shown in Fig. 12. Decreasing Rr (increasing the outer radius) cause the fluid to cooled before reaching the outer cylinder and its velocity will be reduced as shown in Fig. 13. Also, for horizontal annulus, the velocity in the upper region is much higher as compared to that in the lower ...