4.4 Results and Discussion
4.4.2 Effect of Hartmann number
local Nusselt number is decreased from the increasing of Rayleigh numbers are shown in Figure 4.8 (for inclined and vertical position). For lower Rayleigh numbers shape of the curves are almost linear but for higher Rayleigh numbers shape of the curves are nonlinear for all positions.
The local Nusselt number along the horizontal line Y=0.07 of the enclosure with the Rayleigh number and for Ha = 100 are shown in Figure 4.14(For horizontal, inclined and vertical position). A similar pattern is observed but absolute value of local Nusselt numbers are decreased compared with Figure 4.8.
Streamlines (on the left column) and isotherm (on the right column) for different position and Ra = 106 ; Pr = 0.71 are presented in Figures 4.21, 4.22 and 4.23 (for Ha = 0, 50, 100 and 500) to understand the effects of Hartmann number on the flow and temperature fields.
In figure 4.21 (for horizontal position), it is seen in the left column that two vortices are formed on the both side of the strip. The size of upper vortex gradually became larger but the size of lower vortex changes irregularly. It is observed that with the increasing of Hartmann number the flow strength decreases. On the right column, isotherm lines gradually increase on the both side of the strip and condensed to the right side of the strip with the increasing of Hartmann number. In figure 4.22 (for inclined position), It is seen that two vortices are formed also on the both side of the strip. The size of upper vortex gradually increases and flow strength decreases with the increasing of Hartmann number.
Isotherm lines gradually increase on the both side of the strip and concentrated to the right side of the strip with the increasing of Hartmann number. But in the figure 4.23 (for vertical position), there is only one vortex formed on the right side of the strip and the size of vortex became slightly smaller with the increasing of Hartmann number. It is observed here that the flow strength and the value of temperature decrease with the increasing of Hartmann numbers. In the left column, it is seen that isotherm lines are uniformly scattered on the upper side of the cavity. It is also seen that isotherm lines are bending more on the lower left side of the strip in the absence of magnetic field (Ha = 0). The density of isotherms on the right side and below the strip decreases with the increasing of Hartmann numbers.
Velocity Profiles
Variations of the velocity component of different positions along the horizontal line Y=0.07 of the cavity with the Hartmann numbers (Ha = 0, 50, 100 and 500) and for Ra = 104 ; Pr = 0.71 are shown in Figure 4.18. From the Figure 4.18 (horizontal position) it can be observed that the absolute value of maximum and minimum velocity decreases with increasing of Hartmann number. A similar velocity profiles is found with different position of the strip but the absolute value of maximum and minimum velocity increases are shown in Figure 4.18 (for inclined and vertical position). The figure indicates that the velocity profiles become steeper with increasing of Hartmann number. It is clearly seen that magnetic field retards the velocity and trend of velocity profiles changes with Hartmann number.
Variations of the velocity component of different positions along the horizontal line Y=0.07 of the enclosure with the Hartmann numbers (Ha = 0, 50, 100 and 500) and for Ra
= 106; Pr = 0.71 are shown in Figure 4.24. A similar pattern of velocity profiles is seen here to compare with the figures shown in Figure 4.18.
Temperature Profiles
Temperature profiles of different positions along the horizontal line Y=0.07 of the enclosure with the Hartmann numbers (Ha = 0, 50, 100 and 500) and for Ra = 104; Pr = 0.71 are shown in figure 4.19. It is observed that for all values of Hartmann numbers the temperature lines are same which is represented by the line for Ha = 500. It is not found any variation in temperature profiles for different values of Hartmann number.
Temperature profiles of different positions along the horizontal line Y=0.07 of the cavity with the Hartmann numbers (Ha = 0, 50, 100 and 500) and for Ra = 106; Pr = 0.71 are shown in figure 4.25. It is seen in figure 4.25 (horizontal position) that for higher value of Hartmann number the value of temperature is higher. A similar phenomenon is observed for inclination position but the reveres scenario is seen for vertical position. For increasing rotation of heated solid strip temperature value is changed significantly.
Local Nusselt number along the horizontal line
The local Nusselt number for different positions along the horizontal line Y=0.07 of the enclosure with the Hartmann numbers (Ha = 0, 50, 100 and 500) and for Ra = 104; Pr = 0.71 are shown in Figure 4.20. It is found insignificant variation in local Nusselt number for different values of Hartmann number.
The local Nusselt number of different positions along the horizontal line Y=0.07 of the enclosure with the Hartmann numbers (Ha = 0, 50, 100 and 500) and for Ra = 106; Pr = 0.71 are shown in figure 4.26. The values of Local Nusselt number decreases with increasing of Hartmann number up to a certain point, then the values of Local Nusselt numbers change (rate of heat transfer) reciprocally for both horizontal and inclined positions. On the other hand, for vertical position, the values of Local Nusselt number increases with increasing of Hartmann number up to a certain point, then the values of Local Nusselt numbers change reciprocally. For increasing of rotations of solid strip heat transfer rate is increased.
Streamlines Isotherms
Figure 4.3: Streamlines (left column) and isotherms (right column) for horizontal position; Ha = 0 and Pr = 0.71; Ra = 103, 104, 105 and 106
Ra = 103 Ra = 104 Ra = 105 Ra = 106
Streamlines Isotherms
Figure 4.4:Streamlines (left column) and isotherms (right column) for inclined position;
Ha = 0 and Pr = 0.71: Ra = 103, 104, 105 and 106
Ra =103 Ra =104Ra =105Ra =106
Streamlines Isotherms
Figure 4.5: Streamlines (left column) and isotherms (right column) for vertical position;
Ha = 0 and Pr = 0.71: Ra = 103, 104, 105 and 106
Ra =103 Ra =104Ra =105Ra = 106
Figure 4.6: Velocity profiles of different positions along the line y = 0.07, for Ha = 0 and Pr = 0.71
Vertical PositionInclined PositionHorizontal Position
Figure 4.7: Temperature profiles of different positions along the line y = 0.07, for Ha = 0 and Pr = 0.71 Horizontal PositionVertical PositionInclined Position
Figure 4.8: Local Nusselt number of different positions along the line y = 0.07, for Ha = 0 and Pr = 0.71
Vertical Position Inclined Position Horizontal Position
Streamlines Isotherms
Figure 4.9:Streamlines (left column) and isotherms (right column) for horizontal position;
Ha = 100 and Pr = 0.71: Ra = 103, 104, 105 and 106
Ra =103 Ra =104Ra =105 Ra = 106
Streamlines Isotherms
Figure 4.10:Streamlines (left column) and isotherms (right column) for inclined position;
Ha = 100 and Pr = 0.71: Ra = 103, 104, 105 and 106
Ra =103 Ra =104Ra =105Ra = 106
Streamlines Isotherms
Figure 4.11: Streamlines (left column) and isotherms (right column) for vertical position;
Ha = 100 and Pr = 0.71: Ra = 103, 104, 105 and 106
Ra =103 Ra =104Ra =105Ra = 106
Figure 4.12: Velocity profiles of different positions along the line y = 0.07 for Ha = 100 and Pr = 0.71
Horizontal Position Inclined Position Vertical Position
Figure 4.13: Temperature profiles of different positions along the line y = 0.07 for Ha =100 and Pr = 0.71
Vertical Position Horizontal Position Inclined Position
Figure 4.14: Local Nusselt number of different positions along the line y = 0.07 for Ha =100 and Pr = 0.71
Horizontal Position Inclined Position Vertical Position
Streamlines Isotherms
Figure 4.15: Streamlines (left column) and isotherms (right column) for horizontal position;
Ra= 104 and Pr = 0.71: Ha = 0; Ha = 50; Ha = 100 ; Ha = 500
Ha = 0Ha = 50Ha = 100Ha = 500
Streamlines Isotherms
Figure 4.16: Streamlines (left column) and isotherms (right column) for inclined position; Ra = 104 and Pr = 0.71: Ha = 0; Ha = 50; Ha = 100 ; Ha = 500
Ha = 0Ha = 50Ha = 100 Ha = 500
Streamlines Isotherms
Figure 4.17: Streamlines (left column) and isotherms (right column) for vertical position Ra = 104 and Pr = 0.71: Ha = 0; Ha = 50; Ha = 100 ; Ha = 500
Ha = 0Ha = 50Ha = 100Ha = 500
Figure 4.18: Velocity profiles of different positions along the line y = 0.07 for Ra = 104 and Pr = 0.71
Horizontal Position Inclined Position Vertical Position
Figure 4.19: Temperature profiles for different positions along the line y = 0.07 for Ra = 104 and Pr = 0.71
Horizontal Position Inclined Position Vertical Position
Figure 4.20: Local Nusselt number for different positions along the line y = 0.07 for Ra = 104 and Pr = 0.71 Vertical Position Inclined Position Horizontal Position
Streamlines Isotherms
Figure 4.21: Streamlines (left column) and isotherms (right column) for horizontal position; Ra = 106 and Pr = 0.71: Ha = 0; Ha = 50; Ha = 100 ; Ha = 500
Ha = 0Ha = 50Ha = 100Ha = 500
Streamlines Isotherms
Figure 4.22: Streamlines (left column) and isotherms (right column) for inclined position; Ra = 106 and Pr = 0.71: Ha = 0; Ha = 50; Ha = 100 ; Ha = 500
Ha = 0Ha = 50Ha = 100Ha = 500
Streamlines Isotherms
Figure 4.23: Streamlines (left column) and isotherms (right column) for vertical position, Ra = 106 and Pr = 0.71: Ha = 0; Ha = 50; Ha = 100 ; Ha = 500
Ha = 0Ha = 50Ha = 100Ha = 500
Figure 4.24: Velocity profiles different positions along the line y = 0.07 for Ra = 106 and Pr = 0.71
Horizontal Position Inclined Position Vertical Position
Horizontal Position
Figure 4.25: Temperature profiles of different positions along the line y = 0.07 for Ra = 106 and Pr = 0.71
Inclined Position Vertical Position
Figure 4.26: Local Nusselt number of different positions along the line y = 0.07 for Ra = 106 and Pr = 0.71
Horizontal Position Inclined Position Vertical Position