The results of the numerical study on the MHD pump subjected to the mentioned boundary conditions are presented in terms of normal current density, magnetic flux density, volumetric Lorentz force, shear stress and pump flow velocity by varying applied voltage and Hartmann number. For evaluating the MHD pump performance, Cu-water nanofluid with 0.1% volume fraction was considered. In the subsequent sub-sections, the performance of the MHD pump-based microchannel cooling system is presented considering various parameters, including the maximum temperature of a heat dissipating element, heat removal rate, efficiency, thermal field, flow field and Nusselt number. In addition, the heat transfer performance of Cu-water nanofluid is compared with TiO_{2}-water nanofluid and Al_{2}O_{3}-water nanofluid. The study provided an in-depth understanding of the MHD pump functioning and its application in micro-cooling systems.

#### 3.2. Magnetohydrodynamic Pump (MHD) Performance

Figure 5a shows the variation of normal current density with the applied voltage and Hartmann number. The normal current density increased with the increase in applied voltage. For example, as the applied voltage increased from 0.05 V to 0.35 V, at a Hartmann number value of 2.0, the normal current density increased 600%, or 6 times. For the same applied voltage, a higher normal current density is observed for the higher Hartmann number. As the Hartmann number increased from 1.41 to 3.76 at a constant applied voltage of 0.35 V, the normal current density increased 600%, or 6 times. The combined influence of a higher applied voltage and higher Hartmann number are visible with a significant increase in the normal current density.

Figure 5b shows the spatial variation of induced current and it can be seen that the induced current density is higher near electrode area.

Figure 5c shows the variation of the average velocity with respect to current density. The average velocity increased linearly with the increase in current density. The flow rate can be increased either by increasing applied current, keeping magnetic flux constant or by increasing magnetic flux while keeping the applied current constant to enhance the pump performance. Similar trends have been observed by previously conducted studies [

16]. For low Hartmann numbers, the velocity increased with an increase in the Hartmann number. However, the high Hartmann number can have a negative effect on the velocity as well as volumetric flow rate [

11]. For a low Hartmann number, forced convection dominates with higher velocities which is useful for enhancing the pump performance.

Figure 6a shows the variation of magnetic flux along the dimensionless width in the

Y-axis at the center of the magnetohydrodynamic pump. The maximum value of the magnetic flux attained is about 0.25 T at the center of the MHD pump channel. However, the value of the magnetic flux density near the conducting electrode is found to be in the order of 0.11 T. Similar results have been obtained by Aoki et al. [

44]. The magnetic flux showed axisymmetric behavior for the axis passing through the center of the dimensionless width.

Figure 6b shows the magnetic field distribution for the MHD pump on the

XY-plane. As in the present study, the cylindrical permanent magnet is considered for the MHD pump application and the circular magnetic field pattern is observed. The maximum magnetic field value of the order of 100 kA/m is observed. The magnetic field showed radial symmetric behavior for the axis passing through the center of the magnet.

Figure 7 shows the volumetric Lorentz force variation for applied voltage and Hartmann number. The volumetric Lorentz force increased with increase in applied voltage. For example, as the applied voltage increased from 0.05 V to 0.35 V at a Hartmann number value of 2.0, the volumetric Lorentz force increased 600%, or 6 times. For the same applied voltage, a higher volumetric Lorentz force is observed for a higher Hartmann number. As the Hartmann number increased from 1.41 to 3.76, at a constant applied voltage of 0.35 V, the volumetric Lorentz force increased 600%, or 6 times. In the study conducted by Moghaddam on MHD micropumps, the volumetric flow rate increased owing to an increase in the Hartmann number to a value of 40, then volumetric flow rate started to decrease [

11]. Similarly, the volumetric flow rate increased until a Hartmann number of 200, and then decreased in the study conducted on the MHD pump by Yousofvand et al. [

10]. The present study is focused on low Hartmann numbers (Ha < 4) where the volumetric flow rate and Lorentz force increases with the increase in Hartmann number, as the defined Hartman number compares the magnetic force with the viscous force. At low Hartmann numbers, the viscous forces dominate giving a higher volumetric flow rate. As a result, the lower Hartmann number is favorable for the enhancement of heat transfer. However, a higher Hartmann number can have an adverse effect on heat transfer [

10].

Figure 8a shows the shear stress variation along the non-dimensional width at the center of the magnetohydrodynamic pump. The shear stress values for all the Hartmann numbers are compared in the middle section of the channel. Regions of higher shear stress are observed near the wall for all the Hartmann numbers. The values of shear stress in the region near the walls of the channel increased as the Hartmann number increased. Shear stress is directly proportional to the rate of change of velocity. The increase in shear stress at the walls for a higher Hartmann number is observed due to the typical velocity profile of the MHD pump flow inside the channel, where the velocity profile becomes flatter at the center, and a large velocity change is seen near the walls. As the Hartmann number increased from 1.41 to 3.74, the shear stress value near the channel walls increased around 7 times, or 714%. The shear stress variation showed axisymmetric behavior for the axis passing through the center of the dimensionless width.

Figure 8b shows the pressure contours for the flow cross-sectional area at the center of the pump in the

YZ-plane, and it could be seen that higher pressure regions are observed near the wall owing to the Hartmann effect.

Figure 9 shows the variation of the velocity profile along the dimensionless width in the

Y-axis imposed by the Lorentz force at the center of the magnetohydrodynamic pump. The velocity profiles show maximum values near the walls and lower values in the center of the channel owing to the Lorentz force distribution [

44]. The velocity variation showed axisymmetric behavior for the axis passing through the center of a dimensionless width. The M-shape velocity profiles as observed in

Figure 9 are present in many MHD pumps. This can be attributed to the position of conducting electrodes on the two opposite walls to provide the DC power supply. Moreover, the different fluids have responded with a similar velocity profile indicating that it is a geometrically affected phenomenon with the position of the electrode [

45]. It could be seen from

Figure 9 that as the Hartmann number increased, the velocity increased. Moreover, as the value of the Hartmann number increased, the velocity profile became flatter. The plug-like shape remained constant for a large portion of the channel width [

46,

47]. The current flowing in the closed loop generated a non-uniform negative small electromagnetic Lorentz force which counteracted the conducting fluid flow in the magnetic field creating a flat velocity boundary layer [

45]. This phenomenon is called the Hartmann effect. For example, as the value of the Hartmann number increased from 1.41 to 3.74, the maximum velocity increased by 280% at the center of the magnetohydrodynamic pump. Moreover, as the value of the Hartmann number increased, its effect on velocity change was slightly reduced. This is evident from

Figure 9, as the change in maximum velocity for the Hartmann number variation from 3.46 to 3.74 is less as compared to the variation from 1.41 to 2.00.

Figure 10 shows the velocity field variation in the

X-axis along the width at the center of the magnetohydrodynamic pump. The velocity at the center of the channel is higher compared to the channel wall, owing to the high shear stress observed along the channel wall. The average velocity of 0.0034, 0.0061, 0.0085, 0.0106, 0.0126, 0.0145 and 0.0164 m/s are developed for the applied voltage of 0.05, 0.10, 0.15, 0.25, 0.30 and 0.35 V at Hartmann number value of 2.0, respectively.

The increase in average velocity with increase in the applied voltage is attributed to development of higher Lorentz force. It is obvious from Equation (4) that the Lorenz force will increase if the cross product of current density and magnetic field increases.

#### 3.3. MHD-Based Microchannel Cooling System

The magnetohydrodynamic pump has various advantages over traditional pumps including low cost, low electric field and no moving parts. The Lorentz force developed by the interaction between the electric current and magnetic field can be used to propel, stir or manipulate the flow behavior in the channel. This section provided the details of the MHD micropump performance considering the applied voltage and Hartmann number.

Figure 11 shows the variation of the maximum temperature of the heat dissipating element for the varied applied voltage and Hartmann number with Cu-water with volume fraction of 0.1% as coolant. As the applied voltage is increased, the maximum temperature of the heat dissipating element decreased. For example, as the applied voltage increased from 0.05 V to 0.35 V at a Hartmann number value of 2.0, the maximum temperature of the heat dissipating element decreased by 7.7%. For the same applied voltage, lower maximum temperatures of the heat dissipating element are observed for a higher Hartmann number. As the Hartmann number increased from 1.41 to 3.76 at a constant applied voltage of 0.05 V, the maximum temperature of the heat dissipating element decreased by 11.0%. The combined influence of higher applied voltage and higher Hartmann number are visible with significant decrease in the maximum temperature of the heat dissipating element. These findings show that the applied voltage and Hartmann number have a significant effect on maintaining and controlling the maximum temperature of the heat dissipating element.

Figure 12 shows variation of the heat removal rate for the varied applied voltage and Hartmann number with Cu-water with the volume fraction of 0.1% as coolant. As the applied voltage is increased, the heat removal rate increased. For example, as the applied voltage increased from 0.05 V to 0.35 V at a Hartmann number value of 2.0, the heat removal rate increased by 34.5%. For the same applied voltage, higher heat removal rates are observed for a higher Hartmann number. As the Hartmann number increased from 1.41 to 3.76 at a constant applied voltage of 0.05 V, the heat removal rate increased by 39.5%. The combined influence of a higher applied voltage and higher Hartmann number are visible with significant increase in heat removal rate. The increase in heat removal rate with a higher applied voltage is attributed to an increase in the volumetric Lorentz force as shown in

Figure 7, which subsequently results in the higher volumetric flow rate. It can be seen that for a lower Hartmann number, the rate of change heat removal rate is large, whereas for a higher Hartmann number, the rate of change of heat removal rate is small. This is because the dominance of the magnetic force increased as the Hartmann number increased [

10].

Figure 13 shows the variation of efficiency for the varied applied voltage and Hartmann number with Cu-water with volume fraction of 0.1% as coolant. The efficiency is defined as shown in Equation (19).

As shown, the efficiency decreased continuously with increase in applied voltage. This shows that, even though for higher applied voltage the heat removal rate is higher, and the temperature of the heat dissipating element is minimum, the heat removal process is less efficient. Therefore, an optimum operating range considering the heat removal rate, temperature of heat dissipating element and efficiency could be considered. As the applied voltage increased from 0.05 to 0.35 V at a Hartmann number value of 3.46, the efficiency decreased from 204.4 to 4.9. For the same applied voltage, a lower efficiency is observed for a higher Hartmann number. As the Hartmann number increased from 1.41 to 3.76 at a constant applied voltage of 0.35 V, the efficiency decreased from 29.1 to 4.9. The combined influence of the higher applied voltage and higher Hartmann number are visible with a significant decrease in efficiency. These findings show that the applied voltage and Hartmann number have a significant effect on efficiency.

Figure 14 shows the velocity and temperature distribution in the MHD pump microchannel cooling system with Cu-water with volume fraction of 0.1% as coolant. As shown in

Figure 14a, the velocity is uniformly distributed in the microchannel throughout, which makes it an attractive method for the cooling heat dissipating element, especially where space and noise are constraints such as electronic devices. The rate of increase of the developed flow velocity in the magnetohydrodynamic pump cooling system is an indication of cooling performance as a higher velocity development leads to higher cooling performance. However, the increase in flow velocity has limitations owing to applied voltage and applied magnetic field. As expected, the flow velocity in the thin microchannel increased as it passed through the narrow duct of microchannel cooling system [

6]. This is desirable as the heat dissipating element is placed exactly at the center of the microchannel. As shown in

Figure 14b, the temperature of the coolant increased as it passed through microchannel. In the present study, the square microchannel design is investigated considering the manufacturing simplicity of the square duct. The future scope of the study involves the use of different shapes of microchannel including circular and trapezoidal. The temperature field distribution for the MHD pump microchannel at the center plane showed that heat transfer occurred along the edges of the microchannel and heat is taken away as the flow proceeded [

48]. The geometry based microchannel optimization for effective thermal performance could be carried out considering the requirement of cooling performance and these findings can be used to design an effective cooling by optimizing influencing parameters.

Figure 15 shows the variation of the average Nusselt number for the applied voltage and Hartmann number with Cu-water with a volume fraction of 0.1% as the coolant. The Nusselt number is an indication of enhanced heat transfer due to convection as compared to conduction [

49]. The higher Nusselt number indicates the effectiveness of magnetohydrodynamic cooling systems for the heat dissipating element. The average Nusselt number increased with the applied voltage. For example, as the applied voltage increased from 0.05 V to 0.35 V at a Hartmann number value of 2.0, the average Nusselt number increased by 112.6%. For the same applied voltage, a higher average Nusselt number is observed for higher Hartmann numbers. As the Hartmann number increased from 1.41 to 3.76 at a constant applied voltage of 0.25 V, the heat removal rate increased by 100.0%. The combined influence of a higher applied voltage and higher Hartmann number are visible with a significant increase in the average Nusselt number. However, the rate of increase of the average Nusselt number decreased as the applied voltage and Hartmann number increased. The heat transfer performance slightly deteriorated as the value of the Hartmann number increased due to suppression of convection due to the magnetic field [

10,

50]. These findings show that the applied voltage and Hartmann number have a significant effect on the heat transfer performance of MHD micropumps.

#### 3.4. Influence of Various Nanofluids

The thermal performance of the MHD pump is compared using various nanofluids. Three types of nanofluids including Cu-water, TiO_{2}-water and Al_{2}O_{3}-water are considered with a volume fraction of 0.1%. For performance comparison, the volume fraction of nanoparticles in nanofluids is kept constant. To evaluate the thermal performance of MHD pumps with various nanofluids, the heat transfer rate, efficiency and Nusselt number variation are considered.

Figure 16 shows variation of the heat removal rate for the varied Hartmann number. As the Hartmann number is increased, the heat removal rate increased. For example, as the Hartmann number increased from 1.41 to 3.74 at an applied voltage value of 0.35 V, the heat removal rate increased by 18.0% for Cu-water nanofluids. For the same applied voltage, higher heat removal rates are observed for Cu-water nanofluid as compared to TiO

_{2}-water and Al

_{2}O

_{3}-water nanofluids. As previously noted, for a lower Hartmann number, the rate of change heat removal rate is large, whereas for higher Hartmann number, the rate of change of heat removal rate is small. The Cu-based nanofluid showed a better heat transfer rate owing to the high thermal conductivity of copper nanoparticles.

Figure 17 shows variation of the efficiency for the varied Hartmann number. As the Hartmann number is increased, the efficiency decreased. For example, as the Hartmann number increased from 1.41 to 3.74 at applied voltage value of 0.35 V, efficiency decreased from 29.16% to 4.92% for Cu-water nanofluid. For the same applied voltage, higher efficiencies are observed for Cu-water nanofluid as compared to TiO

_{2}-water and Al

_{2}O

_{3}-water nanofluids. For lower Hartmann number, the rate of change efficiency is large, whereas for higher Hartmann number, the rate of change of efficiency is small. This is because the dominance of magnetic force increased as the Hartmann number increased. The Cu-based nanofluid shows better efficiency owing to high thermal conductivity of copper nanoparticles.

Figure 18 shows variation of the average Nusselt number for the varied Hartmann number. As the Hartmann number is increased, the average Nusselt number increased. For example, as the Hartmann number increased from 1.41 to 3.74 at an applied voltage value of 0.35 V, the average Nusselt number increased by 96.5% for Cu-water nanofluid. For the same applied voltage, higher average Nusselt numbers are observed for Cu-water nanofluid as compared to TiO

_{2}-water and Al

_{2}O

_{3}-water nanofluids. Interestingly, the Nusselt number for the TiO

_{2} based nanofluid and Al

_{2}O

_{3} based nanofluid are found to be close. The Cu-based nanofluid showed a better average Nusselt number owing to the high thermal conductivity of copper nanoparticles.