*2.2. Governing Equations and Boundary Conditions*

The modeling of the MHD phenomena involves a multiphysics problem with coupled equations between fluid flow, heat transfer, current flow, and magnetic fields, which are solved numerically. The different fields of physics involved are expressed by partial differential equations, which can be solved via the finite element method. In the present study, the numerical modeling of the MHD phenomena is conducted using COMSOL. The partial differential equations involving multiphysics behavior with coupling between fluid flow, heat transfer, electric current and magnetics are solved using the finite element method. The fluid flow and heat transfer are governed by the Navier–Stokes equation as shown below [29]. Equations (1)–(3) show continuity, momentum and energy conservation, respectively [30], where → *V* is velocity, ρ is density, *p* is pressure and α is thermal diffusivity.

$$
\vec{\nabla} \cdot \vec{V} = \mathbf{0} \tag{1}
$$

$$(\overrightarrow{V}\cdot\nabla)\overrightarrow{V} = \frac{1}{\rho}\nabla P + \nabla^2 \overrightarrow{V} + \frac{1}{\rho}\overrightarrow{F} \tag{2}$$

$$(\overrightarrow{V} \cdot \nabla)T = a\nabla^2 T\tag{3}$$

$$
\vec{F} = \vec{J} \times \vec{B} \tag{4}
$$

$$
\overrightarrow{J} = \sigma \boxdot \overrightarrow{E} + \overrightarrow{V} \times \overrightarrow{B} \tag{5}
$$

*F* is the body force due to Lorentz forces which causes fluid motion as shown in Equation (4) [9]. The electric current density which is defined by Ohm's law is shown in Equation (5) [31], where → *J* is the electric current in y-direction and → *B* is the magnetic field in the z-direction. The electric current and magnetic field are perpendicular which creates a Lorentz force in the x-direction.

The working fluid is Newtonian fluid with flow considered as steady and laminar based on the low Reynolds number. The thermo-physical properties of working fluid, nanoparticle and boundary conditions are presented in Table 2. The heat dissipating element that is acting on the pump's wall is assumed to be a constant volumetric heat generation source. The applied electric voltage is varied from 0.05 V to 0.35 V with an interval of 0.05 V. The Hartmann number is varied from 1.41 to 3.74. The cylindrical type permanent magnets are used for providing the magnetic field intensity. Three different types of nanofluids are considered including Cu-water, TiO2-water, and Al2O3-water nanofluids. The base fluid for all the nanofluids is water. The boundary condition of opening at atmospheric pressure is applied at the coolant inlet and coolant outlet. The density of water is considered as 997.0 kg/m<sup>3</sup> at 25 ◦C and assumed as an incompressible fluid. The thermal conductivity of water is considered as 0.6069 W/m-K at 25 ◦C. The specific heat of water is considered as 4181.7 J/kg-K. The details about the boundary conditions and thermophysical properties of water and nanoparticles are presented in Table 2.


**Table 2.** Boundary conditions and thermophysical properties.
