*2.4. Governing Equations*

The electro-osmotic flow occurs in the channel due to the movement of an upper wall, whereas the flow around the channel of the walls is generated because of an applied electric field. The governing equations for the current flow phenomenon are:

$$\nabla \mathbf{V} = 0\tag{10}$$

$$(\mathbf{V}.\nabla)\mathbf{V} = \frac{1}{\rho\_{nf}}(-\nabla\overline{p} + \nabla.\tau + Body\ force)\tag{11}$$

$$\rho \left( \mathbf{V} \cdot \nabla \right) T = \alpha\_{nf} \nabla^2 T + \frac{1}{\left( \rho \mathbb{C}\_p \right)\_{nf} \sigma\_{nf}} \mathbf{J}^2 + \frac{1}{\left( \rho \mathbb{C}\_p \right)\_{nf}} \Gamma \tag{12}$$

where **V**, *T* and Γ correspondingly represent velocity, temperature and viscous dissipation. The body force contains magnetic, electrical and buoyancy e ffects.

$$
\begin{array}{cccc}
\text{Bady force} = & \underbrace{\begin{pmatrix} \mathbf{J} \times \mathbf{B} \end{pmatrix}}\_{} + & \underbrace{\overline{\rho}\_c \mathbf{E}}\_{} & + \begin{pmatrix} \rho \beta \end{pmatrix}\_{\text{nf}} \mathbf{g}(T - T\_w) \end{array} \tag{13}
$$

Lorentz force External Electric force Buoyancy force

and

$$
\Gamma = \mu\_{nf} \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)^2. \tag{14}
$$

Under the application of Lorentz force, it produces the following Joules heating e ffects.

$$\frac{1}{\sigma\_{nf}}\mathbf{J}^2 = \sigma\_{nf} B\_0 \,^2 \overline{\boldsymbol{\mu}}^2 \text{ and } \mathbf{J} \times \mathbf{B} = \left(-\sigma\_{nf} B\_0 \overline{\boldsymbol{\mu}}, 0, 0\right) \tag{15}$$

*Symmetry* **2019**, *11*, 1038

The governing Equations (10) and (12) in components form after omitting the axial heat conductions at the walls and in the fluid [54] and become:

$$\frac{\partial \overline{\overline{p}}}{\partial \overline{\underline{x}}} = \mu\_{nf} \frac{\partial^2 \overline{u}}{\partial \overline{y}^2} - \sigma\_{nf} B \alpha^2 \overline{u} + (\rho \beta)\_{nf} (T - T\_w) g + \overline{\rho}\_c (\overline{y}) E\_x \tag{16}$$

$$
\overline{u}\frac{\partial T}{\partial \overline{\mathbf{x}}} = \alpha\_{nf}\frac{\partial^2 T}{\partial \overline{y}^2} + \frac{\sigma\_{nf} B\_0}{\left(\rho \mathbb{C}\_p\right)\_{nf}} \overline{u}^2 + \frac{\mu\_{nf}}{\left(\rho \mathbb{C}\_p\right)\_{nf}} \left(\frac{\partial \overline{u}}{\partial \overline{y}}\right)^2. \tag{17}
$$

The associated boundary conditions are:

$$\begin{cases} \text{(At upper wall)}: \overline{u} = \mathcal{U}^\*, k\_f \frac{\partial T}{\partial \overline{y}} = 0 \text{ at } \overline{y} = a\\ \text{(At lower wall)}: \overline{u} = 0, \ -k\_f \frac{\partial T}{\partial \overline{y}} = q\_w \text{ at } \overline{y} = -a \end{cases} \tag{18}$$

The effective density, heat capability, and thermal and electrical conductivities of a nanofluid [55] are respectively given by:

$$\frac{\rho\_{nf}}{\rho\_{\mathcal{P}}} = \left[ (1 - \phi) \frac{\rho\_f}{\rho\_{\mathcal{P}}} + \phi \right] \tag{19}$$

$$\frac{\left(\rho \mathbb{C}\_{p}\right)\_{nf}}{\left(\rho \mathbb{C}\_{p}\right)\_{p}} = \left[ (1 - \phi) \frac{\left(\rho \mathbb{C}\_{p}\right)\_{f}}{\left(\rho \mathbb{C}\_{p}\right)\_{p}} + \phi \right] \tag{20}$$

$$\frac{k\_{nf}}{k\_f} = \left(4.97\phi^2 + 2.72\phi + 1\right) \tag{21}$$

$$\frac{\sigma\_{nf}}{\sigma\_f} = \left[1 + \frac{3\left(\frac{\sigma\_p}{\sigma\_f} - 1\right)\phi}{\left(\frac{\sigma\_p}{\sigma\_f} + 2\right) - \left(\frac{\sigma\_p}{\sigma\_f} - 1\right)\phi}\right].\tag{22}$$

The following dimensionless transformations

$$\begin{aligned} \overline{y} &= ay, \ \overline{u} = u\_{\text{ll}} u, \ \mathcal{U}^\* = u\_{\text{ll}} \mathcal{U}, \ \overline{p} = \rho\_f u\_{\text{m}}^2 p \left( a/u\_{\text{m}} \right)^\eta, \ \theta &= \frac{T - T\_{\text{w}}}{q\_{\text{w}} a/k\_f}, \\ \overline{\rho}\_{\varepsilon} &= -\left( \varepsilon \zeta\_1 / a^2 \right) \rho\_{\varepsilon}, \ \overline{\kappa} = \kappa / a, \ \psi = \overline{\psi} / \zeta\_1 \end{aligned} \tag{23}$$

Transform Equations (16) and (17) by using (23) in the dimensionless form as:

$$\left(123\phi^2 + 7.3\phi + 1\right)\eta \left(\frac{\partial u}{\partial y}\right)^{n-1} \frac{\partial^2 u}{\partial y^2} - A\_4 M^2 u + A\_3 Gr \theta + \beta\_u \rho\_\varepsilon - ReP = 0,\tag{24}$$

$$\left(4.97\phi^2 + 2.72\phi + 1\right)\frac{\partial^2 \theta}{\partial y^2} + Br\left(123\phi^2 + 7.3\phi + 1\right)\left(\frac{\partial u}{\partial y}\right)^{n+1} - B\_1\gamma u A\_4 + BrM^2 u^2 = 0.\tag{25}$$

$$\begin{array}{ccccc} \mu = \mathsf{U}, & \frac{\partial \mathcal{O}}{\partial y} = 0 & \text{at} & y = 1 & \text{(Upper wall)}\\ \mu = 0, & \frac{\partial \mathcal{J}}{\partial y} = -1 & \text{at} & y = -1 & \text{(Lower wall)} \end{array} \tag{26}$$

In which

*Gr* = (ρβ)*f gqwa*<sup>3</sup> δ*k f um aum <sup>n</sup>*−1, *Re* = ρ*f uma* δ *aum <sup>n</sup>*−1, *M*<sup>2</sup> = <sup>σ</sup>*f <sup>B</sup>*02*a*<sup>2</sup> δ *aum n*−1 *Br* = <sup>δ</sup>*um*<sup>2</sup> *qwa aum* <sup>1</sup>−*n*, *UHs* = −εζ1*Ex* δ *aum <sup>n</sup>*−1, β*u* = *UHs um* , γ = *k f uma* <sup>α</sup>*f qw* ∂*T*∂*x* , ρ*e*(*y*) = κ2 Sinh(κ(<sup>1</sup>−*y*))+*<sup>R</sup>*ζSinh(κ(<sup>1</sup>+*y*)) Sin(<sup>2</sup>κ) , <sup>α</sup>*f* = (ρ*Cp*)*f k f A*4 = <sup>σ</sup>*n f* <sup>σ</sup>*f* , *A*3 = (ρβ)*n f* (ρβ)*f* , *B*1 = (ρ*Cp*)*n f* (ρ*Cp*)*f* , *R*ζ = ζ2/ζ1. ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭. (27)

where κ is the electro-osmotic parameter, ρ is density, μ is dynamic viscosity, β is volumetric volume expansion, *Gr* is Grashof number and *Cp* is specific heat. Assume that *um* = −*a*2/2μ*f*<sup>∂</sup>*p*/∂*<sup>x</sup>* is the maximum velocity between two plates, and β*u* is the ratio between electro-osmotic velocity *UHs* and maximum velocity of the fluid. Thermophysical properties of alumina and the base fluid polyvinyl chloride (PVC) are illustrated below in Table 1.


**Table 1.** Physical properties of PVC [56] and Al2O3 [52].

*2.5. Thermophysical Relations*

> Skin friction is defined as:

$$\mathbf{C}\_{f} = \frac{2\tau\_{w}}{\rho\_{f}u\_{m}^{2}} \text{ and } \tau\_{w} = \delta \left( 123\phi^{2} + 7.3\phi + 1 \right) \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right) \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)^{n-1}; \overline{y} = \pm a \tag{28}$$

Moreover, in the present study, different concentration of alumina particles (Al2O3) are used in the polymer solution of polyvinyl alcohol in water; consequently, different values of τ in correlation with nanoparticle volume fraction φ for different concentrations of polyvinyl alcohol are listed in Table 2.

**Table 2.** Properties of the power-law equation and PVC solutions [56].


Hence, the coefficient of skin friction in the dimensionless form for both walls is:

$$C\_f = \frac{2}{Re} (123\phi^2 + 7.3\phi + 1) \left(\frac{\partial u}{\partial y}\right)^n \text{ at } y = \pm 1. \tag{29}$$

#### *2.6. Heat Transfer Rate*

The Nusselt number is defined as:

$$N\mu = \frac{ah}{k\_f} \text{ and } h = \frac{q\_W}{T\_w - T\_m} \tag{30}$$

Here, *h* is heat transfer coefficient, *qw* is wall heat flux and *Tm* is bulk mean temperature, which is given by:

$$T\_m = \frac{\int \rho\_f \overline{u} T dA}{\int \rho\_f \overline{u} dA}.\tag{31}$$

The mean temperature in the dimensionless form becomes:

$$
\theta\_m = \frac{k\_f (T\_m - T\_w)}{q\_w a}.\tag{32}
$$

Therefore, the Nusselt number is:

$$Nu = -\frac{1}{\Theta\_m}.\tag{33}$$

## *2.7. Entropy Generation*

The entropy generation of local volumetric rate can be defined as:

$$S\_G = \frac{k\_{nf}}{T\_w^2} \left(\frac{\partial \overline{T}}{\partial \overline{y}}\right)^2 + \frac{\mu\_{nf}}{T\_w} \left(\frac{\partial \overline{u}}{\partial \overline{y}}\right)^2 + \frac{\sigma\_{nf} B\_0^2 \overline{u}^2}{T\_w} + \frac{\rho\_\varepsilon(\overline{y}) E\_x}{T\_w}.\tag{34}$$

The characteristic entropy generation can be expressed as:

$$S\_0 = \frac{q\_w^2}{k\_f T\_w^2},\tag{35}$$

By using the transformation given in Equation (23), the non-dimensional total entropy generation may be expressed as:

$$\text{Nls} = \frac{S\_G}{S\_0} = \left( 4.97\phi^2 + 2.72\phi + 1 \right) \left( \frac{\partial \theta}{\partial y} \right)^2 + \frac{Br}{\Omega} \left( 123\phi^2 + 7.3\phi + 1 \right) \left( \frac{\partial u}{\partial y} \right)^{n+1} + A\_4 M^2 u^2 + \beta\_n \rho\_t u \Big|\_{y=0} \tag{36}$$

where *M*, *Br*, β*<sup>u</sup>*, ρ*e* and Ω are respectively the magnetic field, Brinkman number, volumetric volume expansion, dimensional electric charge density, and dimensionless temperature difference. These parameters are defined as:

$$\begin{split} M^2 &= \frac{\sigma\_f B\_\rho^2 r^2}{K} \Big(\frac{\mu}{\mu\_m}\Big)^{n-1}, \,\,Br = \frac{K \mu\_m^2}{q\_{wd} d} \Big(\frac{\mu}{\mu\_m}\Big)^{1-n}, \,\,\beta\_u = \frac{l I\_{Hz}}{\mu\_m},\\ \rho\_\varepsilon &= -(\varepsilon \zeta\_1/a^2) \rho\_{\varepsilon \prime} \,\,\,\Omega = \frac{q\_w d}{k\_f \overline{I}\_w} \end{split} \tag{37}$$

$$Be = \frac{\text{Entropy due to heat transfer}}{\text{Total entropy generation}} \tag{38}$$

$$B\varepsilon = \frac{\left(4.97\phi^2 + 2.72\phi + 1\right)\left(\frac{\partial\vartheta}{\partial y}\right)^2}{(4.97\phi^2 + 2.72\phi + 1)\left(\frac{\partial\vartheta}{\partial y}\right)^2 + \frac{Br}{\Pi}\left((123\phi^2 + 7.3\phi + 1)\left(\frac{\partial u}{\partial y}\right)^{n+1} + A\_4M^2u^2 + \beta\_u\rho\_\varepsilon u\right)},\tag{39}$$

#### **3. Discussion of Results**
