**1. Introduction**

Conventional fluids like glycol, acetone, kerosene and water have low heat transfer characteristics and play an important role in laboratories and industrial engineering applications. Several experiments and procedures have been demonstrated to improve heat transfer characteristics. A significant contribution of boosting the heat transfer ability of fluids in an industrial process by the insertion of nanoparticles into the fluids was found. The pioneering work of Choi [1] led to the discovery that heat transfer rate and higher thermal conductivity can be enriched by a combination of base fluids and nanoparticles. Later, Xuan [2] experimentally investigated the enrichment of thermal conductivity in nanofluids. Some core developments have been collated in references [3–9].

Further, power-law fluids have attracted the attention of many scientists as the description of power-law fluids happens to be the best description of fluid behavior (with right choice of power-law index) of shear-dependent fluids. There are several more models that better describe the range of shear rates, but they do so at the cost of simplicity. For this reason, the power law is used to describe fluid behavior as compared to other fluid models. Recently, Ouyang et al. [10] proposed a two-dimensional squirmer model for power law fluids. They concluded that the selection of power law relation is the best choice to demonstrate the rheological properties of three sorts of fluids, namely: (i) Newtonian, (ii) shear-thickening, and (iii) shear-thinning fluids. Also, the exact velocity at the bottom region of shear rate cannot augur the basis of the power law model. After overcoming the insu fficiency of power law, it cannot be used to find the exact velocity profile for the shear rate of the bottom and zero regions [11]. Also, power law fluid in some cases is used to describe non-Newtonian fluids [12,13]

Electro-osmotic flow, also known as electro-osmosis flow, is used in microfluidic devices and electronically controlled fluid flow or any other fluid conduit. Zhao et al. [14] have invoked electro-osmotic flow in a channel with the power law model. Das and Chakraborty [15] investigated the impact of non-Newtonian fluid on electro-osmotic flow. Several attempts regarding electro-osmosis flow can be found in references [16–19].

It is well known that magnetic fluids have significant utilization in applications such as heat transfer, cancer therapy, opto-electronic devices, sensors, cooling of nuclear reactors, liquid metal flow control, high temperature plasmas, solidification of binary alloys, drying processes etc. [20–25]. In addition, a number of researchers have conducted promising investigations on nanoparticles. Examples include the work of Malvandi et al. [26] who have studied MHD mixed convection with nanoparticle migration and Yousif et al. [27] who publicized magnetohydrodynamics (MHD) Carreau nanofluids with internal heat source/sink radiation. Further, magnetic nanofluids also exhibit interesting properties that are used in various microfluidic applications, dipolar nanoparticles, magnetic fluid hyperthermia and magnetic resonance imaging [28–31]. The existing literature bears witness that magnetic fluids with nanoparticles exhibit several interesting structural characteristics depending on the applied magnetic field strength. As a result, several studies have also been undertaken on MHD rheological fluids [32–34].

In recent years, prominent authors have been inspired to study entropy generation by various applications, such as diffusion and Joule heating. Entropy is caused by irreversible processes of a system and can be reduced only when it interacts with some other system whose entropy increases in the process. Bejan [35] studied the entropy generation for a flow system and discussed the main ideas to control the energy loss in flow problems and enhanced the ability of the system. Zeeshan et al. [36] discussed the radiative and electro-magnetohydrodynamic effects of a titanium dioxide/water based nanofluid on entropy generation. Ranjit and Shit [37] presented the effects of entropy generation with MHD on electro-osmotic flow. Numerical analysis for entropy generation on nanofluids with the suspension of nanoparticles (such as copper, Al2O3 and TiO3) in water as a base fluid, which passes through wavy walls, was conducted by Cho et al. [38]. Different researchers have devoted their efforts to explore the impact of entropy generation to real world problems [39–43].

Thus far, the simultaneous effects of entropy generation, MHD and electro-osmosis on power law nanofluid flow has not been studied. In order to fill this gap, the next section is devoted to developing a mathematical formulation of the problem, which comprises mass, momentum, and energy equations. The situation becomes difficult as the resulting equations are not only nonlinear, but also coupled. The analytical findings are developed by a homotopic tactic [44] that has been used effectively for the last two decades [45–50].

The acquired results satisfied the governing equations and boundary conditions. The physical features involve parameters which are adequately determined through various graphs and tables.

## **2. Problem Design**

## *2.1. Physical Considerations*

Incompressible and steady nanofluids pass through horizontal parallel plates as illustrated in Figure 1. The wall at *y* = *a* moves with constant velocity *U*∗ while the second wall remains stationary at *y* = <sup>−</sup>*a*. *B*0 is uniform transverse magnetic field, ξ1 is zeta potential at lower wall, ξ2 is zeta potential at upper wall, ψ is electric double layer potential and 2*a* is the total width of the channel.

**Figure 1.** Schematic representation of electro-osmotic flow of the physical problem.

#### *2.2. Electrical Potential Distribution*

During the process of electro-osmotic flow (EOF), the separation of ions takes place and an electrical double layer (EDL) forms adjacent to the channel walls, thereby developing electric potential distribution. The electric potential ψ within the channel, described by the Poisson–Boltzmann equation [51] in the Cartesian co-ordinate system, can be calculated as:

$$\frac{\partial^2 \overline{\psi}}{\partial \overline{x}^2} + \frac{\partial^2 \overline{\psi}}{\partial \overline{y}^2} = -\frac{\overline{\rho}\_\varepsilon(\overline{y})}{\varepsilon} \tag{1}$$

For small values of electrical potential ψ of the EDL, the Debye–Hückel approximation can be applied and Equation (1) becomes:

$$
\nabla^2 \overline{\psi} = \overline{\kappa}^2 \overline{\psi}, \ \overline{\kappa}^2 = \frac{2n\_0 z\_v^2 e^2}{\varepsilon k\_B \overline{\Upsilon}}.\tag{2}
$$

The plates of the channel are made of different materials and have different zeta potentials.

$$
\overline{\psi} = \zeta\_1 \text{ at } \overline{y} = -a \text{ and } \overline{\psi} = \zeta\_2 \text{ at } \overline{y} = a. \tag{3}
$$

The electrical potential under the action of Equation (3) can be explored as:

$$\overline{\psi}(\overline{y}) = \frac{\zeta\_1 \text{Sink}(\overline{\kappa}(a - \overline{y})) + \zeta\_2 \text{Sink}(\overline{\kappa}(a + \overline{y}))}{\text{Sin}(2a\overline{\kappa})} \tag{4}$$

The electric double layer effects are produced by the external field relation **E** = (*Ex*, 0, 0) and charge density of nanoparticles. The external electric force ρ*e***E**, also called the electro-kinetic force [52,53], is generated outside of the charge particle. Along with <sup>ρ</sup>*e*(*y*), electric charge density is defined as:

$$\overline{\rho}\_c(\overline{y}) = -\epsilon \overline{\kappa}^2 \left( \frac{\zeta\_1 \text{Sink}(\overline{\kappa}(a - \overline{y})) + \zeta\_2 \text{Sink}(\overline{\kappa}(a + \overline{y}))}{\text{Sin}(2a\overline{\kappa})} \right) \tag{5}$$

where ε, κ, *n*0, *zv*, *e*, *kB* and *T*ˆ are the relative permittivity of the medium, Debye parameter, ion density of bulk liquid, valence of ions, electron charge, Boltzmann constant and absolute temperature, respectively.

#### *2.3. Power Law Model*

In the current study, the following power law fluid model [54] is used:

$$
\pi = \mu\_{nf} \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)^{n-1} \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right) = \begin{cases}
\mu\_{nf} \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)^{n} & \text{for } \frac{\partial \overline{u}}{\partial \overline{y}} > 0 \\
\end{cases} \tag{6}
$$

where τ is shear stresses, *n* is the power-law or flow behavior index of the fluid and μ*n f* is the viscosity of the nanofluid [55] and is defined along the consistency index δ as:

$$
\mu\_{nf} = \left(123\phi^2 + 7.3\phi + 1\right)\mu\_f \tag{7}
$$

and the viscosity of the base fluid in the current discussion is taken as:

$$
\mu\_f = \delta \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)^{n-1}. \tag{8}
$$

One signifies a Newtonian fluid for *n* = 1 whereas *n* < 1 and *n* > 1 respectively denote the shear-thinning and shear-thickening of fluids.

To estimate the shear stresses of a fluid, we will illustrate further investigations with shear-thinning properties. Thus, shear stress for the non-Newtonian power-law model can be written as:

$$
\pi = \delta \left( 123\phi^2 + 7.3\phi + 1 \right) \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right) \left( \frac{\partial \overline{u}}{\partial \overline{y}} \right)^{n-1}.\tag{9}
$$
