**2. Problem Description and Formulation**

The steady incompressible two-dimensional MHD flow of a micropolar nanofluid on an exponentially shrinking surface in an Extended-Darcy-Forchheimer porous medium is considered by adding <sup>−</sup> <sup>1</sup> √ *K* <sup>ϑ</sup> √ *<sup>K</sup>* <sup>+</sup> *bu u* in the Navier Stokes equation. The velocity of the shrinking surface in the form of exponential terms is given by *Uw*(*x*) = *U*0*e* 2*x* , while the uniform magnetic field of the strength *B*<sup>0</sup> has been normally applied to it (Figure 1). Due to a small value of the magnetic Reynolds number, the induced magnetic field is ignored. Under the consideration of the mentioned assumptions, the

boundary layer equations of motion for the micropolar nanofluid, heat and concentration equations are expressed as:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}
$$

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \left(\theta + \frac{\kappa}{\rho}\right)\frac{\partial^2 u}{\partial y^2} + \frac{\kappa}{\rho}\frac{\partial N}{\partial y} - \frac{\theta}{K}u - \frac{b}{\sqrt{K}}u^2 - \frac{\sigma B^2 u}{\rho} \tag{2}$$

$$u\frac{\partial N}{\partial \mathbf{x}} + v\frac{\partial N}{\partial y} = \frac{1}{\rho j} \left[ \gamma \frac{\partial^2 N}{\partial y^2} - \kappa \left( 2N + \frac{\partial u}{\partial y} \right) \right] \tag{3}$$

$$
\mu \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} = a \frac{\partial^2 T}{\partial y^2} + \tau\_1 \left[ D\_B \frac{\partial \mathbf{C}}{\partial y} \frac{\partial T}{\partial y} + \frac{D\_T}{T\_\infty} \left( \frac{\partial T}{\partial y} \right)^2 \right] \tag{4}
$$

$$u\frac{\partial \mathcal{C}}{\partial \mathbf{x}} + v\frac{\partial \mathcal{C}}{\partial y} = D\_B \frac{\partial^2 \mathcal{C}}{\partial y^2} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial y^2} \tag{5}$$

subject to the following boundary conditions:

 $w = \sqrt{\frac{\\$lL\_0}{2l}}e^{\frac{\Delta}{2l}}$   $S; u = -lL\_w(x) + B^\*$  $\partial\_{\partial y}^{\underline{u}}; N = -m\frac{\partial u}{\partial y}; T = T\_w$  $\therefore C = C\_w$   $\text{ at } y = 0$   $u \to 0; \ N \to 0; \ T \to T\_{\text{co}}; \ C \to C\_{\text{co}}$   $\text{ as } y \to \infty.$ 

**Figure 1.** Physical model of the flow.

We considered the following similarity transformations, as adopted by Sanjayanand and Khan [39], to solve Equations (1)–(5), with boundary condition (6):

$$\begin{split} \psi &= \sqrt{2\mathcal{S}l\Pi\_0}e^{\frac{\pi}{2l}}f(\mathbf{x},\boldsymbol{\eta}), N = \mathcal{U}\_0e^{\frac{3\pi}{2l}}\sqrt{\frac{lL\_0}{2\mathcal{S}l}}g(\mathbf{x},\boldsymbol{\eta}), \; \mathcal{O}(\mathbf{x},\boldsymbol{\eta}) = \frac{(T-T\_{\rm ev})}{(T\_{\rm w}-T\_{\rm ev})},\\ \mathcal{B}(\mathbf{x},\boldsymbol{\eta}) &= \frac{(\mathbb{C}-\mathbb{C}\_{\rm ev})}{(\mathbb{C}\_{\rm w}-\mathbb{C}\_{\rm ev})}, \; \boldsymbol{\eta} = \mathcal{Y}\sqrt{\frac{lL\_0}{2\mathcal{S}l}}e^{\frac{\pi}{2l}} \end{split} \tag{7}$$

where *u* = ∂ψ ∂*y* and *<sup>v</sup>* <sup>=</sup> <sup>−</sup>∂ψ <sup>∂</sup>*<sup>x</sup>* are components of the velocity along the directions *x* and *y* respectively,ρ is the fluid density, ϑ is the kinematic viscosity, σ is the electrical conductivity of the fluid, *B* = *B*0*e x* <sup>2</sup>*<sup>l</sup>* is the magnetic field with a constant magnetic strength *B*0, *K*<sup>1</sup> is the permeability of the porous medium, *b* is the local inertia coefficient, κ is the vortex viscosity, *N* is the microrotation, γ indicates the spin gradient viscosity, *j* is the ratio of the micro inertia and unit mass, *T* is the fluid temperature, and α is the thermal diffusivity of the micropolar nanofluid. Furthermore, <sup>τ</sup><sup>1</sup> <sup>=</sup> (ρ*c*)*<sup>p</sup>* (ρ*c*)*<sup>f</sup>* is the ratio between the effective heat capacity of the nanoparticle material and the capacity of the fluid, *DB* is the Brownian diffusion coefficient, *DT* is the thermophoretic diffusion coefficient, *Tw* is the temperature of the wall, *T*<sup>∞</sup> is the ambient temperature, *Cw* is the concentration of the wall *C*<sup>∞</sup> is the ambient concentration, and *B*<sup>∗</sup> = *B*1*e* −*x* <sup>2</sup>*<sup>l</sup>* is the velocity slip factor. It might be mentioned that the range of *m* is 0 ≤ *m* ≤ 1; however, *m* is constant. In the case of *m* = 0, we have *N* = 0, which indicates that the strong concentration and micro-elements are near to the wall and are not rotatable. Furthermore, *m* = 0.5 shows a weak concentration, which causes the anti-symmetric part of the stress tensor to vanish. On the other hand, *m* = 1 indicates the turbulent boundary layer flows modeling (see [40,41]).

Using Equation (7) in Equations (2)–(5), we get the following partial differential equations:

$$\left(1+K\right)f\_{\eta\eta\eta} + ff\_{\eta\eta} + Kg\_{\eta} - 2\left(f\_{\eta}\right)^2 - F\_S\left(f\_{\eta}\right)^2 - K\_1f\_{\eta} - Mf\_{\eta} = 2l\left(f\_{\eta}f\_{\eta\xi} - f\_{\eta\eta}f\_{\xi}\right) \tag{8}$$

$$\left(1+\frac{K}{2}\right)\mathbf{g}\_{\eta\eta} + f\mathbf{g}\_{\eta} - 3\mathbf{g}f\_{\eta} - 2\mathbf{K}\mathbf{g} - \mathbf{K}f\_{\eta\eta} = 2l\left(f\_{\eta}\mathbf{g}\_{\mathbf{x}} - \mathbf{g}\_{\eta}f\_{\mathbf{x}}\right) \tag{9}$$

$$\frac{1}{Pr}\theta\_{\eta\eta} + f\theta\_{\eta} + \mathcal{N}\_{\mathsf{b}}\theta\_{\eta}\mathcal{Q}\boldsymbol{\eta} + \mathcal{N}\_{\mathsf{l}}\{\boldsymbol{\Theta}\_{\eta}\}^2 = 2l \Big(f\_{\eta}\theta\_{\mathrm{x}} - \theta\_{\eta}f\_{\mathsf{x}}\Big) \tag{10}$$

$$\mathcal{Q}\_{\eta\eta} + \mathrm{Scf}\mathcal{Q}\_{\eta} + \frac{N\_{\mathrm{f}}}{N\_{\mathrm{b}}} \theta\_{\eta\eta} = 2 \cdot \mathrm{Sc} \cdot \mathrm{l} \Big( f\_{\eta} \mathcal{Q}\_{\mathrm{x}} - \mathcal{Q}\_{\eta} f\_{\mathrm{x}} \Big) \tag{11}$$

Furthermore, many authors considered <sup>γ</sup> = - μ + <sup>κ</sup> 2 *j* = μ - 1 + *<sup>K</sup>* 2 *j*, where κ = μ*K* is the material parameter [41] in their work. In our problem, the terms of the Extended-Darcy-Forchheimer porous medium *K*<sup>1</sup> = *lv* <sup>2</sup>*U*0*K e <sup>x</sup> l* and γ = μ - 1 + *<sup>K</sup>* 2 2ϑ*le* −*x <sup>U</sup>*<sup>0</sup> do not allow it to have self-similar solutions. For this reason, by using the pseudo-similarity variable, a local similarity solution can be obtained by equating the derivative of the functions of *f*, *g*, θ and ∅ with respect to *x* being equal to zero. This implies that *f*(*x*, η) = *f*(η); *g*(*x*, η) = *g*(η); θ(*x*, η) = θ(η) and ∅(*x*, η) = ∅(η) [39]. As a result, all the terms on the right-hand side become zero, and we get the following third-order non-linear quasi-ordinary differential equation:

$$(1+K)f'''' + ff'' + Kg' - 2f'^2 - F\_S f'^2 - K\_1 f' - Mf' = 0\tag{12}$$

$$\left(1+\frac{K}{2}\right)\mathbf{g}^{\prime\prime} + f\mathbf{g}^{\prime} - 3\mathbf{g}f^{\prime} - 2\delta\mathbf{K}\mathbf{g} - \delta\mathbf{K}f^{\prime\prime} = 0\tag{13}$$

$$\frac{1}{Pr}\theta'' + f\theta' + N\_b\mathcal{Q}'\theta' + N\_l(\theta')^2 = 0\tag{14}$$

$$\mathcal{L}'''' + \mathcal{S}cf\mathcal{Q}' + \frac{N\_t}{N\_b}\theta'' = 0\tag{15}$$

subject to the boundary conditions below:

$$\begin{array}{c} f(0) = \text{S; } f'(0) = -1 + \lambda f''(0); \text{ g}(0) = -m f^{\circ \circ}(0); \text{ } \theta(0) = 1; \mathcal{Q}(0) = 1\\ f'(\eta) \to 0; \text{ g}(\eta) \to 0; \text{ } \theta(\eta) \to 0; \mathcal{Q}(\eta) \to 0 \quad \text{as } \eta \to \infty. \end{array} \tag{16}$$

Here, prime stands for the differentiation with respect to the new independent variable η, *K* = <sup>κ</sup> μ is the non-Newtonian parameter, *K*<sup>1</sup> is the permeability parameter, *FS* = <sup>2</sup>*lb* √ *<sup>K</sup>* is the Forchheimmer parameter, *<sup>M</sup>* = <sup>2</sup>*l*σ(*B*0) 2 <sup>ρ</sup>*U*<sup>0</sup> is the Hartmann number, *Pr* <sup>=</sup> <sup>ϑ</sup> <sup>α</sup> is the Prandtl number, *Nt* <sup>=</sup> <sup>τ</sup>1*DT*(*Tw*−*T*∞) <sup>ν</sup>*T*<sup>∞</sup> is the thermophoresis parameter, *Nb* <sup>=</sup> <sup>τ</sup>1*DB*(*Cw*−*C*∞) <sup>ν</sup> is the parameter of Brownian motion, *Sc* <sup>=</sup> <sup>ϑ</sup> *DB* is the Schmidt number, λ = *B*<sup>1</sup> ϑ*U*<sup>0</sup> <sup>2</sup>*<sup>l</sup>* is the velocity slip, and *S* < 0 and *S* > 0 are the mass injunction and suction parameter, respectively.

The physical quantities of interest are the coefficient of the skin friction, the local Nusselt number and local Sherwood number, which are given by:

$$\begin{aligned} \mathsf{C}\_{f} &= \frac{\left[ (\mu + \kappa) \frac{\partial \mathfrak{u}}{\partial \mathbf{y}} + \kappa \mathcal{N} \right]\_{\mathbf{y} = 0}}{\rho \mathcal{U}\_{0}^{2}}; \; \mathcal{N}\_{\mathbf{u}} = \frac{-\mathbf{x} \left( \frac{\partial \mathfrak{f}}{\partial \mathbf{y}} \right)\_{\mathbf{y} = 0}}{(\mathcal{T}\_{\mathbf{u}} - \mathcal{T}\_{\mathbf{v} \mathbf{y}})}; \mathcal{S}\_{\mathbf{h}} = \frac{-\mathbf{x} \left( \frac{\partial \mathfrak{f}}{\partial \mathbf{y}} \right)\_{\mathbf{y} = 0}}{(\mathcal{C}\_{\mathbf{u}} - \mathcal{C}\_{\mathbf{v} \mathbf{y}})}; \\ \mathcal{C}\_{f} (\mathsf{Re}\_{\mathbf{x}})^{\frac{1}{2}} &= \sqrt{2} (1 + (1 - \mathsf{m}) \mathsf{K}) f''(0); \mathcal{N}\_{\mathbf{u}} (\mathsf{Re}\_{\mathbf{x}})^{-\frac{1}{2}} = -\frac{1}{\sqrt{2}} \theta'(0); \mathcal{S}\_{\mathbf{h}} (\mathsf{Re}\_{\mathbf{x}})^{-\frac{1}{2}} = -\frac{1}{\sqrt{2}} \mathcal{O}'(0) \end{aligned} \tag{17}$$

where *Rex* = *luw*/ϑ is the local Reynolds number.
