**2. Mathematical Model and Formulation**

We consider a Ree-Eyring nanofluid flow in a two-dimensional laminar boundary layer with the influence of magnetic dipoles. Furthermore, the Cattaneo-Christov heat flow model is used to analyse heat transmission. The magnetic dipole is located under the sheet, whereas the electrically non-conductive and incompressible Ree-Eyring nanofluid is located above the sheet in the half-space *y* > 0. Figure 1 depicts the flow geometry. By assuming two conflicting and comparable forces along the *x*-*axis*, the sheet is stretched at a proportional rate to the distance between it and the fixed origin *x* = 0. The dipole centre is located on the *y*-*axis* below the *x*-*axis*. It has a powerful magnetic field directed in the positive *x*-*direction*, which increases the magnetic field's intensity enough to feed the Ree-Eyring nanofluid. The stretched sheet is kept at a temperature *Tw* underneath Curie's temperature *Tc*, although the far-flung liquid elements are thought to be at *T* = *Tc*.

**Figure 1.** Physical description flow geometry.

Using the Koo-Kleinstreuer model, the efficacious *kn f* of nanofluids can be displayed as [11]

$$k\_{nf} = k\_{static} + k\_{Brouniau}$$

where

$$k\_{\rm static} = \frac{k\_f \left(k\_s + 2k\_f\right) + 2\phi \left(k\_s - k\_f\right)}{k\_f \left(k\_s + 2k\_f\right) - \phi \left(k\_s - k\_f\right)} \cdot k\_{\rm Brownian} = 5 \times 10^4 \gamma \otimes \left(\rho \mathbb{C}\_p\right)\_f \sqrt{\frac{k\_\beta T}{2\rho\_p r\_p}} \Gamma(T, \phi),$$

where *<sup>k</sup><sup>β</sup>* <sup>=</sup> 1.38 <sup>×</sup> <sup>10</sup>−23m2kgs−2k−<sup>1</sup> is the Boltzmann physical constant and *rp* is the nanoparticle radius.

Particularly,

*γ* = 0.0137(100*φ*) <sup>−</sup>0.8229, where *φ* < 1%; *γ* = 0.0011(100*φ*) <sup>−</sup>0.7272, where *φ* > 1%; 0.01 < *φ* < 0.04 300*K* < *T* < 325*K*.

Taking into account *μn f* reliance on particle volume fraction,

$$\mu\_{nf} = \mu\_{static} + \mu\_{Brownian}$$

where

$$\text{and } \mu\_{\text{static}} = \mu\_f (1 - \phi)^{-2.5} \text{ and } \mu\_{\text{Brownian}} = \frac{k\_{\text{Brownian}}}{k\_f} \frac{\mu\_f}{Pr\_f}$$

The governed equation is formulated as follows [32,36,37]:

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

$$
\mu \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{1}{\rho\_{nf}} \left( \frac{1}{\beta\_1 \varepsilon} + \mu\_{nf} \right) \left( 2 \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 u}{\partial y^2} \right) + \frac{\mu\_0 M}{\rho\_{nf}} \frac{\partial H}{\partial x} \tag{2}
$$

$$\begin{aligned} u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + \lambda\_2 \Omega\_E &= \frac{k\_{nf}}{\left(\rho \mathbb{C}\_p\right)\_{nf}} \left(\frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial x^2}\right) + \frac{1}{\left(\rho \mathbb{C}\_p\right)\_{nf}} \left(\frac{1}{\mathbb{P}\_1 \mathbb{C}} + \mu\_{\mathbb{H}f}\right) \left(\frac{\partial u}{\partial y}\right)^2 \\ &- \frac{\mu\_0 T}{\left(\rho \mathbb{C}\_p\right)\_{nf}} \frac{\partial M}{\partial T} \left(u\frac{\partial H}{\partial x} + v\frac{\partial H}{\partial y}\right) \end{aligned} \tag{3}$$

$$
\mu \frac{\partial \mathcal{C}}{\partial \mathbf{x}} + v \frac{\partial \mathcal{C}}{\partial y} = D\_{\text{nf}} \left( \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial \mathbf{x}^2} \right) - k\_r (\mathcal{C} - \mathcal{C}\_c) \tag{4}
$$

In the above equation, the term Ω*<sup>E</sup>* is defined as

$$
\Omega\_E = u \frac{\partial u}{\partial x} \frac{\partial T}{\partial x} + v \frac{\partial v}{\partial y} \frac{\partial T}{\partial y} + u^2 \frac{\partial^2 T}{\partial x^2} + v^2 \frac{\partial^2 T}{\partial y^2} + 2uv \frac{\partial^2 T}{\partial x \partial y} + u \frac{\partial v}{\partial x} \frac{\partial T}{\partial y} + v \frac{\partial u}{\partial y} \frac{\partial T}{\partial y} \tag{5}
$$

The associated boundary limitations are as follows:

$$\begin{array}{c} \mu = c\mathbf{x}, \ v = 0, \ T = T\_{\text{w}}, \ \mathbf{C} = \mathbf{C}\_{\text{w}} \text{ at } \ y = 0 \\\ \mu \to 0, \ T \to T\_{\text{c}}, \ \mathbf{C} \to \mathbf{C}\_{\text{c}} \text{ at } \ y \to \infty \end{array} \tag{6}$$

The magnetic field affects the presumed liquid flow due to the magnetic dipole, and its magnetic scalar potential is given by

$$\phi\_1 = \frac{\mathbf{x}}{\left(y+a\right)^2 + \mathbf{x}^2} \frac{\gamma}{2\pi} \tag{7}$$

$$H\_y = -\frac{\partial \phi\_1}{\partial y} = \frac{2(y+a)\mathbf{x}}{\left(\left(y+a\right)^2 + \mathbf{x}^2\right)^2} \frac{\gamma}{2\pi},\ H\_y = -\frac{\partial \phi\_1}{\partial \mathbf{x}} = -\frac{\left(y+a\right)^2 - \mathbf{x}^2}{\left(\left(y+a\right)^2 + \mathbf{x}^2\right)^2} \frac{\gamma}{2\pi} \tag{8}$$

where

$$H = \left[ \left( \frac{\partial \phi\_1}{\partial y} \right)^2 + \left( \frac{\partial \phi\_1}{\partial x} \right)^2 \right]^{1/2} \tag{9}$$

We attain that

$$\frac{\partial H}{\partial y} = \left[ \frac{4x^2}{\left(y+a\right)^5} - \frac{2}{\left(y+a\right)^3} \right] \frac{\gamma}{2\pi}, \frac{\partial H}{\partial x} = \left[ -\frac{2x}{\left(y+a\right)^4} \right] \frac{\gamma}{2\pi} \tag{10}$$

Supposing that the applied field *H* is strong enough to saturate the supposed fluid and that the linear equation approximates the variance of magnetisation *M* with temperature *T*,

$$M = K(T\_c - T) \tag{11}$$

The following are some of the similarities:

$$\begin{aligned} (\eta, \xi) &= \sqrt{\frac{\varepsilon}{v\_f}} (y, \mathbf{x}), \ \psi(\eta, \xi) = \begin{pmatrix} \frac{\mu\_f}{\rho\_f} \end{pmatrix} \xi f(\eta) \\\ T = T\_\mathfrak{c} - (T\_\mathfrak{c} - T\_\mathfrak{w}) \theta(\eta, \xi) &= T\_\mathfrak{c} - (T\_\mathfrak{c} - T\_\mathfrak{w}) \left[ \theta\_1(\eta) + \xi^2 \theta\_2(\eta) \right] \end{aligned} \tag{12}$$

The stream function *ψ* is given below:

$$u = \frac{\partial \psi}{\partial y} = \text{cxf}'(\eta), \ v = -\frac{\partial \psi}{\partial x} = -\sqrt{\epsilon \overline{v}\_f} f(\eta) \tag{13}$$

The continuity equation is easily satisfied, while the momentum, thermal equations, and mass transfer are transferred to the relating set of ODEs:

$$\left(\varepsilon\_2 \mathcal{W} \varepsilon + \varepsilon\_1 + \frac{k\_{\text{Brownian}}}{k\_f Pr\_f \varepsilon\_2}\right) f'' - \varepsilon\_2 \frac{2\beta \theta\_1}{\left(\eta + a\right)^4} + f f'' - f'^2 = 0 \tag{14}$$

$$\varepsilon\_3 \frac{k\_{nf}}{k\_f} \frac{1}{Pr} (\theta\_1'' + 2\theta\_2) + f\theta\_1' + \varepsilon\_3 \frac{1}{Pr} \frac{2\lambda\beta}{\left(\eta + a\right)^4} f(\theta\_1 - \varepsilon) - \delta\_t \left(f^2 \theta\_1'' + f f' \theta\_1'\right) = 0 \tag{15}$$

$$\begin{split} \varepsilon\_{3} \frac{k\_{nf}}{k\_f} \frac{1}{Pr} \theta\_2'' + f \theta\_2 \frac{\lambda}{Pr} \Big( \varepsilon\_2 \mathcal{W} \varepsilon + \varepsilon\_1 + \frac{k\_{\text{Reumium}}}{k\_f Pr\_f \varepsilon\_2} \Big) f'^2 - \varepsilon\_3 \frac{\lambda \beta (\theta\_1 - \delta)}{Pr} \Big[ \frac{4f}{\left(\eta + a\right)^5} + \frac{2f'}{\left(\eta + a\right)^4} \Big] \\ + \varepsilon\_3 \frac{1}{Pr} \frac{2\lambda \beta}{\left(\eta + a\right)^3} f \theta\_2 - \delta\_\varepsilon \Big( f^2 \theta\_2'' - 3ff' \theta\_2' - 2ff'' \theta\_2 + 4f'^2 \theta\_2 \Big) \end{split} \tag{16}$$

$$(1 - \phi)^{2.5} \frac{1}{Sc} \left(\chi\_1'' + 2\chi\_2\right) + f\chi\_1' - \sigma\chi\_1 = 0 \tag{17}$$

$$(1 - \phi)^{2.5} \frac{1}{Sc} \chi\_2'' + f \chi\_2' - 2 \chi\_2 f' - \sigma \chi\_2 = 0 \tag{18}$$

where

$$\varepsilon\_1 = \frac{1}{(1-\phi)^{2.5} \left(1-\phi+\phi \frac{\rho\_s}{\rho\_f}\right)}, \; \varepsilon\_2 = \frac{1}{\left(1-\phi+\phi \frac{\rho\_s}{\rho\_f}\right)}, \; \varepsilon\_3 = \frac{1}{\left(1-\phi+\phi \frac{\left(\rho \mathbb{C}\_p\right)\_s}{\left(\rho \mathbb{C}\_p\right)\_f}\right)}\tag{19}$$

Reduced conditions:

$$\begin{array}{c} f(0) = 0, \ f'(0) = 1, \ \theta\_1(0) = 1, \ \theta\_2(0) = 0, \ \chi\_1(0) = 1, \ \chi\_2(0) = 0\\ f'(\infty) \to 0, \ \theta\_1(\infty) \to 0, \ \theta\_2(\infty) \to 0, \ \chi\_1(\infty) \to 0, \ \chi\_2(\infty) \to 0 \end{array} \tag{20}$$

where

$$\begin{cases} \mathbf{a} = \sqrt{\frac{\mathbf{c}}{\mathbf{v}\_f}} \mathbf{a}, \; \boldsymbol{\beta} = \mu\_0 \mathbf{K} \frac{\gamma \mu\_f}{2\pi \mu\_f^2} (T\_\mathbf{c} - T\_\mathbf{w}), \; \mathbf{W} \mathbf{e} = \frac{1}{\overline{\rho\_1 \epsilon} \mu\_f}, \; \delta\_\mathbf{c} = \mathbf{c} \lambda\_\mathbf{2}, \; \delta = \frac{T\_\mathbf{c}}{(T\_\mathbf{c} - T\_\mathbf{w})}\\ \lambda = \frac{\mathbf{c} \mu\_f^2}{k f \rho / (T\_\mathbf{c} - T\_\mathbf{w})}, \; \mathbf{P} \mathbf{r} = \frac{\mu\_f \mathbf{C}\_\mathbf{p}}{k\_f}, \; \mathbf{o} = \frac{k\_\mathbf{r}}{\mathbf{c}}, \; \mathbf{S} \mathbf{c} = \frac{\mathbf{v}\_f}{\mathbf{D}\_f}, \; \mathbf{R} \mathbf{e} = \frac{\mathbf{c} \mathbf{z}^2}{\mathbf{v}\_f} \end{cases} \tag{21}$$

The definitions of the quantities of physical interests are as follows:

$$\mathbf{C}\_{f\_x} = \frac{-2\left(\frac{1}{\delta^x} + \mu\_{nf}\right)\left(\frac{\partial^2 u}{\partial x^2}\right)\_{y=0}}{\rho(cx)^2}, \qquad \mathrm{Nu}\_x = \frac{-\mathrm{xk}\_{nf}\left(\frac{\partial T}{\partial y}\right)\_{y=0}}{\left(T\_w - T\_c\right)},\tag{22}$$

$$\mathrm{Sh}\_x = \frac{-\mathrm{x}\left(\frac{\partial \zeta}{\partial y}\right)\_{y=0}}{\left(\mathrm{C}\_w - \mathrm{C}\_c\right)}\tag{23}$$

The quantities of physical interest corresponding to Equations (12) and (13) transform the following equations:

$$\sqrt{\mathcal{Rce}\_{\mathcal{X}}} \mathcal{C}\_{f\_{\mathcal{X}}} = -\frac{2(1 + W\epsilon)}{\left(1 - \phi\right)^{2.5}} f''(0) \tag{2.3}$$

$$\left(\left(\mathrm{Re}\_{\mathrm{x}}\right)^{-1/2}\mathrm{Nu}\_{\mathrm{x}} = -\frac{k\_{\mathrm{nf}}}{k\_{f}}\left(\theta\_{1}^{\prime}(0) + \xi^{2}\theta\_{2}^{\prime}(0)\right) \tag{24}$$

$$(Re\_x)^{-1/2}Sh\_x = -(1-\phi)^{2.5} \left(\chi\_1'(0) + \mathfrak{J}^2 \chi\_2'(0)\right) \tag{25}$$
