**2. Mathematical Formulation**

The flow of Powell–Eyring nanofluid in a two-dimensional axisymmetric flow across a horizontally stretched sheet is assumed. We are using a system of cylindrical coordinates in which the z-axis, is chosen parallel to the cylinder's axis and the r-axis is chosen perpendicular to the cylindrical surface, as revealed in Figure 1. The cylinder is porous and continually stretches horizontally at *u* = *u<sup>ω</sup>* = *<sup>U</sup>*0*<sup>z</sup> <sup>L</sup>* , where *L* is a characteristic length and *U*<sup>0</sup> > 0. Despite the fact that the moving fluid temperature is set to *T*∞, the cylindrical surface is kept at *Tω*, with the assumption that *T<sup>ω</sup>* > *T*∞. The Buongiorno model and nanofluid contain important slip mechanisms such as thermophoresis diffusion and Brownian motion. The velocity profile for the assumed flow is *V* = [*w*(*r*, *z*), 0, *u*(*r*, *z*)].

**Figure 1.** Sketch for the flow field.

The equations that govern the flow are as follows [10,11]

$$\frac{\partial (rw)}{\partial r} + \frac{\partial (ru)}{\partial z} = 0 \tag{1}$$

$$\begin{split} w \frac{\partial \boldsymbol{u}}{\partial r} + \boldsymbol{u} \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{z}} &= \frac{\mu\_{nf}}{\rho\_{nf}} \left( \frac{\partial^2 \boldsymbol{u}}{\partial r^2} + \frac{1}{r} \frac{\partial \boldsymbol{u}}{\partial r} \right) + \frac{1}{\rho\_{nf} \rho\_{\boldsymbol{c}}} \left( \frac{\partial^2 \boldsymbol{u}}{\partial r^2} + \frac{1}{r} \frac{\partial \boldsymbol{u}}{\partial r} \right) \\ - \frac{1}{6 \rho\_{nf} \beta \boldsymbol{c}^3} \left( \frac{1}{r} \left( \frac{\partial \boldsymbol{u}}{\partial r} \right)^3 + 3 \left( \frac{\partial \boldsymbol{u}}{\partial r} \right)^2 \left( \frac{\partial^2 \boldsymbol{u}}{\partial r^2} \right) \right) - \frac{\mu\_{nf}}{\rho\_{nf} \boldsymbol{K}} \boldsymbol{u} - \frac{\boldsymbol{c}\_b}{\rho\_{nf} \sqrt{\boldsymbol{K}}} \boldsymbol{u}^2 \end{split} \tag{2}$$

$$\begin{split} w\frac{\partial T}{\partial r} + u\frac{\partial T}{\partial z} &= \frac{k\_{nf}}{\left(\rho c\_{p}\right)\_{nf}} \left(\frac{\partial^{2}T}{\partial r^{2}} + \frac{1}{r}\frac{\partial T}{\partial r}\right) + \tau \left(D\_{B}\frac{\partial C}{\partial r}\frac{\partial T}{\partial r} + \frac{D\_{T}}{I\_{nf}}\left(\frac{\partial T}{\partial r}\right)^{2}\right) \\ &+ \frac{1}{\left(\rho c\_{p}\right)\_{nf}} \left(\mu\_{nf}\left(\frac{\partial u}{\partial r}\right)^{2} + \frac{1}{\tilde{\rho}c}\left(\frac{\partial u}{\partial r}\right)^{2} - \frac{1}{6\tilde{\rho}c^{3}}\left(\frac{\partial u}{\partial r}\right)^{4}\right) - \frac{1}{\left(\rho c\_{p}\right)\_{nf}} \frac{\partial}{\partial r} \left(-\frac{16\sigma^{\*}}{3k^{\*}k\_{nf}}T^{3}\frac{\partial T}{\partial r}\right) \end{split} \tag{3}$$

$$
\Delta w \frac{\partial \mathcal{C}}{\partial r} + u \frac{\partial \mathcal{C}}{\partial z} = D\_B \left( \frac{\partial^2 \mathcal{C}}{\partial r^2} + \frac{1}{r} \frac{\partial \mathcal{C}}{\partial r} \right) + \frac{D\_T}{T\_\infty} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right) \tag{4}
$$

With boundary conditions

$$\begin{array}{ccccc} \mu = u\_{\omega} = \frac{\underline{U\_0}z}{L}, w = 0, T = T\_{\omega \prime} \mathbb{C} = \mathbb{C}\_{\omega} \text{ at } & r = R\\ \mu \to 0, T \to T\_{\infty \prime} \mathbb{C} \to \mathbb{C}\_{\infty} \text{ at } & r \to \infty \end{array} \tag{5}$$

Illustrations of the description of the various symbols are shown in Table 1. The fundamental equations can be transformed using the following transforms [21]:

$$\psi(r,z) = \sqrt{u\_{\omega}\mu\_f z}Rf(\eta), \eta = \frac{r^2 - R^2}{2R} \sqrt{\frac{u\_{\omega}}{\mu\_f z}}, \theta(\eta) = \frac{T - T\_{\infty}}{T\_{\omega} - T\_{\infty}}, \phi(\eta) = \frac{\mathcal{C} - \mathcal{C}\_{\infty}}{\mathcal{C}\_{\omega} - \mathcal{C}\_{\infty}} \tag{6}$$

**Table 1.** A description of the multiple symbols that appear in the governing equations is illustrated.


Equation (7) identifies the components of velocity

$$u = \frac{1}{r} \frac{\partial \psi(r, z)}{\partial r}, w = -\frac{1}{r} \frac{\partial \psi(r, z)}{\partial z} \tag{7}$$

The nanofluid expressions are given by [25]:

$$\begin{array}{c} \mu\_{nf} = \mu\_f (1 - \phi)^{-2.5} \\ \rho\_{nf} = \rho\_f (1 - \phi) + \phi \rho\_s \\ \left(\rho c\_P\right)\_{nf} = \left(\rho c\_P\right)\_f (1 - \phi) + \phi \left(\rho c\_P\right)\_s \\ k\_{nf} = k\_f \left[\frac{k\_s + 2k\_f - 2\phi \left(k\_f - k\_s\right)}{k\_s + 2k\_f + 2\phi \left(k\_f - k\_s\right)}\right] \end{array} \tag{8}$$

The base fluid and nanofluid dynamic viscosity are denoted by *μ<sup>f</sup>* and *μn f* , respectively, where *φ* signifies volume fraction.

The equation of incompressibility is fulfilled identically, whereas Equations (2)–(5) are reduced to

$$\begin{aligned} &(1+2\eta\gamma)\left(\frac{1}{\left(1-\phi\_1\right)^{2.5}}+a\right)f''''-\lambda a(1+2\eta\gamma)^2f''^2f'''+2\gamma\left(\frac{1}{\left(1-\phi\_1\right)^{2.5}}+a\right)f''\\ &-\frac{4}{3}a\lambda\gamma(1+2\eta\gamma)f''''-\frac{\beta\_0}{\left(1-\phi\_1\right)^{2.5}}f'-F\_rf'^2+\left(\left(1-\phi\_1\right)+\phi\_1\frac{\rho\_t}{\rho\_f}\right)\left(f f''-f'^2\right)=0 \end{aligned} \tag{9}$$

$$\begin{split} \frac{k\_{\eta}/k\_{f}}{\mathcal{V}} & \left( (1+2\eta\gamma)\theta'' + \gamma\theta' \right) + \mathrm{Ec}(1+2\eta\gamma) \left( \left( \frac{1}{(1-\phi\_{1})^{2}} + a \right) f''^{2} - \frac{1}{3} \lambda a (1+2\eta\gamma) f''^{4} \right) \\ & + \left( (1-\phi\_{1}) + \phi\_{1} \frac{\left( \kappa c\_{\mathrm{p}} \right)\_{s}}{\left( \kappa c\_{\mathrm{p}} \right)\_{f}} \right) \left( \mathrm{Nt}(1+2\eta\gamma)\theta'^{2} + \mathrm{Nb}(1+2\eta\gamma)\theta' \phi' + f\theta' \right) \\ & \times \mathrm{R} \quad \left( \mathrm{ $\omega$ } - \mathrm{ $\omega$ } \right) \left( (\rho(0.0-1) + \lambda^{3})^{\mathrm{d}} + \lambda^{(4)} (\rho(0.-1) + \lambda)^{2} (\rho(-1) + \lambda)^{2} \right) \end{split} \tag{10}$$

+ *Rd Prkn f* /*k <sup>f</sup>* (1 + 2*ηγ*) -(*θ*(*θω* − 1) + 1) 3 *θ* + 3(*θ*(*θω* − 1) + 1) 2 (*θω* − <sup>1</sup>)*θ*<sup>2</sup> = 0 *Nt*

$$\frac{Nt}{Nb}\frac{1}{Sc}\left((1+2\eta\gamma)\theta''+\gamma\theta'\right)+\frac{1}{Sc}((1+2\eta\gamma)\phi''+\gamma\phi')+f\phi'=0\tag{11}$$

$$\begin{array}{l} f(0) = 0, f'(0) = 1, \theta(0) = 1, \phi(0) = 1 \\ f'(\infty) \to 0, \theta(\infty) \to 0, \phi(\infty) \to 0 \end{array} \tag{12}$$

where *γ* = *<sup>v</sup> <sup>f</sup> <sup>L</sup> <sup>U</sup>*0*R*<sup>2</sup> represents the curvature parameter, *<sup>α</sup>* <sup>=</sup> <sup>1</sup> *<sup>μ</sup><sup>f</sup> <sup>β</sup><sup>c</sup>* and *<sup>λ</sup>* <sup>=</sup> *<sup>U</sup>*<sup>0</sup> <sup>3</sup>*z*<sup>2</sup> 2*L*3*c*<sup>2</sup>*vf* are fluid parameters, *β*<sup>0</sup> = *Lv <sup>ρ</sup> <sup>f</sup> <sup>U</sup>*0*<sup>K</sup>* represents the porosity parameter, *Fr* <sup>=</sup> *cbz ρ f* <sup>√</sup>*<sup>K</sup>* is the inertia coefficient, *Ec* = *<sup>u</sup>ω*<sup>2</sup> *cp*(*Tω*−*T*∞) is the Eckert number, *Rd* <sup>=</sup> <sup>16</sup>*σ*∗*T*∞<sup>3</sup> 3*k*∗*k <sup>f</sup>* denotes the radiation parameter, *Pr* = *<sup>μ</sup>cp <sup>k</sup> <sup>f</sup>* denotes the Prandtl number, *θω* <sup>=</sup> *<sup>T</sup><sup>ω</sup> <sup>T</sup>*<sup>∞</sup> is the temperature ratio, *Sc* <sup>=</sup> <sup>V</sup>*<sup>f</sup> DB* is the Schmidt number, *Nt* <sup>=</sup> *<sup>τ</sup>DB*(*Tω*−*T*∞) *vf* and *Nb* = *<sup>τ</sup>DB*(*Cω*−*C*∞) *vf* represent the thermophoresis and Brownian diffusion parameters, respectively.
