**2. Problem Formulation**

The thin layer flow of a nanofluid over a nonlinear radially extending porous disc in an axially symmetric form has been assumed. The extending disc has been kept at *z* = 0. The nanofluid thickness is regulated to the thin layer with the breadth *z* = *h* where *h* is the thin layer thickness (Figure 1). The porous disc is stretching with a nonlinear velocity *Uw* = *ar<sup>n</sup>* where *n* is the integer such that *n* > 0. The applied magnetic field is assumed in a vertical direction to the flow phenomena. The pressure is considered as constant. All others assumptions for the flow phenomena are used as in [24–26]. The leading equations are considered as:

$$\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0 \tag{1}$$

$$
\mu \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} = \frac{\mu}{\rho} \frac{\partial^2 u}{\partial z^2} - \frac{\sigma B\_0^2}{\rho} u - \frac{1}{\rho} \left(\frac{\nu}{k} + Fu\right) u \tag{2}
$$

$$
\mu \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} = \frac{k}{\rho c\_p} \frac{\partial^2 T}{\partial z^2} + \frac{\mu}{\rho c\_p} \left(\frac{\partial u}{\partial z}\right)^2 - \frac{1}{\rho c\_p} \frac{\partial q\_r}{\partial z} + \frac{Q\_0}{\rho c\_p} \left(T\_0 - T\_{\text{ref}}\right) \tag{3}
$$

Here *u*, *v*, *B*0, *F* = <sup>√</sup> *Cb Bx* , *Cb*, *qr*, *<sup>Q</sup>*0, <sup>ρ</sup>*cp*, *Cb*, *<sup>k</sup>*, <sup>σ</sup>, <sup>ρ</sup>, <sup>μ</sup> are the components of velocity in their corresponding directions, induced magnetic strength, inertial coefficient of a permeable medium, drag coefficient, radiative heat flux, heat source/sink, effective heat capacity, thermal conductivity, kinematic viscosity, electrical conductivity, the electrical conductivity, density, and dynamic viscosity, respectively.

**Figure 1.** Geometrical illustration of the problem.

The *qr* is defined as:

$$q\_r = -\frac{4\sigma^\*}{3k^\*} \frac{\partial T^4}{\partial z} \tag{4}$$

By Taylor's expansion, *T*<sup>4</sup> can be written as:

$$T^4 = 4T\_{\text{ref}}^3 T - T\_{\text{ref}}^4 \tag{5}$$

In observation of Equations (4) and (5), Equation (3) is reduced as:

$$u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} = \frac{1}{\rho c\_p} \Big| k + \frac{16\sigma^\* T\_{\text{ref}}^3}{3k^\*} \Big| \frac{\partial^2 T}{\partial z^2} + \frac{\mu}{\rho c\_p} \Big(\frac{\partial u}{\partial z}\Big)^2 + \frac{Q\_0}{\rho c\_p} (T\_0 - T\_{\text{ref}}) \tag{6}$$

The following physical conditions are defined for the nanofluid flow:

$$\begin{array}{l} \mu = ar^n, w = 0, \theta = \theta\_w \text{ at } z = 0\\ \mu \frac{\partial u}{\partial z} = \frac{\partial \theta}{\partial z} = 0, w = \frac{\text{null}}{\text{d}r} \text{ at } z = h \end{array} \tag{7}$$

The <sup>ψ</sup>(*r*, *<sup>z</sup>*) <sup>=</sup> <sup>−</sup>*<sup>r</sup>* √ <sup>2</sup>*Uw* Re *<sup>f</sup>*(η) and <sup>η</sup> <sup>=</sup> *<sup>z</sup> r* √ Re are developed for the similarity transformations in such a way that the components of velocity (*u*, *w*) along the radial direction and axial direction have been converted as:

$$\begin{array}{l} \mu = -\frac{1}{r} \frac{\partial \psi(r, z)}{\partial z} = \mathcal{U}\_{w} f'(\eta) \\ w = \frac{1}{r} \frac{\partial \psi(r, z)}{\partial r} = -\frac{\mathcal{U}\_{w}}{\sqrt{\text{Re}}} \left[ \left( \frac{3 + n}{2} \right) f(\eta) + \left( \frac{n - 1}{2} \right) \eta f'(\eta) \right] \\ T = T\_0 - T\_{\text{ref}} \left( \frac{\mathcal{U}\_{w}^{2}}{2 \text{ref}\_{f}} \right) \mathfrak{d}(\eta) \end{array} \tag{8}$$

here, Re = *rUw* <sup>ν</sup>*<sup>f</sup>* is the Reynolds number.

The transformed velocity and temperature equations are:

$$f^{\prime\prime\prime} + \left(\frac{3+n}{2}\right) f f^{\prime\prime} - n(f^{\prime})^2 - Mf^{\prime} - (\kappa + Frf^{\prime})f^{\prime} = 0\tag{9}$$

$$(1+R)\theta'' + \Pr\left[\left(\frac{3+n}{2}\right)f\theta' - 2n(\theta f')\right] + Ec\Pr\left(f'\right)^2 - \gamma\theta = 0\tag{10}$$

with boundary conditions:

$$f(0) = 0, f'(0) = 1, \theta(0) = 1,\\ f''(\beta) = f(\beta) = \ \theta \nu(\beta) = 0 \tag{11}$$

In Equations (9)–(11), β = *<sup>h</sup>* √ Re *<sup>r</sup>* represents the fluid layer thickness, *<sup>M</sup>* <sup>=</sup> *<sup>r</sup>*σ*B*<sup>2</sup> 0 <sup>ρ</sup>*Uw* indicates the magnetic field parameter, *Fr* = *rCb Uw* √ *Bx* represents the coefficient of inertia, <sup>κ</sup> <sup>=</sup> *<sup>r</sup>*<sup>ν</sup> *kUw* represents the porosity parameter, *<sup>R</sup>* <sup>=</sup> <sup>16</sup>*r*2σ∗*T*<sup>3</sup> ref*T* 3*U*<sup>2</sup> *wkk*<sup>∗</sup> represents the thermal radiation parameter where <sup>σ</sup><sup>∗</sup> is the Boltzmann constant and *<sup>k</sup>*<sup>∗</sup> is the coefficient of absorption, Pr = <sup>μ</sup>*cp <sup>k</sup>* indicates the Prandtl number, Ec <sup>=</sup> *<sup>U</sup>*<sup>2</sup> *w* (*cp*)Δ*<sup>T</sup>* represents the Eckert number, and <sup>γ</sup> <sup>=</sup> *<sup>r</sup>*2*Q*<sup>0</sup> *U*2 *<sup>w</sup>*ρ*cp* represents the heat source/sink.

The skin friction and Nusselt number are defined as:

$$\begin{cases} \frac{\sqrt{\text{Re}}}{2} \mathbb{C}f = -f''(0) \\ \frac{1}{\sqrt{\text{Re}}} \text{N}\mu = -(1+R)\theta'(0) \end{cases} \tag{12}$$

#### **3. HAM Solution**

The HAM technique is used to solve the modeled Equations (9) and (10) with the following procedure.

The primary guesses are picked as follows:

$$f\_0(\mathfrak{n}) = \mathfrak{n} \text{ } \mathfrak{G}\_0(\mathfrak{n}) = 1 \tag{13}$$

The *Lf* and *L*<sup>θ</sup> are selected as:

$$L\_f(f) = f^{\prime\prime\prime},\ L\_\emptyset(\theta) = \theta^{\prime\prime} \tag{14}$$

The resultant non-linear operators *Nf* and *N*<sup>θ</sup> are specified as:

$$\begin{array}{c} N\_f[f(\eta; \mathsf{N})] = \frac{\mathrm{d}^3 f(\eta; \mathsf{N})}{\mathrm{d}\eta^3} + \left(\frac{3+\mathrm{\eta}}{2}\right) f(\eta; \mathsf{N}) \frac{\mathrm{d}^2 f(\eta; \mathsf{N})}{\mathrm{d}\eta^2} - \eta \left(\frac{\mathrm{d} f(\eta; \mathsf{N})}{\mathrm{d}\eta}\right)^2 \\\ -\mathrm{M} \frac{\mathrm{d} f(\eta; \mathsf{N})}{\mathrm{d}\eta} - \left(\mathrm{x} + Fr \frac{\mathrm{d} f(\eta; \mathsf{N})}{\mathrm{d}\eta}\right) \frac{\mathrm{d} f(\eta; \mathsf{N})}{\mathrm{d}\eta} \end{array} \tag{15}$$

$$\begin{split} N\_{\Theta}[\Theta(\eta;\mathsf{N}),f(\eta;\mathsf{N})] &= (1+R)\frac{d^{2}\theta(\eta;\mathsf{N})}{d\eta^{2}} + \mathrm{Pr}\left[\begin{array}{c} \frac{3+n}{2}f(\eta;\mathsf{N})\frac{\mathrm{d}\varPhi(\eta;\mathsf{N})}{\mathrm{d}\eta} \\ -2n\Big(\Theta(\eta;\mathsf{N})\frac{\mathrm{d}f(\eta;\mathsf{N})}{\mathrm{d}\eta}\Big) \end{array}\right] \\ &+ \mathrm{EcPr}\Big(\frac{\mathrm{d}f(\eta;\mathsf{N})}{\mathrm{d}\eta}\Big)^{2} - \gamma\Theta(\eta;\mathsf{N}) \end{split} \tag{16}$$

The zeroth-order problem from Equations (9) and (10) are:

$$\left[ (1 - \aleph) L\_f \left[ f(\mathfrak{n}; \aleph) - f\_0(\mathfrak{n}) \right] \right] = \aleph \hbar\_f N\_f \left[ f(\mathfrak{n}; \aleph) \right] \tag{17}$$

$$\left[ (1 - \aleph) L\_{\emptyset} \left[ \theta(\eta; \aleph) - \theta\_{\emptyset}(\eta) \right] \right] = \aleph \hbar\_{\emptyset} N\_{\emptyset} \left[ \theta(\eta; \aleph), f(\eta; \aleph) \right] \tag{18}$$

The converted boundary conditions are:

$$\begin{array}{c} f(\mathfrak{n}; \mathfrak{N}) \Big|\_{\mathfrak{\eta}=0} = 0, \ \frac{df(\mathfrak{n}; \mathfrak{N})}{\mathrm{d}\mathfrak{\eta}} \Big|\_{\mathfrak{\eta}=0} = 1, \ \mathfrak{d}(\mathfrak{n}; \mathfrak{N}) \Big|\_{\mathfrak{\eta}=0} = 1\\ f(\mathfrak{n}; \mathfrak{N}) \Big|\_{\mathfrak{\beta}} = \frac{\mathrm{d}f^{2}(\mathfrak{\eta}; \mathfrak{N})}{\mathrm{d}\mathfrak{\eta}^{2}} \Big|\_{\mathfrak{\beta}} = \left. \frac{\mathrm{d}\mathfrak{d}(\mathfrak{\eta}, \mathfrak{N})}{\mathrm{d}\mathfrak{\eta}} \right|\_{\mathfrak{\beta}} = 0 \end{array} \tag{19}$$

For ℵ = 0 and ℵ = 1 we can write:

$$\begin{aligned} f(\mathfrak{n};0) &= f\_0(\mathfrak{n}), \; f(\mathfrak{n};1) = f(\mathfrak{n})\\ \Theta(\mathfrak{n};0) &= \Theta(\mathfrak{n}), \; \Theta(\mathfrak{h};1) = \Theta(\mathfrak{n}) \end{aligned} \tag{20}$$

When ℵ fluctuates form 0 to 1, the initial solutions vary to the final solutions. Then, by Taylor's series, we have:

$$\begin{aligned} f(\mathfrak{n}; \mathsf{N}) &= f \mathfrak{o}(\mathfrak{n}) + \sum\_{q=1}^{\infty} f\_{\mathfrak{l}}(\mathfrak{n}) \mathsf{N}^{q} \\ \mathfrak{e}(\mathfrak{n}; \mathsf{N}) &= \mathfrak{e}\_{\mathfrak{0}}(\mathfrak{n}) + \sum\_{q=1}^{\infty} \mathfrak{e}\_{\mathfrak{q}}(\mathfrak{n}) \mathsf{N}^{q} \end{aligned} \tag{21}$$

where

$$f\_{\boldsymbol{\eta}}(\boldsymbol{\eta}) = \left. \frac{1}{q!} \frac{\mathrm{d}f(\boldsymbol{\eta}; \boldsymbol{\aleph})}{\mathrm{d}\boldsymbol{\eta}} \right|\_{\boldsymbol{\pi}=\boldsymbol{0}} \text{ and } \theta\_{\boldsymbol{q}}(\boldsymbol{\eta}) = \left. \frac{1}{q!} \frac{\mathrm{d}\theta(\boldsymbol{\eta}; \boldsymbol{\aleph})}{\mathrm{d}\boldsymbol{\eta}} \right|\_{\boldsymbol{\pi}=\boldsymbol{0}} \tag{22}$$

The series (21) at ℵ = 1 converges, we obtain:

$$\begin{aligned} f(\mathfrak{n}) &= f\_0(\mathfrak{n}) + \sum\_{q=1}^{\infty} f\_q(\mathfrak{n}) \\ \Theta(\mathfrak{n}) &= \Theta\_0(\mathfrak{n}) + \sum\_{q=1}^{\infty} \Theta\_q(\mathfrak{n}) \end{aligned} \tag{23}$$

The *qth*− order gratifies the succeeding:

$$\begin{aligned} L\_f \left[ f\_q(\eta) - \chi\_q f\_{q-1}(\eta) \right] &= \hbar\_f V\_q^f(\eta) \\ L\_\Theta \left[ \Theta\_q(\eta) - \chi\_q \Theta\_{q-1}(\eta) \right] &= \hbar\_\Theta V\_q^\Theta(\eta) \end{aligned} \tag{24}$$

with boundary conditions:

$$\begin{cases} f\_q(0) = f\_q'(0) = 0, \theta\_q(0) = 0\\ f\_q''(\pounds) = f\_q(\pounds) = \theta\_q'(\pounds) = 0 \end{cases} \tag{25}$$

here

$$V\_q^f(\eta) = f\_{q-1}^{\prime\prime} + \left(\frac{3+n}{2}\right) \sum\_{k=0}^{q-1} f\_{q-1-k} f\_k^{\prime\prime} - n \left(f\_{q-1}^{\prime}\right)^2 - M f\_{q-1}^{\prime} - \left(\kappa + F r f\_{q-1}^{\prime}\right) f\_{q-1}^{\prime} \tag{26}$$

$$V\_q^{\mathfrak{G}}(\eta) = (1+R)\mathfrak{G}\_{q-1}^{\prime\prime} + \text{Pr}\left[ \left( \frac{3+n}{2} \right) \sum\_{k=0}^{q-1} f\_{q-1-k} \mathfrak{G}\_k^{\prime} - 2n \left| \sum\_{k=0}^{q-1} \mathfrak{G}\_{q-1-k} f\_k^{\prime} \right| \right] + \text{EcPr} \{f\_{q-1}^{\prime\prime} \}^2 - \gamma \mathfrak{G}\_{q-1} \tag{27}$$

where

$$\chi\_{\eta} = \begin{cases} 0, \text{ if } \aleph \le 1 \\ 1, \text{ if } \aleph > 1 \end{cases} \tag{28}$$
