**2. Mathematical Formulation**

In the mathematical model presented herein, we incorporated the time-dependent 2D mixed convective flow of H2O\C2H6O<sup>2</sup> based γ − Al2O<sup>3</sup> nanoparticles through a stretched vertical sheet. The viscous dissipation, non-linear radiation and non-uniform heat source/sink were taken as an extra assumption in the energy equation. It was also presumed that the flow was incompressible and that the nanoparticles and the base fluid were in thermal equilibrium. The applied magnetic field (MF) was taken to be timedependent *B* = *B*0/ √ 1 − *Ct* and normal to the flow of surface. In addition, there was no polarization effect, and thus the external electric field was presumed to be zero and the magnetic Reynolds number was presumed to be small (in comparison to the applied MF, the induced MF was negligible). The demarcated values of the thermo-physical properties of the aforementioned nanofluids are shown in Table 1.


**Table 1.** Thermo-physical properties of nanoparticle and base fluids [47].

The coordinate system is assumed in Cartesian form (*x*, *y*, *t*), where the x-axis is run along the stretching sheet and the y-axis is orthogonal to it; *t* symbolizes the time. The velocity and temperature at the stretching sheet are respectively presented as *U<sup>w</sup>* = *ax*/(1 − *Ct*) and *<sup>T</sup><sup>f</sup>* <sup>=</sup> *<sup>T</sup>*<sup>∞</sup> <sup>+</sup> *bx*2/(<sup>1</sup> <sup>−</sup> *Ct*) 2 , where *a*, *b*, are the constants and the capital letter *C* is used for the decelerated and accelerated sheet when *C* < 0 and *C* > 0, respectively. Under these hypotheses, the governing equations for the momentum and heat transfer of nanofluids with thermo-physical properties and unsteady boundary layer convective flow can be explained as:

$$\frac{\partial u\_1}{\partial x} + \frac{\partial v\_1}{\partial y} = 0 \tag{1}$$

$$\frac{\partial u\_1}{\partial t} + v\_1 \frac{\partial u\_1}{\partial y} + u\_1 \frac{\partial u\_1}{\partial x} = \frac{\mu'\_{nf}}{\rho'\_{nf}} \frac{\partial^2 u\_1}{\partial y^2} - \frac{\sigma'\_{nf} \mathbb{B}^2}{\rho'\_{nf}} u\_1 + g' \frac{(\rho' \beta')\_{nf}}{\rho'\_{nf}} (T\_1 - T\_\infty) \tag{2}$$

$$\frac{\partial T\_1}{\partial t} + v\_1 \frac{\partial T\_1}{\partial y} + u\_1 \frac{\partial T\_1}{\partial x} = \frac{k'\_{nf}}{\left(\rho' \varepsilon'\_p\right)\_{nf}} \frac{\partial^2 T\_1}{\partial y^2} - \frac{1}{\left(\rho' \varepsilon'\_p\right)\_{nf}} \left(\frac{\partial q'\_{nf}}{\partial y}\right) + \frac{\mu'}{\left(\rho' \varepsilon'\_p\right)\_{nf}} \left(\frac{\partial u\_1}{\partial y}\right)^2 + \frac{Q\_0}{\left(\rho' \varepsilon'\_p\right)\_{nf}}\tag{3}$$

The approximation of Rosseland for the term nonlinear radiative heat flux is given as:

$$q\_{\;\;r}^{\prime} = -\frac{4\sigma^{\prime\prime}}{3k^{\prime\ast}} \frac{\partial T\_1^4}{\partial y} = -\frac{16\sigma^{\prime\ast}}{3k^{\prime\ast}} T\_1^3 \frac{\partial T\_1}{\partial y} \tag{4}$$

Utilizing Equation (4) in Equation (3), it can defined as:

$$\begin{split} \frac{\partial T\_1}{\partial t} + v\_1 \frac{\partial T\_1}{\partial y} + u\_1 \frac{\partial T\_1}{\partial x} &= \frac{k'\_{nf}}{\left(\rho' c'\_{p}\right)\_{nf}} \frac{\partial^2 T\_1}{\partial y^2} + \frac{16 \sigma' \prime T\_1^2}{3k' \left(\rho' c'\_{p}\right)\_{nf}} \left( T\_1 \frac{\partial}{\partial y} \left(\frac{\partial T\_1}{\partial y}\right) + 3 \left(\frac{\partial T\_1}{\partial y}\right)^2 \right) + \\ \frac{\mu'\_{nf}}{\left(\rho' c'\_{p}\right)\_{nf}} \left(\frac{\partial u\_1}{\partial y}\right)^2 + \frac{Q\_0}{\left(\rho' c'\_{p}\right)\_{nf}}, \end{split} \tag{5}$$

where the last term represents the erratic heat sink/source and is defined as:

$$Q\_0 = \frac{k'\_f \left(T\_f - T\_\infty\right) \mathcal{U}\_w(\mathbf{x}, t)}{\mathbf{x} \nu'\_f} \left(A\_0 f' + B\_0 \left(\frac{T\_1 - T\_\infty}{T\_f - T\_\infty}\right)\right) \tag{6}$$

The boundary conditions are:

$$-k'\_{\,nf} \frac{\partial T\_1}{\partial y} = h\_f \left(T\_f - T\_1\right), \,\mu\_1 = \mathcal{U}\_{\le}(\mathbf{x}, t), \,\upsilon\_1 = 0, \,\text{at } y = 0,\tag{7}$$

$$T\_1 \to T\_{\approx \prime} \,\,\mu\_1 \to 0 \,\,\text{as } y \to \infty.$$

Here, *T*<sup>1</sup> is the temperature, *T*<sup>∞</sup> is the free stream or the cold temperature moving on the right side of the sheet, with a zero free stream velocity, while the left side of the sheet is heated at temperature *T<sup>f</sup>* from a hot fluid owing convection, which offers a coefficient of heat transfer *h<sup>f</sup>* and comprising the expression of thermo-physical properties revealed in Table 2. The interpretations of the rest of the symbols or notations and the mathematical letters in Equation (1) to Equation (7) are presented in Table 3.


**Table 2.** Thermo-physical properties of gamma nanofluids.

Following the non-dimensional similarity variables are:

$$\begin{split} u\_1 &= a x (1 - \mathcal{C}t)^{-0.5} F', \ v\_1 = -\left( v'\_f a (1 - \mathcal{C}t)^{-0.5} \right)^{\frac{1}{2}} F, \\ \eta &= y \left( \frac{a (1 - \mathcal{C}t)^{-0.5}}{v'\_f} \right)^{\frac{1}{2}}, \theta = \frac{T\_1 - T\_\infty}{T\_f - T\_\infty}. \end{split} \tag{8}$$

Using Equation (8) in Equation (2) to Equation (6), along with the boundary condition (7) we get the dimensional form of the momentum equations, as follows:

$$\begin{aligned} \left[ \mathbf{K}\_1 \mathbf{F}'' + \left[ \mathbf{K}\_2 \left( F \mathbf{F}'' - F'^2 - \varepsilon \left( \frac{\eta}{2} F'' + F' \right) \right) - \mathbf{K}\_3 M \mathbf{F}' + \mathbf{K}\_4 \lambda \theta \right] = 0 \\ \text{(for } \gamma \text{Al}\_2 \mathbf{O}\_3 - \text{H}\_2 \mathbf{O}) \end{aligned} \right. \tag{9}$$

$$\begin{aligned} \mathbf{K}\_5 \mathbf{F}'' + \left[ \mathbf{K}\_2 \left( F \mathbf{F}'' - F'^2 - \varepsilon \left( \frac{\eta}{2} F'' + F' \right) \right) - \mathbf{K}\_3 M \mathbf{F}' + \mathbf{K}\_4 \lambda \theta \right] &= \mathbf{0} \\ \qquad \text{(for } \gamma \text{Al}\_2 \mathbf{O}\_3 - \mathbf{C}\_2 \mathbf{H}\_6 \mathbf{O}\_2) \end{aligned} \tag{10}$$


**Table 3.** The list of symbols used and their interpretation.

In which:

$$\begin{aligned} \mathbf{K}\_1 &= \{123\boldsymbol{\phi}^2 + 7.3\boldsymbol{\phi} + 1\}, \mathbf{K}\_2 = (1 - \boldsymbol{\phi} + \boldsymbol{\phi}\left(\frac{\boldsymbol{\rho}\_s'}{\boldsymbol{\rho}\_f'}\right)\}, \mathbf{K}\_3 = \left[\frac{3\boldsymbol{\phi}\left(\frac{\boldsymbol{\sigma}\_s'}{\boldsymbol{\sigma}\_f'} - 1\right)}{\left(\frac{\boldsymbol{\sigma}\_s'}{\boldsymbol{\sigma}\_f'} + 2\right) - \left(\frac{\boldsymbol{\sigma}\_s'}{\boldsymbol{\sigma}\_f'} - 1\right)\boldsymbol{\phi}} + 1\right], \\ \mathbf{K}\_4 &= (1 - \boldsymbol{\phi}) + \boldsymbol{\phi}\frac{(\boldsymbol{\rho}'\boldsymbol{\beta})\_s}{(\boldsymbol{\rho}'\boldsymbol{\beta}')\_f}, \mathbf{K}\_5 = \{306\boldsymbol{\phi}^2 - 0.19\boldsymbol{\phi} + 1\}. \end{aligned}$$

while the corresponding dimensional form of the energy equations for the γAl2O<sup>3</sup> nanoparticle are given as:

$$\begin{aligned} \theta'' \left[ 1 + \frac{4}{3} \mathsf{R}\_d \mathsf{K}\_6 (1 + (\theta\_w - 1)\theta)^3 \right] + 4 \mathsf{R}\_d \mathsf{K}\_6 \left[ \left( 1 + (\theta\_w - 1)\theta \right)^2 \theta'^2 (\theta\_w - 1) \right] + \\ \mathsf{K}\_7 \left\{ \left( F\theta' - 2F'\theta \right) - \varepsilon \left( 2\theta + \frac{\eta}{2}\theta' \right) \right\} + \mathsf{K}\_6 \left( A\_0 \mathsf{F}' + B\_0 \theta \right) + \mathsf{Pr}\_f \mathsf{K}\_1 \mathrm{Ec}(\mathcal{F}'')^2 = 0 \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( \text{for } \gamma \text{Al}\_2 \mathrm{O}\_3 - \mathrm{H}\_2 \mathrm{O} \right) \end{aligned} \tag{11}$$

$$\begin{aligned} \boldsymbol{\theta}^{\nu} \left[ \mathbf{1} + \frac{4}{3} \mathbf{R}\_d \mathbf{K}\_8 (\mathbf{1} + (\boldsymbol{\theta}\_w - \mathbf{1}) \boldsymbol{\theta})^3 \right] + 4 \mathbf{R}\_d \mathbf{K}\_8 \left[ \left( \mathbf{1} + \boldsymbol{\theta} (\boldsymbol{\theta}\_w - \mathbf{1}) \right)^2 \boldsymbol{\theta}^{\prime 2} (\boldsymbol{\theta}\_w - \mathbf{1}) \right] + \\ \mathbf{K}\_9 \left\{ \left( F \boldsymbol{\theta}^{\prime} - 2F^{\prime} \boldsymbol{\theta} \right) - \boldsymbol{\varepsilon} \left( 2 \boldsymbol{\theta} + \frac{\eta}{2} \boldsymbol{\theta}^{\prime} \right) \right\} + \mathbf{K}\_8 \left( A\_0 \mathbf{F}^{\prime} + B\_0 \boldsymbol{\theta} \right) + \mathbf{Pr}\_f \mathbf{K}\_5 \boldsymbol{\varepsilon} \boldsymbol{\varepsilon} (F^{\prime})^2 = \mathbf{0} \\ \boldsymbol{\varepsilon} \boldsymbol{\varepsilon} \end{aligned} \tag{\text{for } \gamma \text{Al}\_2 \mathbf{O}\_3 - \mathbf{C}\_2 \mathbf{H}\_6 \mathbf{O}\_2} \tag{\text{f}}$$

and the appropriate boundary conditions are:

$$\begin{cases} \theta'(0) = -\mathcal{K}\_6 \mathfrak{J} (1 - \theta(0)), \, F'(0) = 1, \, F(0) = 0 \text{ at } \eta = 0, \\\theta(\eta) \to 0, \, F'(\eta) \to 0 \text{ as } \eta \to \infty. \\\ \qquad \text{(for } \gamma \text{Al}\_2\mathcal{O}\_3 - \text{H}\_2\mathcal{O}) \end{cases} \}, \tag{13}$$

$$\begin{cases} \theta'(0) = -\mathsf{K}\_{8}\tilde{\varsigma}(1-\theta(0)), F'(0) = 1, F(0) = 0 \text{ at } \eta = 0, \\ \theta(\eta) \to 0, F'(\eta) \to 0, \text{ as } \eta \to \infty. \\ \text{ (for } \gamma \text{Al}\_2\mathrm{O}\_3 - \mathrm{C}\_2\mathrm{H}\_6\mathrm{O}\_2) \end{cases} \right\}. \tag{14}$$

Where:

$$\begin{aligned} \mathbf{K}\_6 &= \frac{1}{4.97 \phi^2 + 2.72 \phi + 1}, \mathbf{K}\_7 = \frac{\Pr\_f \left( 1 - \phi + \phi \left( \frac{\rho\_s'}{\rho\_f'} \right) \right) \left( 82.1 \phi^2 + 3.95 \phi + 1 \right)}{123 \phi^2 + 7.3 \phi + 1}, \\\ \mathbf{K}\_8 &= \frac{1}{28.905 \phi^2 + 2.8273 \phi + 1}, \mathbf{K}\_9 = \frac{\Pr\_f \left( 1 - \phi + \phi \left( \frac{\rho\_s'}{\rho\_f'} \right) \right) \left( 254.3 \phi^2 - 3 \phi + 1 \right)}{306 \phi^2 - 0.19 \phi + 1}. \end{aligned}$$

For the above equations, the interpretations of the various dimensional parameters are given in Table 4 (for Equation (9) to Equation (14)). The remaining two parameters are the local mixed convection parameter (ratio of the Grashof number and Reynolds number) and the convective parameter and are demarcated as follows:

 $\lambda = \frac{Gr\_{\mathbf{x}}}{\operatorname{Re}^2\_{\times}}, \operatorname{Re}\_{\mathbf{x}} = \frac{\operatorname{xII}\_{\mathbf{u}}}{\nu\_f^{\prime}}, \text{ }$ 
$$Gr\_{\mathbf{x}} = \mathbf{g}^{\prime} \boldsymbol{\beta}^{\prime}\_{\, f} \left( T\_f - T\_{\infty} \right) \mathbf{x}^3 / \nu^{\prime 2}\_{\, f}, \text{ Bi} = \frac{h\_f \sqrt{\mathbf{v}\_f^{\prime}} (1 - \mathbb{C}t)}{k\_f^{\prime} \sqrt{\mathbf{a}}} \tag{15}$$

**Table 4.** The list of parameters used and their values.


In order to find the similarity solution for Equations (9)–(12), it is presumed that [48]

$$\boldsymbol{\beta}'\_{f} = \boldsymbol{m}\_1 \boldsymbol{\mathfrak{x}}^{-1} \text{ and } \boldsymbol{h}\_f = \boldsymbol{m}\_2 (1 - \mathbb{C}t)^{-0.5} \tag{16}$$

where *m*1, *m*<sup>2</sup> are the constants.
