**2. Mathematical Formulation**

Assume a two-dimensional laminar boundary layer incompressible flow with steady characteristics in which Al2O3 and *γ*-Al2O3 nanofluids move across a stretched sheet of various base fluids, such as water and C2H6O2, see from Figure 1. The flow that is related to nanofluids is produced as a result of the sheet being stretched along the *x*-axis by two forces that are identical in magnitude but act in opposite directions. The stretching velocity *uw*(*x*) is used to suppose that the flow is confined to *y* > 0 and external velocity to boundary layer flow is *<sup>U</sup>*∞, such that *<sup>U</sup>*<sup>∞</sup> <sup>=</sup> *ax*, where *a* is constant. It is observed that the ratio of the stretching surface velocity to the inviscid flow at the stagnation point determines the shape of a boundary layer produced in a stagnation-point flow of an incompressible viscous fluid towards a stretching surface. Therefore, the stagnation-point flow is represented by

the term, *<sup>U</sup>*-∞ *dU*-∞ *dx* in the momentum equation of the fluid flow where *<sup>U</sup>*-∞ is the velocity distribution for the free stream which is far away from the surface. The temperature profile at the surface which is being stretched is *<sup>T</sup><sup>ω</sup>* <sup>=</sup> *<sup>T</sup>*<sup>∞</sup> <sup>+</sup> *bx*2, with *b* is constant and *T*∞ is the ambient temperature. In addition, it is assumed that the nanoparticles and the base fluids are in a state of thermal equilibrium and there is a slip between them [21,23,45]. Table 3 summarizes the considered thermo-physical features of nanofluids.

**Figure 1.** Physical description to mathematical model.

**Table 3.** Thermophysical characteristics of Nanofluids.


Under these conditions, we can write down the steady boundary equation that controls the convective flow and heat transfer of nanofluids as:

$$
\mathfrak{u}\_x + \mathfrak{v}\_y = 0,
$$

$$
\overline{u}\overline{u}\_x + \overline{v}\overline{u}\_y = \frac{\mu\_{nf}}{\rho\_{nf}} \frac{\partial^2 \overline{u}}{\partial y^2} + \overline{U}\_{\infty} \frac{d\overline{U}\_{\infty}}{dx} \tag{2}
$$

$$
\hbar \bar{T}\_x + \mathcal{E}\bar{T}\_y = \frac{k\_{nf}}{(\rho \mathbb{C}\_p)\_{nf}} \frac{\partial^2 \mathcal{T}}{\partial y^2} - \frac{1}{\bar{\rho} \mathbb{C}\_p} \frac{\partial \bar{q}\_r}{\partial y},\tag{3}
$$

where *u*˜ and *v*˜ are velocity components along *x* and *y* directions, respectively. Moreover, *ρ*˜ is the density of the fluid and *C*˜ *<sup>p</sup>* represents specific heat at constant pressure. For the reason that the intensity of radiant emission increases with increasing absolute temperature a very crucial factor in the heat transfer process. Therefore, the term <sup>1</sup> *ρ*˜*C*˜ *<sup>p</sup> ∂q*˜*r <sup>∂</sup><sup>y</sup>* represents the thermal radiation in which *q*˜*<sup>r</sup>* is the radiative heat flux. Such heat flux is defined by Rosseland approximation,

$$d\tilde{q}\_r = -\left(\frac{4\bar{\sigma}}{3\bar{k}}\right)\frac{\partial T^4}{\partial y},\tag{4}$$

where ˜ *k* is the Rosseland coefficient of mean absorption and *σ*˜ is the Stefan Boltzmann constant. For the construction of the linear function of temperature *T*4, apply the Taylor series about the free stream temperature *T*∞ and neglect the higher power, we have the following equation,

$$T^4 \approx (4T - 3T\_{\infty})T\_{\infty}^3. \tag{5}$$

Therefore, the radiative heat flux can be expressed as;

$$\eta\_r = -\left(\frac{16\vartheta}{3\bar{k}}\right) T\_{\infty}^3 \frac{\partial T}{\partial y}.\tag{6}$$

Boundary conditions are:

$$
\mu \mathfrak{u} = \mathfrak{u}\_w + \mathfrak{d} \frac{\partial \mathfrak{u}}{\partial y'}, \\
\mathfrak{v} = 0, \\
\mathfrak{T} = T\_\omega (T\_\omega = T\_\infty + \tilde{b}\mathfrak{x}^2) \text{ at } y = 0,\tag{7}
$$

*<sup>u</sup>*˜ <sup>→</sup> 0, *<sup>T</sup>*˜ <sup>→</sup> *<sup>T</sup>*<sup>∞</sup> *as y* <sup>→</sup> <sup>∞</sup> (8)

where, *u* is the tangential velocity of the free fluid which is exterior normal to the stretching sheet and *a*˘ = √ *K*˘ *<sup>α</sup>*˘ in which *<sup>K</sup>*˘ is the permeability, *uw*(*x*) stretching velocity and *α*˘ is a dimensionless parameter which depends only on the properties of the fluid and permeable material.
