**2. Problem Formulation**

In the current exploration, the buoyancy flow through a slender revolution bodies containing Molybdenum-Disulfide Graphene Oxide along with the generalized hybrid nanofluid entrenched in a saturated porous medium is shown schematically in Figure 1. To optimize the heat transport, the impact of radiation in the occurrence of the opposing and the assisting flows is analyzed. It is assumed that the ambient velocity is considered as *ue*(*x*) = *U*∞*x <sup>m</sup>* with constant *<sup>U</sup>*∞, while *<sup>T</sup>*<sup>∞</sup> the constant free stream temperature. The temperature of the slender revolution body is taken as *Tw*(*x*) with *Tw*(*x*) > *T*<sup>∞</sup> utilizes for (ASSF) and *Tw*(*x*) < *T*<sup>∞</sup> uses for (OPPF). The system of coordinate from the slender body vortex is at origin, while the distance along the revolution body and normal to the slender revolution body is presented by the cylindrical coordinate (*x*,*r*). Applying the single-phase model suggested by the Tiwari and Das [30] with the approximations of Boussinesq and boundary layer scaling, the leading equations are

$$\frac{\partial(ru)}{\partial \mathbf{x}} + \frac{\partial(vr)}{\partial r} = \mathbf{0} \tag{1}$$

$$\frac{\mu\_{\rm HNF}}{\mu\_{\rm F}} \frac{\partial \mu}{\partial r} = \frac{\rm Kg(\rho \beta)\_{\rm F}}{\mu\_{\rm F}} \frac{(\rho \beta)\_{\rm HNF}}{(\rho \beta)\_{\rm F}} \frac{\partial T}{\partial r} \tag{2}$$

$$u\frac{\partial T}{\partial \mathbf{x}} + v\frac{\partial T}{\partial r} = \frac{k\_{\rm HNF}}{\left(\rho c\_p\right)\_{\rm HNF} r} \frac{\partial}{\partial r} \bigg(r\frac{\partial T}{\partial r}\bigg) - \frac{1}{\left(\rho c\_p\right)\_{\rm HNF}} \frac{\partial}{\partial r} (r\eta\_r) \tag{3}$$

with the corresponding boundary conditions

$$\begin{cases} \boldsymbol{v} = \mathbf{0}, \; T = T\_w(\mathbf{x}) = T\_{\infty} + \mathbf{A} \mathbf{x}^{\lambda} \text{ at } \mathbf{r} = \mathbf{R}(\mathbf{x}),\\ \boldsymbol{u} = \boldsymbol{u}\_{\varepsilon}(\mathbf{x}) = \mathbf{U}\_{\infty} \mathbf{x}^{m}, \; T = T\_{\infty} \text{ as } \mathbf{r} \to \infty. \end{cases} \tag{4}$$

Integrating Equation (2) and utilizing Equation (4), it becomes

$$\frac{\mu\_{\rm HNF}}{\mu\_{\rm F}}u = \frac{\mu\_{\rm HNF}}{\mu\_{\rm F}}u\_{\varepsilon} + \frac{\rm Kg\beta\_{\rm F}}{\nu\_{\rm F}} \frac{(\rho\beta)\_{\rm HNF}}{(\rho\beta)\_{\rm F}}(T - T\_{\infty}) \tag{5}$$

Here *v*, *u* signify Darcy's law velocity components in (*r*, *x*) directions, *T* temperature of the hybrid nanofluid, *g* acceleration owing to gravity, µ*HNF* hybrid viscosity, µ*<sup>F</sup>* base fluid viscosity, *K* permeability of the porous medium, *<sup>k</sup>HNF* hybrid nanofluid thermal conductivity, ρ*c<sup>p</sup> HNF* hybrid nanofluid heat capacitance, (ρβ)*HNF* hybrid nanofluid thermal expansion, and *R*(*x*) surface shape of the axisymmetric body.

**Figure 1.** Physical diagram of the problem.

For radiation effect, the Rosseland approximation is illustrated as

$$q\_r = -\frac{4\gamma\_1}{3k\_1} \left(\frac{\partial T^4}{\partial r}\right) \tag{6}$$

where *k*<sup>1</sup> and γ<sup>1</sup> signify coefficient of mean absorption and Stefan–Boltzmann, respectively. Applying Taylor series to expand *T* <sup>4</sup> about *<sup>T</sup>*<sup>∞</sup> and prohibiting the terms involving higher-order, one gets

$$T^4 \cong 4TT\_{\infty}^3 - 3T\_{\infty}^4 \tag{7}$$

The similarity variables for further analysis are introduced as

$$\begin{split} \psi &= \alpha\_{\text{F}} \text{xF}(\eta), \ \eta = Pe\_{\text{x}} \frac{r^{2}}{\mathfrak{x}^{2}} = \frac{\mathcal{U}\_{\text{os}} r^{2} \mathfrak{x}^{m-1}}{\alpha\_{\text{F}}}, \ \theta(\eta) = (T - T\_{\text{os}}) / (T\_{\text{w}} - T\_{\text{os}}), \\ \mathcal{P}e\_{\text{F}} &= \frac{\mathcal{U}\_{\text{os}} \mathfrak{x}^{m+1}}{\alpha\_{\text{F}}}, \ \mathfrak{u} = 2u\_{\text{c}} \mathcal{F}, \ \mathcal{v} = \frac{\alpha\_{\text{F}}}{r} \eta (1 - m) \mathcal{F}' - \frac{\alpha\_{\text{F}}}{r} \mathcal{F}, \ \mathcal{R}\_{\text{d}} = \frac{4\gamma\_{1} T\_{\text{c}}^{3}}{k\_{1} k\_{\text{F}}}. \end{split} \tag{8}$$

Equation (1) is identically true and Equation (3) and Equation (5) are transformed to

$$\frac{\mu\_{\rm HNF}}{\mu\_{\rm F}}(2F'-1) - \frac{\mathcal{K}g\mathcal{\mathcal{S}}\_{\rm F}\mathfrak{x}^{\lambda-m}}{\nu\_{\rm F}\mathcal{U}\_{\rm \infty}}\frac{(\rho\mathcal{\beta})\_{\rm HNF}}{(\rho\mathcal{\beta})\_{\rm F}}\theta = 0\tag{9}$$

$$\left(\frac{k\_{\rm HNF}}{k\_{\rm F}} + \frac{4}{3}\mathcal{R}\_{\rm d}\right) \left(2\eta \theta^{\prime\prime} + 2\theta^{\prime}\right) + \frac{\left(\rho c\_{p}\right)\_{\rm HNF}}{\left(\rho c\_{p}\right)\_{\rm F}} \left(\theta^{\prime}\mathcal{F} - \lambda \mathcal{F}^{\prime}\theta\right) = 0\tag{10}$$

It is perceptible that Equation (9) and Equation (10) will consent the similarity solutions if the power of *x* in Equation (9) disappears, i.e.,:

$$
\mathfrak{m} = \lambda \tag{11}
$$

*Crystals* **2020**, *10*, 771

With this classified condition, Equation (9) can be rewritten as

$$\frac{\mu\_{\rm HNF}}{\mu\_{\rm F}}(2F'-1) - \xi \frac{(\rho \beta)\_{\rm HNF}}{(\rho \beta)\_{\rm F}} \theta = 0 \tag{12}$$

where the dimensionless constraint involved in the aforementioned equations is mathematically expressed as

$$\begin{array}{l} \xi = \frac{Ra}{Pe\_{\mathcal{X}}} = \frac{Kg\beta\_{F}(T\_{\mathcal{w}} - T\_{\infty})\mathbf{x}}{\nu\_{F}\alpha\_{F}} \frac{\alpha\_{F}}{L l\_{\infty}\mathbf{x}^{m+1}} = \frac{Kg\beta\_{F}A}{\nu\_{F}lL\_{\infty}}, \; Ra = \frac{Kg\beta\_{F}(T\_{\mathcal{w}} - T\_{\infty})\mathbf{x}}{\nu\_{F}\alpha\_{F}},\\ \text{Pr}\_{\mathcal{X}} = \frac{\mathcal{U}\_{\infty}\mathbf{x}^{m+1}}{\alpha\_{F}}, \; \text{Pr} = \frac{\nu\_{F}}{\alpha\_{F}}, \; \mathcal{R}\_{d} = \frac{4\mathcal{V}\_{1}T\_{\infty}^{3}}{k\_{1}k\_{F}}. \end{array}$$

and the interpretation of these constraints are the mixed convection parameter, the local Rayleigh number for a porous medium, the Peclet number, the Prandtl number and the radiation parameter, respectively.

Placing η = *b*, where *b* is constant and utilized for a slender body, it is numerically small. Equation (9) stipulated the body size as well as body shape with surface is defined via

$$R(\mathbf{x}) = \left(\frac{\nu\_F \alpha\_F b \mathbf{R} a}{P e\_\mathbf{x} K g \beta\_F A}\right) \frac{1}{\mathbf{x}} \mathbf{x} \frac{(1 - \lambda)}{2} \text{ OR}\left(\frac{b}{P e\_\mathbf{x}}\right) \overline{\mathbf{2}} \mathbf{x} \tag{13}$$

The problems concerning the realistic interest, the amount of λ ≤ 1. For instance, λ = 1, λ = 0 and λ = −1 represent the cylinder, paraboloid, and cone shape bodies.

The boundary restrictions are

$$\begin{aligned} b(1-m)F' - F &= 0, \; \theta = 1 \text{ at } \eta = b, \\ F' &\to 0.5, \; \theta \to 0 \text{ as } \eta \to \infty. \end{aligned} \tag{14}$$

If Equation (10) and Equation (12) are merged, one gets

$$\left(\frac{k\_{\rm HNF}}{k\_F} + \frac{4}{3}\mathcal{R}\_d\right) \mathrm{(4\eta F'''' + 4F')} + \frac{\left(\rho c\_p\right)\_{\rm HNF}}{\left(\rho c\_p\right)\_F} \mathrm{\left(2FF'' - 2\lambda F'^2 + \lambda F'\right)} = 0\tag{15}$$

along with the modified boundary restrictions

$$b(1-m)F'(b) - F(b) = 0,\\ \frac{\mu\_{\rm HNF}}{\mu\_{\rm F}}(2F'(b) - 1) = \xi \frac{(\rho \beta)\_{\rm HNF}}{(\rho \beta)\_{\rm F}},\\ F'(\infty) \to 0.5. \tag{16}$$

The quantities of practical interest and to measure the liquid behaviors is the skin friction which is explained as

$$\mathbf{C}\_{\rm F} = \frac{\tau\_{\rm w}}{\rho u\_{\varepsilon}^{2}} = \frac{\mu\_{\rm HNF} \frac{\partial u}{\partial r}\Big|\_{r=R(\mathbf{x})}}{\rho u\_{\varepsilon}^{2}} \tag{17}$$

The skin friction in the dimensional form is

$$\frac{P e\_\chi^{0.5}}{\text{Pr}} \mathbf{C}\_F = 4 \frac{\mu\_{HNF}}{\mu\_F} b \frac{1}{2} F'(b). \tag{18}$$
