Mixed Convective Radiative Flow through a Slender Revolution Bodies Containing Molybdenum-Disulfide Graphene Oxide along with Generalized Hybrid Nanoparticles in Porous Media

Abstract

The current framework tackles the buoyancy flow via a slender revolution bodies comprising Molybdenum-Disulfide Graphene Oxide generalized hybrid nanofluid embedded in a porous medium. The impact of radiation is also provoked. The outcomes are presented in this analysis to examine the behavior of hybrid nanofluid flow (HNANF) through the cone, the paraboloid, and the cylinder-shaped bodies. The opposing flow (OPPF) as well as the assisting flow (ASSF) is discussed. The leading flow equations of generalized hybrid nanoliquid are worked out numerically by utilizing bvp4c solver. This sort of the problem may meet in the automatic industries connected to geothermal and geophysical applications where the sheet heat transport occurs. The impacts of engaging controlled parameters of the transmuted system on the drag force and the velocity profile are presented through the graphs and tables. The achieved outcomes suggest that the velocity upsurges due to the dimensionless radius of the slender body parameter in case of the assisting flow and declines in the opposing flow. Additionally, an increment is observed owing to the shaped bodies as well as in type A nanofluid and type B hybrid nanofluid.

1. Introduction

The research regarding the convective-flow entrenched in porous media widely has been utilized owing to its vast engineering applications as solar collectors, heat exchangers, post-accidental heat exclusion in nuclear reactors, building construction, drying processes, oil recovery and geothermal, ground water pollution, etc. Nield [1] analyzed the liquid flow of stability ensuing through a vertical mass and thermal gradients via a horizontal-layer immersed in porous media. Bejan and Khair [2] explored the marvel of mass and heat transfer through a vertical sheet entrenched in a porous medium and they have taken unvarying concentration as well as temperature. The impact of mixed as well as free convective flows with heat transport through a slender revolution of the body in a porous medium was examined by Lai et al. [3] and they concluded that the temperature gradient shrinks due to dimensional radius. The thermal and mass diffusion through a cone embedded in porous medium was scrutinized by Yih [4]. Bano and Singh [5] explored the radiation influence on mass and heat transport from a radiated thin needle in a saturated porous medium by utilizing a technique of Von Karman-integral. Singh and Chandarki [6] inspected the free convective flow with mass and heat transfer through a vertical cylinder occupied in porous media. The mixed convective flow through a vertical flat surface in a porous medium with nanoliquid was examined by Ahmad and Pop [7]. Talebizadeh et al. [8] scrutinized the impact of radiation on natural convective flow through a porous vertical surface and found numerical as well as exact solutions. Moghimi et al. [9] discussed the MHD (magnetohydrodynamic) influence on natural convective flow via a sphere saturated in a porous medium. Moghimi et al. [10] obtained an exact solution of flow through a flat surface with constant heat flux and slip effect by utilizing DQM (differential quadrature method) and HAM (homotopy analysis method). Raju and Sandeep [11] inspected the magnetic influence on flow of mass and heat transport containing Casson fluid through a rotated vertical cone/plate in porous media with micro-organisms. They perceived that the mass and heat transfer from a cone is superior compared to flow over a plate. Raju et al. [12] applied the Buongiorno model to inspect the phenomenon of mass and heat transport through a radiated revolution slender body in porous media. They perceived that the behavior of temperature shrinks in a flow revolution over a cone compared to flow revolution through the cylinder and paraboloid.

Several recent explorations divulged that nanofluids have superior capability of heat transport than convectional fluids. Thus, it is likely to swap conventional heat transport fluids via nanofluids in the numerous designs of heat transport like heat exchanger, heat generators, and cooling systems. Choi [13] observed that, by scattering metallic nanometer sized particles in regular heat transport liquids, the ensuing nanoliquids hold greater thermal conductivity than those of presently utilized ones. Further, Eastman et al. [14] discovered that the shape of the particle has a stronger effect on nanofluid effective thermal conductivity than the size of particle or thermal conductivity of the particle. To augment the nanofluids heat transport owing to nanoparticles migration and the resulting boundary-layer disturbance was experimentally examined by Wen and Ding [15]. The Boltzmann technique to inspect the magnetic impact on the natural convective flow comprising nanoliquid through a cylindrical annulus was studied by Ashorynejad et al. [16]. The impact of distinct shapes of nanoscale particles in EG (entropy generation) based aqueous solution was inspected by Ellahi et al. [17]. Inspired by the importance of nanofluids, several researchers recently were engaged with the debate of flow with heat transport to nanoliquids via different perspective [18,19,20,21].

Recently, a novel type of fluid, suggested hybrid nanofluids has been utilized to augment the heat transport in applications of thermal [22,23,24]. Hybrid nanoliquids consist of two or more different nanoparticles in either mixture of non-composite forms. Hybrid nanofluids envisages in the fields of heat transport as electronic and generator cooling, thermal storage, biomedical, cooling of transformer, lubrication, solar heating, spacecraft and aircraft, welding, protection, refrigeration, and heat pump. Minea [25] inspected the estimations of distinct viscosity of hybrid nanoliquid by scattering the water-based TiO₂, Al₂O₃, and SiO₂ nanomaterials. The impact of nonlinear radiation on the magneto flow of micropolar hybrid (Cu-Al₂O₃) dusty nanofluid from a stretched sheet was examined by Ghadikolaei et al. [26]. Sheikoleslami et al. [27] scrutinized the modeling of the porous domain with Lorentz forces and radiation impact and obtained the solution by CVFEM (control volume based finite element method). They explored that temperature gradient has greater influence due to the greater buoyancy parameter. Gholinia et al. [28] inspected the steady magneto flow of hybrid CNTs (carbon nanotubes) nanofluid over a permeable stretched cylinder. Recently, Khan et al. [29] discussed the influence of magnetic function comprising ethylene glycol-based hybrid nanofluid from a stretched/shrinking wedge with mixed convection and stability analysis performed.

The earlier review literature reveals that the mixed convective generalized hybrid (MoS₂-Go) nanofluid flow through a slender revolution bodies in porous media has been highlighted as a mostly unknown area. The radiation impact with generalized hybrid single phase model is discussed here, which is not considered in the earlier published results [7,11]. The leading PDEs (partial differential equations) are altered into ODEs (ordinary differential equation) through suitable transformations and then tackled via three stage Lobatto formula. Impacts of the pertinent variables are portrayed and investigated through the graphs.

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 $u_e\left(x\right)=U_{\infty }{x}^m$ with constant $U_{\infty }$, while $T_{\infty }$ the constant free stream temperature. The temperature of the slender revolution body is taken as $T_w\left(x\right)$ with $T_w\left(x\right)>T_{\infty }$ utilizes for (ASSF) and $T_w\left(x\right)<T_{\infty }$ 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 $\left(x,r\right)$. 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

$$\frac{\partial \left(ru\right)}{\partial x}+\frac{\partial \left(vr\right)}{\partial r}=0$$

(1)

$$\frac{{\mu }_{HNF}}{{\mu }_F}\frac{\partial u}{\partial r}=\frac{Kg{\left(\rho \beta \right)}_F}{{\mu }_F}\frac{{\left(\rho \beta \right)}_{HNF}}{{\left(\rho \beta \right)}_F}\frac{\partial T}{\partial r}$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{k_{HNF}}{{\left(\rho c_p\right)}_{HNF}r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)-\frac{1}{{\left(\rho c_p\right)}_{HNF}}\frac{\partial }{\partial r}\left(r{q}_r\right)$$

(3)

with the corresponding boundary conditions

$$\begin{array}{l}v=0, T=T_w\left(x\right)=T_{\infty }+\mathrm{A}{x}^{\lambda } \mathrm{at} r=R\left(x\right),\\ u=u_e\left(x\right)=U_{\infty }{x}^m,T=T_{\infty } \mathrm{as} r\to \infty .\end{array}$$

(4)

Figure 1. Physical diagram of the problem.

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

$$\frac{{\mu }_{HNF}}{{\mu }_F}u=\frac{{\mu }_{HNF}}{{\mu }_F}{u}_e+\frac{Kg{\beta }_F}{{\nu }_F}\frac{{\left(\rho \beta \right)}_{HNF}}{{\left(\rho \beta \right)}_F}\left(T-T_{\infty }\right)$$

(5)

Here $v$, $u$ signify Darcy’s law velocity components in $\left(r,x\right)$ directions, $T$ temperature of the hybrid nanofluid, $g$ acceleration owing to gravity, ${\mu }_{HNF}$ hybrid viscosity, ${\mu }_F$ base fluid viscosity, $K$ permeability of the porous medium, $k_{HNF}$ hybrid nanofluid thermal conductivity, ${\left(\rho c_p\right)}_{HNF}$ hybrid nanofluid heat capacitance, ${\left(\rho \beta \right)}_{HNF}$ hybrid nanofluid thermal expansion, and $R\left(x\right)$ surface shape of the axisymmetric body.

For radiation effect, the Rosseland approximation is illustrated as

$$q_r=-\frac{4{\gamma }_1}{3k_1}\left(\frac{\partial T^4}{\partial r}\right)$$

(6)

where $k_1$ and ${\gamma }_1$ signify coefficient of mean absorption and Stefan–Boltzmann, respectively. Applying Taylor series to expand $T^4$ about $T_{\infty }$ and prohibiting the terms involving higher-order, one gets

$$T^4\cong 4T{T}_{\infty }^3-3T_{\infty }^4$$

(7)

The similarity variables for further analysis are introduced as

$$\begin{array}{l}\psi ={\alpha }_{F}xF\left(\eta \right),\eta =P{e}_x\frac{r^2}{x^2}=\frac{U_{\infty }{r}^2{x}^{m-1}}{{\alpha }_F},\theta \left(\eta \right)=\left(T-T_{\infty }\right)/\left(T_w-T_{\infty }\right),\\ P{e}_x=\frac{U_{\infty }{x}^{m+1}}{{\alpha }_F},u=2u_e{F}^{\prime },v=\frac{{\alpha }_F}{r}\eta \left(1-m\right)F^{\prime }-\frac{{\alpha }_F}{r}F,R_d=\frac{4{\gamma }_1{T}_{\infty }^3}{k_1{k}_F}.\end{array}$$

(8)

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

$$\frac{{\mu }_{HNF}}{{\mu }_F}\left(2F^{\prime }-1\right)-\frac{Kg{\beta }_F{x}^{\lambda -m}}{{\nu }_F{U}_{\infty }}\frac{{\left(\rho \beta \right)}_{HNF}}{{\left(\rho \beta \right)}_F}\theta =0$$

(9)

$$\left(\frac{k_{HNF}}{k_F}+\frac{4}{3}{R}_d\right)\left(2\eta {\theta }^{″}+2{\theta }^{\prime }\right)+\frac{{\left(\rho c_p\right)}_{HNF}}{{\left(\rho c_p\right)}_F}\left({\theta }^{\prime }F-\lambda F^{\prime }\theta \right)=0$$

(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.,:

$$m=\lambda$$

(11)

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

$$\frac{{\mu }_{HNF}}{{\mu }_F}\left(2F^{\prime }-1\right)-\xi \frac{{\left(\rho \beta \right)}_{HNF}}{{\left(\rho \beta \right)}_F}\theta =0$$

(12)

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

$$\begin{array}{l}\xi =\frac{Ra}{P{e}_x}=\frac{Kg{\beta }_F\left(T_w-T_{\infty }\right)x}{{\nu }_F{\alpha }_F}\frac{{\alpha }_F}{U_{\infty }{x}^{m+1}}=\frac{Kg{\beta }_{F}A}{{\nu }_F{U}_{\infty }},Ra=\frac{Kg{\beta }_F\left(T_w-T_{\infty }\right)x}{{\nu }_F{\alpha }_F},\\ P{e}_x=\frac{U_{\infty }{x}^{m+1}}{{\alpha }_F},\mathrm{Pr}=\frac{{\nu }_F}{{\alpha }_F},R_d=\frac{4{\gamma }_1{T}_{\infty }^3}{k_1{k}_F}.\end{array}$$

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 $\eta =b$, 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

$$R\left(x\right)={\left(\frac{{\nu }_F{\alpha }_{F}bRa}{P{e}_{x}Kg{\beta }_{F}A}\right)}^{\frac{1}{2}}{x}^{\left(\frac{1-\lambda }{2}\right)}\hspace{1em}\mathrm{OR}\hspace{1em}{\left(\frac{b}{P{e}_x}\right)}^{\frac{1}{2}}x$$

(13)

The problems concerning the realistic interest, the amount of $\lambda \le 1.$ For instance, $\lambda =1$, $\lambda =0$ and $\lambda =-1$ represent the cylinder, paraboloid, and cone shape bodies.

The boundary restrictions are

$$\begin{array}{l}b\left(1-m\right)F^{\prime }-F=0, \theta =1\mathrm{at}\eta =b,\\ F^{\prime }\to 0.5, \theta \to 0\mathrm{as}\eta \to \infty .\end{array}$$

(14)

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

$$\left(\frac{k_{HNF}}{k_F}+\frac{4}{3}{R}_d\right)\left(4\eta F^{‴}+4F^{″}\right)+\frac{{\left(\rho c_p\right)}_{HNF}}{{\left(\rho c_p\right)}_F}\left(2F{F}^{″}-2\lambda F{\prime }^2+\lambda F^{\prime }\right)=0$$

(15)

along with the modified boundary restrictions

$$b\left(1-m\right)F^{\prime }\left(b\right)-F\left(b\right)=0,\frac{{\mu }_{HNF}}{{\mu }_F}\left(2F^{\prime }\left(b\right)-1\right)=\xi \frac{{\left(\rho \beta \right)}_{HNF}}{{\left(\rho \beta \right)}_F},F^{\prime }\left(\infty \right)\to 0.5.$$

(16)

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

$$C_F=\frac{{\tau }_w}{\rho u_e^2}=\frac{{\left.{\mu }_{HNF}\frac{\partial u}{\partial r}\right|}_{r=R\left(x\right)}}{\rho u_e^2}$$

(17)

The skin friction in the dimensional form is

$$\frac{P{e}_x^{0.5}}{\mathrm{Pr}}{C}_F=4\frac{{\mu }_{HNF}}{{\mu }_F}{b}^{\frac{1}{2}}{F}^{″}\left(b\right).$$

(18)

3. Model of Generalized Hybrid Nanoliquid

In numerical and experimental investigations on the behaviors of nanofluid, modeling their physical quantities using condensed mathematical relationships between solid particles and regular liquid is a common procedure. Numerous experiments have been performed to validate such terms for nanofluids dilute in scattering of a single sort of solid material [31] and mixtures of two kinds of particles (Suresh et al. [32]). Devi and Devi [33] recommended a collection of correlation for hybrid nanofluid physical-quantities. They approached the liquid involving a single sort of nanoparticle as the regular liquid and the other sort of nanoparticle as the individual particle. The relationship of thermal conductivity and viscosity matched Suresh et al.’s [32] experimental outcomes. In the approach of Devi and Devi [33], there exists the non-linear terms owing to the communication of two sorts of distinct nanoparticles. However, in dilute mixtures where the volumetric fractions of nanoparticle are generally tiny, the impacts of these non-linear conditions may not be important. Thus, it realistically ignores the non-linear terms in the model of Devi and Devi. The hybrid nanofluid model in simplified form and the models of normal nanofluid and Devi and Devi are scheduled in Table 1, where Type A signifies the conventional nanoliquid model, Types B and C, respectively; indicate the hybrid nanoliquid model of Devi and Devi and the model of hybrid nanoliquid in simplified form. Devi and Devi [33] used the approach of the recurrence formulae to signify the density, viscosity, thermal conductivity, and specific heat of the hybrid nanoliquid related generally to the N-th type of nanoparticles as

$${\rho }_{HNF}\equiv {\rho }_{HN{F}_N}=\left(1-{\varphi }_N\right){\rho }_{HN{F}_{N-1}}+{\varphi }_N{\rho }_{S_N}$$

(19)

$${\mu }_{HNF}\equiv {\mu }_{HN{F}_N}=\frac{{\rho }_{HN{F}_{N-1}}}{{\left(1-{\varphi }_N\right)}^{2.5}}$$

(20)

$$k_{HNF}\equiv k_{HN{F}_N}=\frac{k_{S_N}+k_{HN{F}_{N-1}}\left(M-1\right)-{\varphi }_N\left(M-1\right)\left(k_{HN{F}_{N-1}}-k_{S_N}\right)}{k_{S_N}+k_{HN{F}_{N-1}}\left(M-1\right)+{\varphi }_N\left(k_{HN{F}_{N-1}}-k_{S_N}\right)}$$

(21)

$${\left(\rho c_p\right)}_{HNF}\equiv {\left(\rho c_p\right)}_{HN{F}_N}={\varphi }_N{\left(\rho c_p\right)}_{S_N}+\left(1-{\varphi }_N\right){\left(\rho c_p\right)}_{HN{F}_{N-1}}$$

(22)

$${\beta }_{HNF}\equiv {\beta }_{HN{F}_N}=\left(1-{\varphi }_N\right){\beta }_{HN{F}_{N-1}}+{\varphi }_N{\beta }_{S_N}$$

(23)

Table 1. Thermo-physical models of nanofluid and hybrid nanofluid.

Property	Types	Correlation
Density	A	${\rho }_{NF}=\left(1-\varphi \right){\rho }_F+\varphi {\rho }_S$
	B	${\rho }_{HNF}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\rho }_F+{\varphi }_1{\rho }_{S_1}\right]+{\varphi }_2{\rho }_{S_2}$
	C	${\rho }_{HNF}=\left(1-{\varphi }_1-{\varphi }_2\right){\rho }_F+{\varphi }_1{\rho }_{S_1}+{\varphi }_2{\rho }_{S_2}$
Viscosity	A	${\mu }_{NF}=\frac{{\mu }_F}{{\left(1-\varphi \right)}^{2.5}}$
	B	${\mu }_{HNF}=\frac{{\mu }_F}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}}$
	C	${\mu }_{HNF}=\frac{{\mu }_F}{{\left(1-{\varphi }_1-{\varphi }_2\right)}^{2.5}}$
Heat Capacity	A	${\left(\rho c_p\right)}_{NF}=\left(1-\varphi \right){\left(\rho c_p\right)}_F+\varphi {\left(\rho c_p\right)}_S$
	B	${\left(\rho c_p\right)}_{HNF}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left(\rho c_p\right)}_F+{\varphi }_1{\left(\rho c_p\right)}_{S_1}\right]+{\varphi }_2{\left(\rho c_p\right)}_{S_2}$
	C	${\left(\rho c_p\right)}_{HNF}=\left(1-{\varphi }_1-{\varphi }_2\right){\left(\rho c_p\right)}_F+{\varphi }_1{\left(\rho c_p\right)}_{S_1}+{\varphi }_2{\left(\rho c_p\right)}_{S_2}$
Thermal conductivity	A	$k_{NF}=\frac{k_S+\left(M-1\right)k_F-\left(M-1\right)\varphi \left(k_F-k_S\right)}{k_S+\left(M-1\right)k_F+\varphi \left(k_F-k_S\right)}$
	B & C	$k_{HNF}=\frac{k_{S_2}+\left(M-1\right)k_{MF}-\left(M-1\right){\varphi }_2\left(k_{MF}-k_{S_2}\right)}{k_{S_2}+\left(M-1\right)k_{MF}+{\varphi }_2\left(k_{MF}-k_{S_2}\right)}$ where $k_{MF}=\frac{k_{S_1}+\left(M-1\right)k_F-\left(M-1\right){\varphi }_1\left(k_F-k_{S_1}\right)}{k_{S_1}+\left(M-1\right)k_F+{\varphi }_1\left(k_F-k_{S_1}\right)}{k}_F$
Thermal expansion	A	${\beta }_{NF}=\left(1-\varphi \right){\beta }_F+\varphi {\beta }_S$
	B	${\beta }_{HNF}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\beta }_F+{\varphi }_1{\beta }_{S_1}\right]+{\varphi }_2{\beta }_{S_2}$
	C	${\beta }_{HNF}=\left(1-{\varphi }_1-{\varphi }_2\right){\beta }_F+{\varphi }_1{\beta }_{S_1}+{\varphi }_2{\beta }_{S_2}$
Pr	6.2	-	-

From the above correlations, ignoring the non-linear terms, one obtains

$${\rho }_{HNF}=1-{\rho }_F{\displaystyle \sum _{a=1}^N{\varphi }_a}+{\displaystyle \sum _{a=1}^N{\varphi }_a}{\rho }_{S_a}$$

(24)

$${\mu }_{HNF}=\frac{{\mu }_F}{{\left(1-{\displaystyle \sum _{a=1}^N{\varphi }_a}\right)}^{2.5}}$$

(25)

$${\left(\rho c_p\right)}_{HNF}=1-{\left(\rho c_p\right)}_F{\displaystyle \sum _{a=1}^N{\varphi }_a}+{\displaystyle \sum _{a=1}^N{\varphi }_a}{\left(\rho c_p\right)}_{S_a}$$

(26)

$${\beta }_{HNF}=1-{\beta }_F{\displaystyle \sum _{a=1}^N{\varphi }_a}+{\displaystyle \sum _{a=1}^N{\varphi }_a}{\beta }_{S_a}$$

(27)

It is worth mentioning that for $k_{HNF}$, we remain the recurrence Formula (21) because of the connections amid dissimilar particles can barely be uttered by the Maxwell equations. Additionally, throughout the research the values of $M=3$ is taken which implies that the shape of the particle is spherical.

5. Concluding Remarks

The key points of the current study are summarized as:

• The velocity augments for the (ASSF) and declines for the (OPPF) owing to magnifying the dimensionless radius of the slender body parameter $b$ while the change behavior is detected in response of the higher nanoparticle volume fraction ${\varphi }_2$.
• The persistent effect of $b$, the velocity upsurges for the flow of the shape bodies like paraboloid shape body ($\lambda =0$), cylindrical shape body ($\lambda =1$) as well as cone shape body ($\lambda =-1$) while the same behavior of the velocity is seen in the type A nanofluid and type B hybrid nanofluid.
• The velocity field decreases for the type A nanofluid as well as for the type B hybrid nanofluid and also for the flow over the different shape bodies owing to ${\varphi }_2$.
• Due to $R_d$, the velocity upsurges for the type A nanofluid as well as for the type B hybrid nanofluid while for the type B hybrid nanofluid, the velocity is lesser relative to the type A nanofluid.
• The skin friction under the distinct shape bodies and along the type A nanofluid and type B hybrid nanofluid are augmented for both parameter $b$ and ${\varphi }_2$ along the x-axis of the slender sheet in the range $0\le \xi <\infty$ while vice versa in the range $-\infty <\xi \le 0$.

This sort of problem may be useful in geothermal and geophysical applications. Thus, the outcomes of the problem will be obliging in the evaluation and assessment of resources in geothermal energy.