## Abstract

This study used numerical analysis to investigate the effects of nonlinear radiation and variable viscosity on free convection of a power-law nanofluid over a vertical truncated cone in porous media with Rosseland diffusion approximation considering zero nanoparticles flux and internal heat generation. The internal heat generation is of an exponential decaying form and the viscosity of the fluid is assumed to follow Reynolds viscosity model. The surface boundary conditions of vertical truncated cone is maintained at the uniform wall temperature (UWT) and the zero nanoparticle flux (ZNF) to cause the results to be more realistic and useful. The nanofluid model considered the effects of Brownian motion and thermophoresis. The nonsimilar governing equations are obtained by using a suitable coordinate transformation and then solved by Keller box method (KBM). Comparisons with previously published work obtained good agreement. Graphical and tabular presentations of numerical data for the dimensionless temperature profile and the local Nusselt number were presented for main parameters: dimensionless streamwise coordinate, thermophoresis parameter, Lewis number, radiation parameter, surface temperature parameter, viscosity parameter, power-law index of the non-Newtonian fluid, and internal heat generation coefficient. The local Nusselt number increased when the following parameters were increased: radiation parameter, surface temperature parameter, viscosity parameter, power-law index of the non-Newtonian fluid, and dimensionless streamwise coordinate. In contrast, the local Nusselt number decreased when the following parameters were increased: internal heat generation coefficient, thermophoresis parameter, and Lewis number. Besides, the physical aspects of the problem are discussed in details.

## Introduction

The study of convective heat and mass transfer in a saturated porous medium has important applications such as geothermal reservoirs, nuclear reactor cooling system, and underground disposal of nuclear wastes. A recent monograph by Nield and Bejan [1] comprehensively discussed the available information in this field. Concerning the aspect of pure heat transfer in porous media, Cheng et al. [2] first studied natural convection of a Darcian fluid about a vertical full core by similarity solution or truncated cone by local nonsimilarity solution. Chamkha [3] studied non-Darcy hydromagnetic free convection from a cone and a wedge in porous media. Chamkha et al. [4] performed a numerical analysis of free convection flow over a truncated cone embedded in a porous medium saturated with pure or saline water at low temperatures. Regarding combined heat and mass transfer, Yih [5] extended the work of Cheng et al. [2] to analyze the double-diffusion from a vertical truncated cone with the both variable wall temperature/concentration (VWT/VWC) and variable heat/mass flux (VHF/VMF) boundary conditions in a saturated porous medium using Keller box method (KBM). An integral method for heat and mass transfer by free convection from a vertical VWT/VWC truncated cone with in porous media was presented by Cheng [6]. Takhar et al. [7] investigated combined heat and mass transfer along a vertical moving cylinder with a free stream. Hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid-saturated porous medium was analyzed by Chamkha and Khaled [8]. Takhar et al. [9] presented unsteady mixed convection flow from a rotating vertical cone with a magnetic field. Later, Cheng [10] extended the research of Yih [5] to examine Soret and Dufour effects on heat and mass transfer by natural convection from a vertical VWT/VWC truncated cone in porous media by the cubic spline collocation method.

Numerous industrially important fluids present the behavior of non-Newtonian. The convection flow of non-Newtonian fluids in a saturated porous medium is extremely important because of its engineering applications, e.g., oil recovery and food processing. As for the study of non-Newtonian fluids in a porous medium, Chen and Chen [11] investigated free convection flow of non-Newtonian fluids along a vertical plate. Boundary-layer flow and heat transfer of non-Newtonian fluids in porous media was studied by Wang and Tu [12]. Yih [13] extended the work of Chen and Chen [11] to examine uniform lateral mass flux effect on free convection of non-Newtonian fluids over the vertical plate and cone in porous media by utilizing KBM. Cheng [14] presented natural convection heat transfer of non-Newtonian fluids in porous media from a vertical cone under mixed thermal boundary conditions by the cubic spline collocation method. Free convection of coupled heat and mass transfer about a vertical truncated cone in a porous medium saturated with a non-Newtonian fluid was studied by Cheng [15] for the VWT/VWC case.

The internal heat generation effect must be considered in reactor safety analysis, fire and combustion studies, and the storage of radioactive materials. Regarding research in the internal heat generation effect in a porous medium, natural convection in non-Newtonian fluid-saturated porous media with an exponential decaying internal heat generation was investigated over a vertical flat plate in Groşan and Pop [16] and over a vertical cone in Groşan et al. [17]. The similarity equations are solved numerically using a version of the shooting method. Chamkha and Al-Mudhaf [18] presented unsteady heat and mass transfer from a rotating vertical cone with a magnetic field and heat generation or absorption effects. Al-Mudhaf and Chamkha [19] studied similarity solutions for magnetohydrodynamic (MHD) thermosolutal Marangoni convection over a flat surface considering heat generation or absorption effects. The effect of heat generation or absorption on thermophoretic free convection boundary layer from a vertical flat plate embedded in a porous medium was investigated by Chamkha et al. [20]. Damseh et al. [21] analyzed the combined effects of heat generation or absorption and first-order chemical reaction on micropolar fluid flows over a uniformly stretched permeable surface. Khedr et al. [22] examined MHD flow of a micropolar fluid past a stretched permeable surface with heat generation or absorption. A full analytical solution for the combined effect of heat generation or absorption and first-order chemical reaction on micropolar fluid flows over a uniformly stretched permeable surface was reported in Magyari and Chamkha [23]. An analytical solution for natural convection past a vertical cone in a non-Newtonian fluid-saturated porous medium with internal heat generation effect was reported in Rashidi and Rastegari [24]. The similarity equations are solved analytically using a homotopy analysis method. Recently, Yih and Huang [25,26] analyzed internal heat generation effect on natural convection heat and mass transfer of non-Newtonian fluids flow over a vertical plate [25] and a vertical truncated cone [26] in porous media for the case of VWT/VWC. Huang [27] examined internal heat generation and Soret/Dufour effects on natural convection of non-Newtonian fluids about a vertical permeable cone in porous media.

The effect of radiation on convection flow has many important applications, including space technology and processes involving high temperatures, e.g., geothermal engineering and nuclear reactor cooling systems. Chamkha [28] analyzed solar radiation-assisted convection in uniform porous medium supported by a vertical flat plate. Coupled heat and mass transfer by natural convection about a truncated cone in the presence of magnetic and radiation effects were presented by Chamkha [29]. Unsteady MHD natural convection from a heated vertical porous plate in a micropolar fluid with Joule heating, chemical reaction, and radiation effects was investigated by Chamkha et al. [30]. Yih [31] used the Rosseland diffusion approximation to extend the work of Cheng et al. [2] in an analysis of the effect of thermal radiation on combined free and forced convection over an isothermal cone in porous media. In an extension of the work of Groşan et al. [17], Mahmoud [32] presented the effect of thermal radiation on natural convection of a non-Newtonian fluid about a vertical cone embedded in a porous medium in the presence of exponentially decaying internal heat generation. Recently, Huang [33,34] extended the work of Mahmoud [32] and used the KBM to investigate numerically the Soret/Dufour effects on coupled heat and mass transfer by free convection over a vertical cone with internal heat generation and thermal radiation influences [33] and uniform blowing/suction and thermal radiation effects [34] in porous media, respectively. To simplify their analysis, articles [32–34] assumed that, because of the small temperature difference within the flow, may be expressed as $T4=4T\u221e3T\u22123T\u221e4$ (linear function of temperature *T*) by a Taylor series expansion about *T*_{∞} and neglecting higher-order terms. In the energy equation in Ref. [31], however, the term *T*^{4} (nonlinear) is retained.

In the above studies [2–34], fluid viscosity was assumed to be constant. However, viscosity is known to change significantly with temperature. Therefore, in order to predict the fluid flow behavior accurately, it is necessary to consider the viscosity variation. In the aspect of variable viscosity effect on heat transfer in a porous medium, Gary et al. [35] described how large viscosity changes affect convective heat transfer transport in water-saturated porous medium. Bagai [36] used the Reynolds viscosity model to solve the problem of effect of variable viscosity on free convection over a nonisothermal axisymmetric body in a porous medium with internal heat generation (the viscosity decreases exponentially with temperature). Kairi et al. [37] presented the effect of viscous dissipation on natural convection in a non-Darcy porous medium saturated with non-Newtonian fluid of variable viscosity. Bagai and Nishad [38] extended the work of Bagai [36] by analyzing the effect of variable viscosity on free convective heat transfer over a non-isothermal body with an arbitrary shape in a non-Newtonian fluid-saturated porous medium considering internal heat generation. They observed that the heat transfer rate increases for a less viscous fluid. The effects of viscosity and fluid suction/injection on free convection flow from a vertical plate in a porous medium saturated with a pseudo-plastic fluid were studied by Achemlal et al. [39]. In the couple heat and mass transfer aspect, Cheng [40] examined nonsimilar boundary layer analysis of double-diffusive convection from a vertical truncated cone in a porous medium with variable viscosity. Mahdy et al. [41] extended Cheng [40] by including magnetic field and radiation effects. Narayana et al. [42] analyzed the Soret effect on the natural convection from a vertical plate in a thermally stratified porous medium saturated with non-Newtonian liquid.

Nanotechnology applications have been reported in the biological sciences, physical sciences, electronic cooling, and advanced nuclear systems. The term “nanofluid” refers to a liquid containing a suspension of submicronic solid particles (nanoparticles). The heat transfer characteristic of nanofluid is very important because these fluids may be applicable for increasing the heat transfer rate. Buongiorno [43] comprehensively surveyed convection in nanofluids and derived the conservation equations for nanofluids involving both the Brownian motion and thermophoresis effects. Recently, Kakaç and Pramuanjaroenkij [44] and Mahdi et al. [45] reviewed the literature on convection heat transfer and fluid flow in porous media with nanofluid. The following studies are the most important numerical studies of nanofluids convective boundary-layer flow in porous media filled with a nanofluid. Nield and Kuznetsov [46] used the Buongiorno nanofluid model to study the Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Cheng investigated natural convection boundary-layer flow over a truncated cone in a porous medium saturated by Newtonian nanofluids in Ref. [47] and by non-Newtonian nanofluids in Ref. [48], respectively. Rashad et al. [49] extended the work of Yih [13] in a study of natural convection boundary layer of a non-Newtonian fluid about a permeable vertical cone in porous media saturated with a nanofluid. The effects of heat generation/absorption on natural convective boundary-layer flow from a vertical cone embedded in a porous medium filled with a non-Newtonian nanofluid were analyzed by Hady et al. [50]. Chamkha et al. [51] presented the effect of suction/injection on free convection along a vertical plate in a nanofluid saturated non-Darcy porous medium with internal heat generation. Chamkha et al. [52] extended the research of Yih [31] by examining the nonlinear effects of thermal radiation on mixed convection about a cone embedded in a porous medium filled with a nanofluid. Recently, variable viscosity effect on non-aligned MHD stagnation point flow of nanofluids past a stretching sheet with radiative heat was presented by Khan et al. [53].

However, the model in the above articles [46–53] controlled for nanoparticle volume fraction at the boundary. Therefore, Kuznetsov and Nield [54] developed a revised model and repeated the experiments in Nield and Kuznetsov [46] for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. In their revised model, the nanoparticle volume fraction on the boundary is passively rather than actively controlled, i.e., the zero nanoparticle flux (ZNF), leading the solutions to be more realistic and useful. They observed that the Nusselt number decreases independently of the Brownian motion parameter, which is different from the results of Nield and Kuznetsov [46]. Cheng [55] also used the revised model to study free convection about a vertical full cone in a porous medium saturated by a nanofluid. Brownian motion parameter and the buoyancy ratio had almost no effect on the local Nusselt number. However, increasing the thermophoresis parameter decreases the local Nusselt number. The problem of MHD boundary-layer flow of a power-law nanofluid with new mass flux condition was solved by Khan and Khan [56]. Khan et al. [57] investigated the triple convective-diffusion boundary layer along a vertical flat plate in a porous medium saturated by a water-based nanofluid. Kameswaran et al. [58] solved the problem of mixed convection from a wavy surface embedded in a thermally stratified nanofluid saturated porous medium with nonlinear Boussinesq approximation. Gorla and Chamkha [59] studied natural convective boundary-layer flow over a nonisothermal vertical plate embedded in a porous medium saturated with a nanofluid. RamReddy et al. [60] examined the Soret effect on mixed convection flow in a nanofluid under the condition of a convective boundary. Ghalambaz et al. [61] analyzed effects of nanoparticles diameter and concentration on natural convection of the Al_{2}O_{3}–water nanofluids considering variable thermal conductivity around a vertical cone in porous media. Sudarsana Reddy and Chamkha [62] studied Soret and Dufour effects on MHD convective flow of Al_{2}O_{3}–water and TiO_{2}–water nanofluids past a stretching sheet in porous media with heat generation/absorption. The MHD boundary-layer flow, heat and mass transfer analysis over a rotating disk and through a porous medium saturated with Cu-water and Ag-water nanofluid with chemical reaction was presented by Reddy et al. [63]. Rasool et al. [64] studied entropy generation and consequences of binary chemical reaction on MHD Darcy-Forchheimer Williamson nanofluid flow over nonlinearly stretching surface.

Therefore, the objective of the present work was to extend works by Cheng et al. [2], Groşan et al. [17], Bagai and Nishad [38], Cheng [48], Chamkha et al. [52], and Cheng [55] by using the Reynolds viscosity model to investigate the effects of nonlinear thermal radiation and variable viscosity on natural convection of a power-law nanofluid over a vertical truncated cone in porous media with zero nanoparticles flux and exponential decaying internal heat generation. The governing numerical equations were solved by KBM. The results are obtained and discussed for various values of the main parameters in tabular and graphic forms.

## Analysis

Consider the problem of the influences of zero nanoparticles flux, internal heat generation, nonlinear radiation, and variable viscosity on free convection of a non-Newtonian nanofluid past a vertical truncated cone (with half angle *γ*) embedded in a fluid-saturated porous medium filled with a nanofluid. Figure 1 illustrates the flow model and physical coordinate system. The origin of the coordinate system is placed at the vertex of the full cone, where *x* is the coordinate along the surface of cone measured from the origin and *y* is the coordinate normal to the surface, respectively. *r* is the local radius of the vertical truncated cone. The *x*_{o} is the distance from the origin to the leading edge of the vertical truncated cone. The boundary conditions are maintained at the UWT *T*_{w} and ZNF; *T*_{w} is higher than the ambient temperature *T*_{∞}. *C*_{∞} is the ambient nanoparticle volume fraction. *g* is the gravitational acceleration. The effects of Brownian motion and thermophoresis on the nanofluids are also incorporated into the model. All the fluid properties are assumed to be constant except for the viscosity of fluid and the density variation in the buoyancy term. The viscous dissipation effect is neglected for the low velocity.

For the boundary layer and Oberbeck–Boussinesq approximations, the governing equations and the boundary conditions based on the Darcy law [48,52,55] can be written as follows:

Here, *u* and *v* denote the volume-averaged velocity components in the *x*- and *y*-directions, respectively. The *T* is the volume-averaged temperature. The *C* is the nanoparticle volume fraction. The *K*(*n*) is the modified permeability of the porous medium. *n* is the power-law index of the non-Newtonian fluid; *ρ*_{f}, *μ*, and *β* are the density, viscosity, and volumetric expansion coefficient of the fluid. The *ρ*_{p} is the density of the nanoparticle. *α* is the thermal diffusivity of the porous medium. The heat capacity ratio between nanoparticle and fluid is defined as *τ* = (*ρc*)_{p}/(*ρc*)_{f}. (*ρc*)_{p} and (*ρc*)_{f} are the effective heat capacity of the nanoparticle material and the heat capacity of the fluid, respectively. *c* is the specific heat at constant pressure. *q*″′ is the internal heat generation rate per unit volume; *σ*_{o}, *a*_{r}, and *σ*_{s} are the Stefan-Boltzmann constant, the Rosseland mean extinction coefficient, and the scattering coefficient, respectively. The *D*_{B} and *D*_{T} are the Brownian diffusion coefficient and the thermophoresis diffusion coefficient, respectively. The third term of Eq. (5) is the ZNF. Hence, the nanoparticle volume fraction on the boundary is passively controlled.

Since the boundary layer is sufficiently thin in comparison with the local radius of the vertical truncated cone. The local radius to a point in the boundary layer, therefore, can be replaced by the radius of the vertical truncated cone, i.e., *r* = *x* sin *γ*. Equations (1)–(6) are valid in *x*_{o} ≤ *x* < ∞.

*d*is the particle diameter while

*ɛ*is the porosity.

Note that the power-law fluid index *n* < 1 corresponds to pseudo-plastic fluids (for example, the polymer solution), *n* = 1 to Newtonian fluids (for instance, air and water), and *n* > 1 to dilatant fluids (for example, the suspensions of sand) [13–15].

*ψ*is defined by

*q*″′ is modeled according to the exponential decaying form [24–27]:

*k*is the equivalent thermal conductivity.

*μ*

_{∞}is the ambient viscosity of the medium and Ω is the viscosity parameter depending on the nature of the fluid. This model can be applicable in many processes where pre-heating of the fuel is used as a means to enhanced heat transfer effect. For some fluids, such as lubricants, polymers, an appropriate constitutive relation where viscosity is a function of temperature should be used.

In the above, primes denote differentiation with respect to *η*. Equation (17) is obtained by integrating Eq. (2) once with the help of Eq. (6).

*x*- and

*y*-directions are, respectively, given by

Here, *R*_{d}, *H*, *Nr*, *Nb*, *Nt*, and Le denote the radiation parameter, the surface temperature parameter, the buoyancy ratio, the Brownian motion parameter, the thermophoresis parameter, and the Lewis number, respectively.

*q*

_{w}is defined as follows:

*Nr*= 0 (pure heat transfer),

*n*= 1, $Nt=Nb=Rd=A*=\Omega =0$, Eqs. (17) and (18), (20) and (21) are reduced as described in Cheng et al. [2] where a nonsimilar solution was obtained before. (The boundary value problem for

*φ*then becomes ill-posed and is of no physical significance.) When $A*=1$,

*ξ*=

*γ*= 0 (

*ξ*→ ∞), i.e., vertical flat plate (vertical cone), Eqs. (17) and (18), (20) and (21) are reduced as reported previously in Groşan and Pop [16] (Groşan et al. [17]) where similar solutions were obtained previously. However, in the case of

*n*= 1, $Rd=A*=\Omega =0$, and

*ξ*=

*γ*= 0 (

*ξ*→ ∞), Eqs. (17)–(21) are reduced to those of Kuznetsov and Nield [54] (Cheng [55]) where a similar solution was obtained previously.

## Numerical Method

This analysis integrates Eqs. (17)–(21) by using the implicit finite difference approximation and the Keller box method developed by Cebeci and Bradshaw [67]. First, differential equations are converted into a system of five first-order equations. These first-order equations are expressed in finite difference forms, and the equations and their boundary conditions are then solved by an iterative scheme. This approach gives a better rate of convergence and reduces the numerical computational times.

Computations were performed on a personal computer with Δ*ξ* = 0.001(0 ≤ *ξ* ≤ 0.01), Δ*ξ* = 0.01(0.01 ≤ *ξ* ≤ 0.1), Δ*ξ* = 0.1(0.1 ≤ *ξ* ≤ 1), Δ*ξ* = 1(1 ≤ *ξ* ≤ 10), Δ*ξ* = 10(10 ≤ *ξ* ≤ 100), Δ*ξ* = 100(100 ≤ *ξ* ≤ 1000), Δ*ξ* = 1000(1000 ≤ *ξ* ≤ 10000) in *ξ*-direction. The first step size Δ*η*_{1} = 0.01 and the variable grid parameter is 1.01 in *η*-direction. The value of boundary-layer edge *η*_{∞} is adjusted for *R*_{d}, *H*, Ω, and *n*. When the errors in computing the dimensionless wall temperature gradient $\theta w\u2032$ and the dimensionless wall nanoparticle volume fraction *φ*_{w} in the next procedure become less than 10^{−5}, the iterative procedure is stopped to give the final dimensionless temperature and nanoparticle volume fraction profiles.

## Results and Discussion

The accuracy of the method was verified by comparing the results with those of Cheng et al. [2], Chamkha et al. [4], Yih [5], Cheng [15], Yih and Huang [26], Chen and Chen [11], Wang and Tu [12], Groşan and Pop [16], Cheng [48], Achemlal et al. [39], Hady et al. [50], Groşan et al. [17], Cheng [14], Rashad et al. [49], Mahmoud [32], Huang [34], Chamkha et al. [52], Bagai and Nishad [38], Bagai [36], and Kuznetsov and Nield [54]. Table 1 compares the values of −*θ*′(*ξ*, 0) for $A*$ with *Nr* = *Nt* = *Nb* = 0, *n* = 1, *R*_{d} = 0, Ω = 0. Tables 2 and 3 compare the values of (a) −*θ*′(0, 0) (vertical plate), (b) −*θ*′(∞, 0) (vertical cone) for $A*$ with *Nr* = *Nt* = *Nb* = 0, *n* = 1, *R*_{d} = 0, Ω = 0, respectively. Table 4 compares the values of $Nux*/Rax*1/(2n)$ for *R*_{d}, *H*, $A*$, and *ξ* with *Nr* = *Nt* = *Nb* = 0, *n* = 1, Ω = 0. Tables 5 and 6 compare the values of (a) −*θ*′(0, 0) (vertical plate), (b) −*θ*′(*ξ*, 0) for Ω and *n* with *Nr* = *Nt* = *Nb* = 0, *R*_{d} = 0, $A*=1$, respectively. Table 7 compares the values of −*θ*′(*ξ*, 0) for *Nr*, *Nt*, *Nb*, Le, and $A*$ with *n* = 1, Ω = 0, *R*_{d} = 0. Tables 1–7 reveal the excellent agreement obtained for all of the above cases.

ξ | −θ′(ξ, 0) | ||||||
---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | ||||||

Cheng et al. [2] | Chamkha et al. [4] | Yih [5] | Cheng [15] | Present results | Yih and Huang [26] | Present results | |

0 | 0.4437 | 0.4444 | 0.4439 | 0.4439 | 0.4437 | −0.2153 | −0.2153 |

0.5 | 0.5412 | 0.5294 | 0.5285 | 0.5285 | 0.5286 | – | −0.1195 |

1 | 0.5991 | 0.5812 | 0.5807 | 0.5807 | 0.5808 | – | −0.0557 |

2 | 0.6572 | 0.6399 | 0.6373 | 0.6373 | 0.6373 | – | 0.0165 |

6 | 0.7219 | 0.7130 | 0.7123 | 0.7116 | 0.7123 | – | 0.1176 |

10 | 0.7391 | 0.7336 | 0.7330 | 0.7326 | 0.7330 | – | 0.1463 |

20 | 0.7532 | 0.7507 | 0.7500 | 0.7490 | 0.7500 | – | 0.1700 |

40 | 0.7607 | 0.7596 | 0.7592 | 0.7587 | 0.7591 | – | 0.1827 |

∞ | 0.7685 | 0.7690 | 0.7686 | 0.7686 | 0.7685 | 0.1957 | 0.1957 |

ξ | −θ′(ξ, 0) | ||||||
---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | ||||||

Cheng et al. [2] | Chamkha et al. [4] | Yih [5] | Cheng [15] | Present results | Yih and Huang [26] | Present results | |

0 | 0.4437 | 0.4444 | 0.4439 | 0.4439 | 0.4437 | −0.2153 | −0.2153 |

0.5 | 0.5412 | 0.5294 | 0.5285 | 0.5285 | 0.5286 | – | −0.1195 |

1 | 0.5991 | 0.5812 | 0.5807 | 0.5807 | 0.5808 | – | −0.0557 |

2 | 0.6572 | 0.6399 | 0.6373 | 0.6373 | 0.6373 | – | 0.0165 |

6 | 0.7219 | 0.7130 | 0.7123 | 0.7116 | 0.7123 | – | 0.1176 |

10 | 0.7391 | 0.7336 | 0.7330 | 0.7326 | 0.7330 | – | 0.1463 |

20 | 0.7532 | 0.7507 | 0.7500 | 0.7490 | 0.7500 | – | 0.1700 |

40 | 0.7607 | 0.7596 | 0.7592 | 0.7587 | 0.7591 | – | 0.1827 |

∞ | 0.7685 | 0.7690 | 0.7686 | 0.7686 | 0.7685 | 0.1957 | 0.1957 |

*ξ* → ∞: denotes *ξ* = 10^{4}.

n | −θ′(0, 0) (vertical plate) | ||||||||
---|---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | ||||||||

Chen and Chen [11] | Wang and Tu [12] | Groşan and Pop [16] | Cheng [48] | Present results | Groşan and Pop [16] | Achemlal et al. [39] | Hady et al. [50] | Present results | |

0.5 | 0.3768 | – | 0.3777 | 0.3771 | 0.3765 | −0.2754 | −0.2755 | −0.2754 | −0.2756 |

0.8 | 0.4238 | 0.424 | 0.4240 | 0.4240 | 0.4237 | −0.2288 | −0.2288 | −0.2288 | −0.2290 |

1.0 | 0.4437 | 0.444 | 0.4439 | 0.4439 | 0.4437 | −0.2152 | −0.2152 | −0.2152 | −0.2153 |

1.5 | 0.4752 | 0.475 | 0.4754 | 0.4755 | 0.4754 | −0.1921 | – | −0.1921 | −0.1923 |

2.0 | 0.4938 | – | 0.4938 | 0.4938 | 0.4938 | −0.1778 | – | – | −0.1780 |

n | −θ′(0, 0) (vertical plate) | ||||||||
---|---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | ||||||||

Chen and Chen [11] | Wang and Tu [12] | Groşan and Pop [16] | Cheng [48] | Present results | Groşan and Pop [16] | Achemlal et al. [39] | Hady et al. [50] | Present results | |

0.5 | 0.3768 | – | 0.3777 | 0.3771 | 0.3765 | −0.2754 | −0.2755 | −0.2754 | −0.2756 |

0.8 | 0.4238 | 0.424 | 0.4240 | 0.4240 | 0.4237 | −0.2288 | −0.2288 | −0.2288 | −0.2290 |

1.0 | 0.4437 | 0.444 | 0.4439 | 0.4439 | 0.4437 | −0.2152 | −0.2152 | −0.2152 | −0.2153 |

1.5 | 0.4752 | 0.475 | 0.4754 | 0.4755 | 0.4754 | −0.1921 | – | −0.1921 | −0.1923 |

2.0 | 0.4938 | – | 0.4938 | 0.4938 | 0.4938 | −0.1778 | – | – | −0.1780 |

n | −θ′(∞, 0) (vertical cone) | ||||||||
---|---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | ||||||||

Groşan et al. [17] | Cheng [14] | Rashad et al. [49] | Mahmoud [32] | Huang [34] | Present results | Groşan et al. [17] | Hady et al. [50] | Present results | |

0.5 | 0.6527 | 0.6521 | 0.6522 | 0.6527 | 0.6521 | 0.6521 | 0.0939 | 0.0939 | 0.0931 |

0.8 | 0.7340 | 0.7339 | 0.7339 | 0.7339 | 0.7338 | 0.7338 | 0.1646 | 0.1647 | 0.1641 |

1.0 | 0.7686 | 0.7686 | 0.7686 | 0.7686 | 0.7685 | 0.7685 | 0.1962 | 0.1962 | 0.1957 |

1.5 | 0.8233 | 0.8233 | 0.8233 | 0.8233 | 0.8232 | 0.8233 | 0.2477 | 0.2478 | 0.2473 |

2.0 | 0.8552 | 0.8552 | 0.8552 | 0.8552 | 0.8552 | 0.8552 | 0.2787 | – | 0.2783 |

n | −θ′(∞, 0) (vertical cone) | ||||||||
---|---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | ||||||||

Groşan et al. [17] | Cheng [14] | Rashad et al. [49] | Mahmoud [32] | Huang [34] | Present results | Groşan et al. [17] | Hady et al. [50] | Present results | |

0.5 | 0.6527 | 0.6521 | 0.6522 | 0.6527 | 0.6521 | 0.6521 | 0.0939 | 0.0939 | 0.0931 |

0.8 | 0.7340 | 0.7339 | 0.7339 | 0.7339 | 0.7338 | 0.7338 | 0.1646 | 0.1647 | 0.1641 |

1.0 | 0.7686 | 0.7686 | 0.7686 | 0.7686 | 0.7685 | 0.7685 | 0.1962 | 0.1962 | 0.1957 |

1.5 | 0.8233 | 0.8233 | 0.8233 | 0.8233 | 0.8232 | 0.8233 | 0.2477 | 0.2478 | 0.2473 |

2.0 | 0.8552 | 0.8552 | 0.8552 | 0.8552 | 0.8552 | 0.8552 | 0.2787 | – | 0.2783 |

R_{d} | H | $Nux*/Rax*1/(2n)$ | ||||||
---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | |||||||

ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | |||

Present results | Present results | Chamkha et al. [52] | Present results | Present results | Present results | Present results | ||

0.5 | 1.1 | 0.5993 | 0.7843 | 1.0380 | 1.0379 | −0.1163 | 0.0952 | 0.4095 |

5 | 1.1 | 1.3486 | 1.7651 | 2.3359 | 2.3358 | 0.5107 | 0.9542 | 1.5714 |

5 | 2 | 2.6824 | 3.5107 | 4.6461 | 4.6458 | 1.7748 | 2.6249 | 3.7924 |

5 | 3 | 4.5757 | 5.9896 | 7.9256 | 7.9258 | 3.6324 | 5.0604 | 7.0181 |

10 | 3 | 6.4553 | 8.4493 | 11.1814 | 11.1811 | 5.4965 | 7.5008 | 10.2489 |

R_{d} | H | $Nux*/Rax*1/(2n)$ | ||||||
---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | |||||||

ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | |||

Present results | Present results | Chamkha et al. [52] | Present results | Present results | Present results | Present results | ||

0.5 | 1.1 | 0.5993 | 0.7843 | 1.0380 | 1.0379 | −0.1163 | 0.0952 | 0.4095 |

5 | 1.1 | 1.3486 | 1.7651 | 2.3359 | 2.3358 | 0.5107 | 0.9542 | 1.5714 |

5 | 2 | 2.6824 | 3.5107 | 4.6461 | 4.6458 | 1.7748 | 2.6249 | 3.7924 |

5 | 3 | 4.5757 | 5.9896 | 7.9256 | 7.9258 | 3.6324 | 5.0604 | 7.0181 |

10 | 3 | 6.4553 | 8.4493 | 11.1814 | 11.1811 | 5.4965 | 7.5008 | 10.2489 |

Ω | −θ′(0, 0) (vertical plate) | ||||||
---|---|---|---|---|---|---|---|

n = 0.5 | n = 1 | n = 2 | |||||

Bagai and Nishad [38] | Present results | Bagai [36] | Bagai and Nishad [38] | Present results | Bagai and Nishad [38] | Present results | |

0 | −0.2574 | −0.2576 | −0.2152 | −0.2152 | −0.2153 | −0.1780 | −0.1780 |

0.1 | −0.2090 | −0.2091 | −0.19 | −0.1891 | −0.1892 | −0.1644 | −0.1644 |

0.2 | −0.1586 | −0.1587 | −0.16 | −0.1625 | −0.1626 | −0.1507 | −0.1507 |

0.5 | 0.0070 | 0.0070 | −0.08 | −0.0791 | −0.0792 | −0.1086 | −0.1086 |

0.75 | 0.1659 | 0.1658 | −0.0055 | −0.0053 | −0.0054 | −0.0725 | −0.0725 |

Ω | −θ′(0, 0) (vertical plate) | ||||||
---|---|---|---|---|---|---|---|

n = 0.5 | n = 1 | n = 2 | |||||

Bagai and Nishad [38] | Present results | Bagai [36] | Bagai and Nishad [38] | Present results | Bagai and Nishad [38] | Present results | |

0 | −0.2574 | −0.2576 | −0.2152 | −0.2152 | −0.2153 | −0.1780 | −0.1780 |

0.1 | −0.2090 | −0.2091 | −0.19 | −0.1891 | −0.1892 | −0.1644 | −0.1644 |

0.2 | −0.1586 | −0.1587 | −0.16 | −0.1625 | −0.1626 | −0.1507 | −0.1507 |

0.5 | 0.0070 | 0.0070 | −0.08 | −0.0791 | −0.0792 | −0.1086 | −0.1086 |

0.75 | 0.1659 | 0.1658 | −0.0055 | −0.0053 | −0.0054 | −0.0725 | −0.0725 |

Ω | −θ′(ξ, 0) | |||||
---|---|---|---|---|---|---|

ξ = 1 | ξ → ∞: vertical cone | |||||

n = 0.5 | n = 1 | n = 2 | n = 0.5 | n = 1 | n = 2 | |

0 | −0.1233 | −0.0557 | 0.0005 | 0.0931 | 0.1957 | 0.2783 |

0.1 | −0.0678 | −0.0257 | 0.0161 | 0.1607 | 0.2323 | 0.2975 |

0.2 | −0.0096 | 0.0049 | 0.0319 | 0.2322 | 0.2699 | 0.3169 |

0.5 | 0.1849 | 0.1016 | 0.0805 | 0.4740 | 0.3890 | 0.3766 |

0.75 | 0.3758 | 0.1882 | 0.1224 | 0.7154 | 0.4966 | 0.4283 |

Ω | −θ′(ξ, 0) | |||||
---|---|---|---|---|---|---|

ξ = 1 | ξ → ∞: vertical cone | |||||

n = 0.5 | n = 1 | n = 2 | n = 0.5 | n = 1 | n = 2 | |

0 | −0.1233 | −0.0557 | 0.0005 | 0.0931 | 0.1957 | 0.2783 |

0.1 | −0.0678 | −0.0257 | 0.0161 | 0.1607 | 0.2323 | 0.2975 |

0.2 | −0.0096 | 0.0049 | 0.0319 | 0.2322 | 0.2699 | 0.3169 |

0.5 | 0.1849 | 0.1016 | 0.0805 | 0.4740 | 0.3890 | 0.3766 |

0.75 | 0.3758 | 0.1882 | 0.1224 | 0.7154 | 0.4966 | 0.4283 |

Nr | Nt | Nb | Le | −θ′(ξ, 0) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | |||||||||

ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | |||||

Kuznetsov and Nield [54] | Present results | Present results | Present results | Present results | Present results | Present results | ||||

0.1 | 0.1 | 0.1 | 10 | 0.4334 | 0.4329 | 0.5666 | 0.7498 | −0.2260 | −0.0683 | 0.1815 |

0.1 | 0.5 | 0.1 | 10 | 0.3914 | 0.3916 | 0.5126 | 0.6783 | −0.2679 | −0.1180 | 0.1266 |

0.5 | 0.5 | 0.5 | 1000 | 0.3620 | 0.3643 | 0.4767 | 0.6309 | −0.2766 | −0.1264 | 0.1128 |

Nr | Nt | Nb | Le | −θ′(ξ, 0) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

$A*=0$ | $A*=1$ | |||||||||

ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | ξ = 0: vertical plate | ξ = 1 | ξ → ∞: vertical cone | |||||

Kuznetsov and Nield [54] | Present results | Present results | Present results | Present results | Present results | Present results | ||||

0.1 | 0.1 | 0.1 | 10 | 0.4334 | 0.4329 | 0.5666 | 0.7498 | −0.2260 | −0.0683 | 0.1815 |

0.1 | 0.5 | 0.1 | 10 | 0.3914 | 0.3916 | 0.5126 | 0.6783 | −0.2679 | −0.1180 | 0.1266 |

0.5 | 0.5 | 0.5 | 1000 | 0.3620 | 0.3643 | 0.4767 | 0.6309 | −0.2766 | −0.1264 | 0.1128 |

The local Nusselt number is almost independent of the Brownian motion parameter *Nb* and the buoyancy ratio *Nr* [54,55]. Hence, the numerical results are presented for *Nb* = *Nr* = 0.5 with the dimensionless streamwise coordinate *ξ* (range, 0–10^{4}), the thermophoresis parameter *Nt* (range, 0.1–0.5), the Lewis number Le (range, 10–1000), the radiation parameter *R*_{d} (range, 0.5–3), the surface temperature parameter *H* (range, 1.1–1.5), the viscosity parameter Ω (range, 0–0. 5), the power-law index of the non-Newtonian fluid *n* (range, 0.5–1.5), and the internal heat generation coefficient $A*$ (range, 0–1).

Figures 2 and 3 illustrate the effects of two values of the radiation parameter *R*_{d} (*R*_{d} = 1, 3) and the surface temperature parameter *H* (*H* = 1.1, 1.5) on the dimensionless temperature profile *θ* and the dimensionless nanoparticle volume fraction profile *φ* with *Nr* = *Nb* = 0.5, *Nt* = 0.3, Le = 10, $A*=1$, *n* = 0.5, Ω = 0.2, *ξ* = 1, respectively. Figure 2 shows that, when both the radiation parameter *R*_{d} and the surface temperature parameter *H* increase, the dimensionless temperature profile becomes wide, but the dimensionless wall temperature gradient [−*θ*′(*ξ*, 0)] becomes narrow. This occurs because the value of *R*_{d} or *H* increases, the radiation absorption in the boundary layer increases, causing the dimensionless temperature profile to become large. However, Fig. 3 shows that the dimensionless nanoparticle volume fraction profile reduces with increasing the radiation parameter and the surface temperature parameter.

Table 8 illustrates the values of the local Nusselt number $Nux*/Rax*1/(2n)$ for various values of *ξ*, *R*_{d}, and *H* with *Nr* = *Nb* = 0.5, *Nt* = 0.3, Le = 10, $A*=1$, *n* = 0.5, Ω = 0.2. For the fixed value of *ξ*, the local Nusselt number tends to enhance as the radiation parameter *R*_{d} and the surface temperature parameter *H* are increased. In the pure convection heat transfer, the local Nusselt number is only proportional to the dimensionless wall temperature gradient [−*θ*′(*ξ*, 0)]. However, Fig. 2 shows that, in the coupled convection and radiation, for the case of large *R*_{d} and *H* (radiation effect becomes significant), although the value of [−*θ*′(*ξ*, 0)] is low, the local Nusselt number is still large. This is because the local Nusselt number is found to be more sensitive to *R*_{d} and *H* than [−*θ*′(*ξ*, 0)], as revealed in Eq. (32). Moreover, when *R*_{d} and *H* are fixed, increasing the dimensionless streamwise coordinate *ξ* increases the local Nusselt number. Additionally, the local Nusselt number approaches to constant value when *ξ* is small (*ξ* = 0: vertical flat plate) and large (*ξ* → ∞: vertical full cone).

ξ | $Nux*/Rax*1/(2n)$ | |||||
---|---|---|---|---|---|---|

R_{d} = 1 | R_{d} = 3 | |||||

H = 1.1 | H = 1.3 | H = 1.5 | H = 1.1 | H = 1.3 | H = 1.5 | |

0 | 0.0055 | 0.0912 | 0.1930 | 0.3041 | 0.4920 | 0.7039 |

0.001 | 0.0058 | 0.0916 | 0.1935 | 0.3047 | 0.4926 | 0.7047 |

0.01 | 0.0092 | 0.0955 | 0.1979 | 0.3098 | 0.4987 | 0.7119 |

0.1 | 0.0408 | 0.1323 | 0.2404 | 0.3577 | 0.5560 | 0.7793 |

1 | 0.2449 | 0.3667 | 0.5085 | 0.6600 | 0.9151 | 1.2000 |

10 | 0.5309 | 0.6895 | 0.8731 | 1.0683 | 1.3951 | 1.7586 |

100 | 0.5925 | 0.7586 | 0.9506 | 1.1547 | 1.4962 | 1.8759 |

1000 | 0.5991 | 0.7660 | 0.9590 | 1.1641 | 1.5071 | 1.8885 |

∞ | 0.5998 | 0.7668 | 0.9598 | 1.1650 | 1.5082 | 1.8898 |

ξ | $Nux*/Rax*1/(2n)$ | |||||
---|---|---|---|---|---|---|

R_{d} = 1 | R_{d} = 3 | |||||

H = 1.1 | H = 1.3 | H = 1.5 | H = 1.1 | H = 1.3 | H = 1.5 | |

0 | 0.0055 | 0.0912 | 0.1930 | 0.3041 | 0.4920 | 0.7039 |

0.001 | 0.0058 | 0.0916 | 0.1935 | 0.3047 | 0.4926 | 0.7047 |

0.01 | 0.0092 | 0.0955 | 0.1979 | 0.3098 | 0.4987 | 0.7119 |

0.1 | 0.0408 | 0.1323 | 0.2404 | 0.3577 | 0.5560 | 0.7793 |

1 | 0.2449 | 0.3667 | 0.5085 | 0.6600 | 0.9151 | 1.2000 |

10 | 0.5309 | 0.6895 | 0.8731 | 1.0683 | 1.3951 | 1.7586 |

100 | 0.5925 | 0.7586 | 0.9506 | 1.1547 | 1.4962 | 1.8759 |

1000 | 0.5991 | 0.7660 | 0.9590 | 1.1641 | 1.5071 | 1.8885 |

∞ | 0.5998 | 0.7668 | 0.9598 | 1.1650 | 1.5082 | 1.8898 |

Figures 4 and 5 plot the dimensionless temperature profile and the dimensionless nanoparticle volume fraction profile for two values of the thermophoresis parameter *Nt* (*Nt* = 0.1, 0.5) and the Lewis number Le (Le = 10, 1000), respectively, with *Nr* = *Nb* = 0.5, *R*_{d} = 0.5, *H* = 1.2, $A*=1$, *n* = 1, Ω = 0.5, *ξ* = 0.1. In Fig. 4, for the fixed Le, increasing the thermophoresis parameter *Nt* widens the dimensionless temperature profile and narrows the dimensionless wall temperature gradient. This occurs because the thermophoresis force generated by the temperature gradient moves the fluid fast away from the surface. Therefore, the nanoparticles from the hot surface are driven toward the ambient, which has a larger effect on the fluid and causes a larger dimensionless temperature profile. However, for the given *Nt*, the dimensionless wall temperature gradient slightly decreases when the Lewis number Le increases. In Fig. 5, it is apparent when *Nt* increases or Le decreases, dimensionless nanoparticle volume fraction profile increases.

Table 9 lists the local Nusselt number $Nux*/Rax*1/(2n)$ for various values of *ξ*, Le, and *Nt* with *Nr* = *Nb* = 0.5, *R*_{d} = 0.5, *H* = 1.2, $A*=1$, *n* = 1, Ω = 0.5. Results show that for the fixed *ξ* and Le, increasing the thermophoresis parameter *Nt* reduces the local Nusselt number. Figure 4 shows that this occurs because the increase in the thermophoresis parameter tends to decrease the dimensionless wall temperature gradient, which reduces the local Nusselt number. Besides, for the given *ξ* and *Nt*, as the Lewis number Le is enhanced, the local Nusselt number a little decreases.

ξ | $Nux*/Rax*1/(2n)$ | |||||
---|---|---|---|---|---|---|

Le = 10 | Le = 1000 | |||||

Nt = 0.1 | Nt = 0.3 | Nt = 0.5 | Nt = 0.1 | Nt = 0.3 | Nt = 0.5 | |

0 | 0.0687 | 0.0496 | 0.0310 | 0.0676 | 0.0465 | 0.0261 |

0.1 | 0.1060 | 0.0862 | 0.0668 | 0.1048 | 0.0828 | 0.0615 |

1 | 0.3227 | 0.2988 | 0.2755 | 0.3208 | 0.2937 | 0.2675 |

10 | 0.6275 | 0.5989 | 0.5711 | 0.6244 | 0.5905 | 0.5579 |

∞ | 0.7007 | 0.6712 | 0.6423 | 0.6974 | 0.6618 | 0.6277 |

ξ | $Nux*/Rax*1/(2n)$ | |||||
---|---|---|---|---|---|---|

Le = 10 | Le = 1000 | |||||

Nt = 0.1 | Nt = 0.3 | Nt = 0.5 | Nt = 0.1 | Nt = 0.3 | Nt = 0.5 | |

0 | 0.0687 | 0.0496 | 0.0310 | 0.0676 | 0.0465 | 0.0261 |

0.1 | 0.1060 | 0.0862 | 0.0668 | 0.1048 | 0.0828 | 0.0615 |

1 | 0.3227 | 0.2988 | 0.2755 | 0.3208 | 0.2937 | 0.2675 |

10 | 0.6275 | 0.5989 | 0.5711 | 0.6244 | 0.5905 | 0.5579 |

∞ | 0.7007 | 0.6712 | 0.6423 | 0.6974 | 0.6618 | 0.6277 |

Figures 6 and 7 present the effects of two values of the viscosity parameter Ω (Ω = 0, 0.5) and the internal heat generation coefficient $A*$$(A*=0,1)$ on the dimensionless temperature profile and the dimensionless nanoparticle volume fraction profile with *Nr* = *Nb* = 0.5, *Nt* = 0.2, Le = 5, *R*_{d} = 2, *H* = 1.5, *n* = 0.5, *ξ* = 10, respectively. Figure 6 shows that increasing the viscosity parameter or decreasing the internal heat generation coefficient enhances the dimensionless wall temperature gradient. Figure 7 illustrates that the dimensionless wall nanoparticle volume fraction enhances with an increase in the viscosity parameter or a decrease in the internal heat generation coefficient.

Table 10 lists the values of the local Nusselt number $Nux*/Rax*1/(2n)$ for various values of *ξ*, Ω, and $A*$ with *Nr* = *Nb* = 0.5, *Nt* = 0.2, Le = 5, *R*_{d} = 2, *H* = 1.5, *n* = 0.5. First, when *ξ* and $A*$ are fixed, as the viscosity parameter Ω is enhanced, the local Nusselt number increases. This occurs because enhancing the viscosity parameter tends to decrease the viscosity of the fluid, thus accelerating the velocity of fluid and increasing the local Nusselt number. Second, when *ξ* and Ω are fixed, the local Nusselt number tends to decrease when the internal heat generation coefficient $A*$ increases. Increasing the internal heat generation coefficient increases the thermal boundary layer thickness, as revealed in Fig. 6. As the thermal boundary layer thickness increase, the local Nusselt number decreases.

ξ | $Nux*/Rax*1/(2n)$ | |||||
---|---|---|---|---|---|---|

Ω = 0 | Ω = 0.5 | |||||

$A*=0$ | $A*=0.5$ | $A*=1$ | $A*=0$ | $A*=0.5$ | $A*=1$ | |

0 | 1.0736 | 0.6652 | 0.2585 | 1.6184 | 1.2480 | 0.8808 |

0.1 | 1.1242 | 0.7178 | 0.3131 | 1.6948 | 1.3260 | 0.9603 |

1 | 1.4051 | 1.0110 | 0.6188 | 2.1181 | 1.7605 | 1.4062 |

10 | 1.7734 | 1.3988 | 1.0262 | 2.6734 | 2.3371 | 2.0042 |

∞ | 1.8594 | 1.4897 | 1.1221 | 2.8029 | 2.4725 | 2.1454 |

ξ | $Nux*/Rax*1/(2n)$ | |||||
---|---|---|---|---|---|---|

Ω = 0 | Ω = 0.5 | |||||

$A*=0$ | $A*=0.5$ | $A*=1$ | $A*=0$ | $A*=0.5$ | $A*=1$ | |

0 | 1.0736 | 0.6652 | 0.2585 | 1.6184 | 1.2480 | 0.8808 |

0.1 | 1.1242 | 0.7178 | 0.3131 | 1.6948 | 1.3260 | 0.9603 |

1 | 1.4051 | 1.0110 | 0.6188 | 2.1181 | 1.7605 | 1.4062 |

10 | 1.7734 | 1.3988 | 1.0262 | 2.6734 | 2.3371 | 2.0042 |

∞ | 1.8594 | 1.4897 | 1.1221 | 2.8029 | 2.4725 | 2.1454 |

Figures 8 and 9 illustrate the dimensionless temperature profile and the dimensionless nanoparticle volume fraction profile for three values of the power-law index *n* (*n* = 0.5, 1, and 1.5) with *Nr* = *Nb* = 0.5, *Nt* = 0.4, Le = 50, *R*_{d} = 2.5, *H* = 1.3, $A*=1$, Ω = 0.1, *ξ* = 100, respectively. Figure 8 shows that increasing the power-law index causes both the dimensionless temperature profile and the thermal boundary thickness to become thin, and thus enhancing the dimensionless wall temperature gradient. However, Fig. 9 shows that an increase in the power-law index decreases the dimensionless nanoparticle volume fraction profile.

Table 11 gives the values of the local Nusselt number $Nux*/Rax*1/(2n)$ for various values of *ξ* and *n* with *Nr* = *Nb* = 0.5, *Nt* = 0.4, Le = 50, *R*_{d} = 2.5, *H* = 1.3, $A*=1$, Ω = 0.1. The table indicates that, for the fixed *ξ*, enhancing the power-law index *n* tends to increase the local Nusselt number because an increase in the power-law index tends to reduce the thermal boundary layer thickness, as shown in Fig. 8. As the thinner the thermal boundary layer thickness decreases, the local Nusselt number increases.

ξ | $Nux*/Rax*1/(2n)$ | ||||
---|---|---|---|---|---|

n = 0.5 | n = 0.8 | n = 1 | n = 1.2 | n = 1.5 | |

0 | 0.2901 | 0.3594 | 0.3863 | 0.4115 | 0.4377 |

0.1 | 0.3449 | 0.4178 | 0.4467 | 0.4732 | 0.5008 |

1 | 0.6526 | 0.7458 | 0.7860 | 0.8189 | 0.8549 |

10 | 1.0648 | 1.1847 | 1.2388 | 1.2805 | 1.3274 |

∞ | 1.1620 | 1.2881 | 1.3452 | 1.3890 | 1.4383 |

ξ | $Nux*/Rax*1/(2n)$ | ||||
---|---|---|---|---|---|

n = 0.5 | n = 0.8 | n = 1 | n = 1.2 | n = 1.5 | |

0 | 0.2901 | 0.3594 | 0.3863 | 0.4115 | 0.4377 |

0.1 | 0.3449 | 0.4178 | 0.4467 | 0.4732 | 0.5008 |

1 | 0.6526 | 0.7458 | 0.7860 | 0.8189 | 0.8549 |

10 | 1.0648 | 1.1847 | 1.2388 | 1.2805 | 1.3274 |

∞ | 1.1620 | 1.2881 | 1.3452 | 1.3890 | 1.4383 |

## Conclusions

A steady, two-dimensional, laminar boundary layer analysis was performed to study the effect of zero nanoparticles flux (ZNF) on natural convection over an isothermal vertical truncated cone embedded in Darcy porous media filled with a nanofluid. Thermophoresis and Brownian motion effects are also considered. After the coordinate transformation is performed, the transformed governing equations are obtained and solved by KBM. Comparisons with previously published work revealed good agreement. The numerical analysis results of present study are summarized as follows:

The local Nusselt number increases with increasing the radiation parameter

*R*_{d}, the surface temperature parameter*H*, and the dimensionless streamwise coordinate*ξ*.Increasing the thermophoresis parameter

*Nt*and the Lewis number Le reduces the local Nusselt number.Increasing the viscosity parameter Ω increases the local Nusselt number. However, enhancing the internal heat generation coefficient $A*$ has tendency to decrease the local Nusselt number.

Enhancing the power-law index

*n*tends to increase the local Nusselt number.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

## Nomenclature

*c*=specific heat at constant pressure

*d*=particle diameter

*f*=dimensionless stream function

*g*=gravitational acceleration

*k*=equivalent thermal conductivity

*r*=local radius of the vertical truncated cone

*u*=velocity in the

*x*-direction*v*=velocity in the

*y*-direction*x*=streamwise coordinate

*y*=transverse coordinate

*C*=nanoparticle volume fraction

*H*=surface temperature parameter

*T*=temperature of fluid

*a*_{r}=Rosseland mean extinction coefficient

- $hx*$ =
local convective heat transfer coefficient

*x*_{o}=distance of the leading edge of vertical truncated cone measured from the origin

*D*_{B}=Brownian diffusion coefficient

*D*_{T}=thermophoretic diffusion coefficient

*R*_{d}=radiation parameter

*q*″′ =internal heat generation rate per unit volume

- $x*$ =
distance measured from the leading edge of the vertical truncated cone

- $A*$ =
internal heat generation coefficient

*K*(*n*) =modified permeability of the porous medium

- Le =
Lewis number

*Nb*=Brownian motion parameter

*Nr*=buoyancy ratio

*Nt*=thermophoresis parameter

- $Nux*$ =
local Nusselt number

- $Rax*$ =
local Rayleigh number

### Greek Symbols

*α*=equivalent thermal diffusivity

*β*=volumetric expansion coefficient

*γ*=half angle of the vertical truncated cone

*ɛ*=porosity

*η*=pseudo-similarity variable

*θ*=dimensionless temperature

*μ*=viscosity of fluid

*ξ*=dimensionless streamwise coordinate

*ρ*_{f}=fluid density

*ρ*_{p}=nanoparticle mass density

- (
*ρc*)_{f}= heat capacity of the fluid

- (
*ρc*)_{p}= heat capacity of nanoparticle material

*σ*_{o}=Stefan–Boltzmann constant

*σ*_{s}=scattering coefficient

*τ*=heat capacity ratio between nanoparticle and fluid

*φ*=dimensionless nanoparticle volume fraction

*ψ*=stream function

- Ω =
viscosity parameter

### Subscripts

## References

_{2}O

_{3}–Water Nanofluids Considering Variable Thermal Conductivity Around a Vertical Cone in Porous Media

_{2}O

_{3}–Water and TiO

_{2}–Water Nanofluids Past a Stretching Sheet in Porous Media With Heat Generation/Absorption