Numerical Study of Natural Convection Flow of Nanoﬂuid Past a Circular Cone with Cattaneo–Christov Heat and Mass Flux Models

: The objective of this study is to analyze the natural convection ﬂow of nanoﬂuid along a circular cone placed in a vertical direction. The generalized heat ﬂux and mass ﬂux models are commonly known as the Cattaneo–Christov heat ﬂux model and mass ﬂux models. In the present study, these models are used for both heat and mass transfers analysis in nanoﬂuid ﬂow. For the governing equations, the Buongiorno transport model is used in which two important slip mechanism, namely thermophoresis and Brownian motion parameters, are discussed. The resulting governing equations in the form of partial di ﬀ erential equations (PDEs) are converted into ordinary di ﬀ erential equations (ODEs) due to similar ﬂow along the surface of a circular cone. To solve these ODEs, a numerical algorithm based on implicit ﬁnite di ﬀ erence scheme is utilized. The e ﬀ ects of dimensionless parameters on heat and mass transfer in nanoﬂuid ﬂow are discussed graphically in the form of velocity proﬁle, temperature proﬁle, Sherwood number and Nusselt number. It is noted that in the presence of the Cattaneo–Christov heat ﬂux model and mass ﬂux model, the heat transfer rate decreases by increasing both thermal and concentration relaxation parameters; however, Sherwood number decreases by increasing the thermal relaxation parameter, and increases by increasing the concentration relaxation parameter.


Introduction
Improvement in the efficiency of cooling systems is essential in various industrial and engineering processes. The excellent functioning of a cooling system requires a thermally efficient coolant. However, the relatively poor thermal conductivity of conventional coolants is a significant constraint on the emergence of highly effective cooling systems. In most of the heat transfer processes, conventional fluids or base fluids like water, engine oil, and ethylene glycol, etc. are used as coolants. Due to low thermal conductivity of these fluids, they do not provide efficient results in a cooling system. An inventive procedure for enhancing the heat transfer rate in the conventional fluids is by colloidal suspension of nanosized particles in base fluid, resulting in a mixture known as nanofluid [1,2]. Nanofluid exhibits excellent potential, considering the substantial increase in heat transfer rates in a variety of applications. Nanotechnology has been the source for the creation of nanoparticles in dimensions of nanometers, which possess distinctive chemical and physical properties. Nanofluid possesses high thermal conductivity and promotes the heat transfer rate. Nanofluid helps in manufacturing light and smart heat exchangers. Due to nano-sized particles, the fluid mixture is homogenous and stable without having the problems of sedimentation and clogging. Nanofluids are most suitable for heating and cooling systems. Therefore, for heating, nanofluids can be used to improve the heat transfer phenomena in diverse thermal systems; and for cooling, its applications include engine cooling, refrigeration, use in the petroleum industry, cancer therapy, nano-drug delivery, environmental remediation, inkjet printing, etc. The work of the nanofluid fundamentally depends upon the size, percentage of nanoparticle concentration, ability to stay suspended in base fluid, shape, and chemical unreactive in the base fluid.
Heat transfer is an important phenomenon in nature, which occurs due to the temperature difference between two bodies or within the same body. The heat transfer characteristics have wide-ranging demands in numerous industrial and engineering processes, such as: nuclear reactors, fuel cells, transportation, microelectronics, etc. Fourier's law of heat conduction explains the heat transfer phenomena in various practical processes. Fourier's law gives a parabolic energy equation in which the whole system is instantly affected by the initial disturbance. To handle this situation, a modified version of Fourier's heat conduction law was introduced by Cattaneo including a relaxation time. Christov incorporated Cattaneo's theory and replaced the time derivative with an Oldroyd upper-convected derivative. The derivative model of Cattaneo's law is termed as the Cattaneo-Christov heat flux model. Many researchers and scientists studied self-similar solutions for natural convection flow over a vertical cone. Hering and Grosh [3,4] studied natural convection flow along the vertical cone and reported the similarity solution. Roy [5] investigated the heat transfer phenomena for a large Prandtl number in natural convection flow along the vertical cone. Vajravelu and Nayfeh [6] studied heat transfer analysis in a viscous heat source fluid along a cone and wedge surface and concluded that the flow and heat transfer rates have smaller values along the cone surface, as compared to the wedge surface. The study of Kafoussias [7] is related to the isothermal vertical cone. He grasped from the results, that bouncy parameter and Schmidt number strongly affect the heat and mass transfer rates.
Yih [8] studied the radiation effects along a truncated isothermal cone and found the significant contribution in the enhancement of Nusselt numbers. Behrang et al. [9] performed heat transfer analysis along a vertical cone saturated in a porous medium. He established a new approach named the hybrid neural network-particle swarm optimization method and concluded that values of the Nusselt numbers found a good agreement with the numerically computed values. Cheng [10] discussed natural convection flow of Newtonian fluid along the non-isothermal permeable vertical cone with suction and variable properties. Through results, he concluded that the Nusselt number becomes high with the increase of suction and viscosity variation parameters. Duwairi et al. [11] considered magnetohydrodynamic (MHD) mixed convection flow over a cone and observed that the Nusselt number increases with the increase of cone angle in a porous medium. Elbashbeshy et al. [12] examined natural convection flow phenomena along a vertical circular cone in the presence of pressure work, variable heat flux and heat generation. They observed that the skin friction increases, and the Nusselt number decreases with the increments in heat generation parameters. Braun et al. [13] studied free convection similarity flows along the families of bodies with closed lower ends and observed that the body shape parameter enhances the heat transfer rate. Grosan [14] examined free convection flow over a vertical cone in a viscoelastic fluid with a heat source in a porous medium. Chamkha et al. [15] discussed the effects of a combined chemical reaction and pressure work in natural convection flow. They found that the Nusselt number decayed with the enhancement of heat generation, chemical reaction parameter, and the Schmidt number. Sohouli et al. [16] analyzed the free convection analytically by using the Homotopy Analysis Method (HAM) in the Darcian fluid along a vertical cone.
Several researchers and scientists examined different techniques and models to explain the heat transfer phenomena of nanofluid. There are two types of models available in the literature, namely, the homogenous and the non-homogenous. Among homogenous theoretical models, the Buongiorno transport model [17,18] and Tiwari and Das model [19] are the most famous ones. Buongiorno (2006) developed a theoretical model for convective transport in nanofluid by incorporating Brownian motion and thermophoresis effects, whereas thermophysical properties of the nanoparticles were introduced by Tiwari and Das [19] in their proposed model.
In the recent past, attention has been given to the boundary layer flow of nanofluid over a cone. Mahdy [20] numerically computed the Sherwood and Nusselt numbers for the case of natural convection flow along a vertical cone. He noticed with the variation of Brownian motion and thermophoresis parameters, the Nusselt number decreased and the Sherwood number increased. Behseresht et al. [21] discussed the free convection flow of nanofluid along a vertical cone using the Buongiorno model in a porous medium. They noticed that the change in the heat transfer rate is negligible due to the migration of nanoparticles in comparison with convection and heat conduction phenomenon. Noghrehabadi et al. [22] discussed the natural convection flow of a nanofluid past an isothermal vertical cone in a non-Darcy porous medium and noted that both heat and mass transfer rates reduce with increasing non-Darcy parameters. Keshtkar and Hadizadeh [23] investigated boundary layer nanofluid flow along a vertical cone in a porous medium. Fauzi et al. [24] studied mixed convection nanofluid flow along a vertical cone. In a series of papers, Khan et al. [25,26] presented mathematical models to investigate the natural convection flow of a water-based nanofluid containing gyrotactic microorganisms over a truncated cone with a convective boundary condition at the surface. They found that in case of non-Newtonian nanofluids, the local Nusselt and the local Sherwood numbers are found to be higher for dilatant nanofluids than pseudoplastic and Newtonian fluid.
Straughan [27] investigated thermal convection phenomena with the Cattaneo-Christov heat flux model. Tibulle and Zampoli [28] examined the Cattaneo-Christov heat flux model for incompressible fluid flows. Kumar et al. [29] studied the MHD flow over a cone and a wedge with the Cattaneo-Christov heat flux model and shows that the heat transfer rate in the fluid flow over a cone is higher than that of the flow over a wedge. Further, numerous researchers [30][31][32][33][34][35] used the Cattaneo-Christov heat flux model to formulate energy equations and discuss the flow and heat transfer phenomenon for different types of non-Newtonian fluids.
With such an intensive literature review, we came to know that a study on natural convection flow along a vertical cone under the effects of Brownian motion, thermophoresis parameter, buoyancy force and the presence of Cattaneo-Christov heat flux model, has not yet been examined. To deal with this theoretical investigation, self-similar transformations are used and obtained by a coupled system of non-linear ordinary differential equations. The problem is solved numerically by applying the Keller-box scheme [36,37]. The impact of different involved parameters on the concentration, temperature and velocity profiles, Nusselt number, Sherwood number and skin friction are presented through graphs. The numerical values of the Nusselt and the Sherwood numbers are computed and reported in the form of Tables.

Mathematical Formulation
Steady two-dimension flow problem along a vertical cone of a circular base with radius r is considered. The symbol γ is used to represent the internal half-angle cone. The temperature and concentration at the surface of the circular cone are kept constant, T w i.e., (T w > T ∞ ) and C w (C w > C ∞ ). The symbols T ∞ and C ∞ are used to represent the constant ambient temperature and concentration far away from the surface. The flow is developed in an upward direction. The x-axis is taken along the surface of the cone, and the y-axis is taken normal to it as depicted in Figure 1. Brownian motion and thermophoresis effects of nanoparticles are considered, which are studied using the Buongiorno nanofluid model. According to the Buongiorno's model [21,22], the mass, momentum and energy conservation laws after the consideration of the above assumptions are written as: Mass Momentum The generalized heat and mass flux models for the thermal and concentration diffusions are commonly termed as Cattaneo-Christov heat and mass flux models. These models are defined as: where and are the relaxation time parameter for heat flux and mass flux, k represents the thermal conductivity, and is used to represent the mass diffusion coefficient. The abovegeneralized flux models become to Fourie and Fick's laws, if = 0, and = 0, that is: Hence after the incorporation of Cattaneo-Christov heat and mass flux models, energy and concentration equations can be written as [19,20]: Energy Concentration where and Brownian motion and thermophoresis effects of nanoparticles are considered, which are studied using the Buongiorno nanofluid model. According to the Buongiorno's model [21,22], the mass, momentum and energy conservation laws after the consideration of the above assumptions are written as: Mass Momentum The generalized heat and mass flux models for the thermal and concentration diffusions are commonly termed as Cattaneo-Christov heat and mass flux models. These models are defined as: where λ E and λ C are the relaxation time parameter for heat flux and mass flux, k represents the thermal conductivity, and D m is used to represent the mass diffusion coefficient. The above-generalized flux models become to Fourie and Fick's laws, if λ E = 0, and λ C = 0, that is: Hence after the incorporation of Cattaneo-Christov heat and mass flux models, energy and concentration equations can be written as [19,20]: Energy Concentration where The used symbols are defined as: The local radius r of the cone surface is described as r(x) = xSinγ, as shown in Figure 1. The appropriate boundary conditions are written as: The stream function in the polar form is defined as rv = −∂ψ/∂x , ru = ∂ψ/∂y, which satisfies the continuity equation identically. To achieve the dimensionless form, the following transformations are utilized (see [6]): which transforms the governing PDEs (2)-(8) to dimensionless ODEs i.e., 1 1 Le where are the Grashof number, Lewis number, Brownian motion parameter, thermophoresis parameter, and Prandtl number respectively. Accordingly, the boundary conditions are also transformed and written as The aforementioned prime symbol " ' " is used to represent the differentiation with respect to η. The wall shear stress (τ w = µ(∇u ) y=0 ,) the surface heat flux (q w = −κ(∇T) y=0 ) and the surface mass flux (q m = −D B (∇C) y=0 ) are the quantities of physical interest. The coefficient of skin friction, the Nusselt number and the Sherwood number in the non-dimensional form are where C f x = τ w /ρU 2 , Nu x = xq w /κ(T w − T ∞ ) and Sh x = xq m /D B (C w − C ∞ ) denote the coefficient of skin friction, the Nusselt number, and the Sherwood number respectively.

Method of Solution
The Keller-box [36,37] method, based on the finite difference scheme, is used to find the solution in numerical form for Equations (13)-(15) with the given boundary conditions (Equation (17)). The implementation of the Keller-box method can be described in the following steps: (i) The higher-order differential equations are transformed into the first-order. For this purpose, lets us consider then Equations (13)- (15) can take the following form and the boundary conditions are where δ() i represents the small increment in the function's values. The second and higher orders' terms in δ() i are neglected. (iv) Linearized algebraic equations are finally solved through block-tridiagonal elimination method.
The present results of the skin friction coefficient C f x (Gr/x) 1 4 and heat transfer rate Nu x Grx 3 −1/4 have been compared with the available results of Yih et al. [8] in Table 1, and excellent agreement is achieved. This validates our current solution scheme and allows for further analysis.

Results and Discussion
The graphical representations of the dimensionless velocity f (η), temperature θ(η), and nanoparticle concentration ϕ(η) profiles, in addition to the local Sherwood number Sh = −ϕ (0) and local Nusselt number Nu= = −θ (0) are made in this Section. The impact of the involved parameters on the graphs of the quantities named above are discussed in detail. Some numerical values of Sh and Nu against various values of parameters γ 1 , γ 2 , Nb, Le, Nt, and Pr are given in Tables 2-4. Table 2 reveals that the Nu reduces with the increase of Nt and Nb; however, Sh rises with increasing Nt and Nb. Table 2 shows that Sh decreases for high values of Nt and Nb. Table 3 provides numerical results for the impact of Pr and Le on heat transfer rate Nu and mass transfer rate Sh. It is noted that Nu decreases and Sh increases with increasing Pr and Le. Table 4 is prepared for the numerical data of the local Nusselt number Nu and Sherwood number Sh for various values of the thermal relaxation parameter δ 1 and concentration relaxation parameter δ 2 . It is observed that Nu has lower values when the larger values of δ 1 and δ 2 are taken. It is further observed that numerical values of Sh are higher for increasing values of δ 2 and lower for higher values of δ 1 .    Figure 2 illustrates the impact of thermal and concentration relaxation parameters δ 1 and δ 2 on the dimensionless profiles i.e., (η), θ(η), and ϕ(η). It is observed that the thermal relaxation parameter reduces the velocity profile, whereas no significant change occurs in the case of the concentration relaxation parameter. It is further observed the temperature within the nanofluid reduces with increasing the values of δ 1 , but enhancement is observed in the dimensionless concentration profile. Figure 2 also shows that δ 2 temperature profile increases whereas concentration distribution decreases. Additionally, reduction in the concentration profile is more prominent, as compared to enhancement in the temperature distribution. Overall, temperature and concentration distributions within nanofluid become higher in the case of δ 2 and δ 1 . Figure 3 depicts the influence of the parameters Nb and Nt on f (η), θ(η), and ϕ(η) distributions. From Figure 3, it is seen that by increasing Nb and Nt parameters, the velocity inside the boundary layer decreases. Figure 3 also shows that an increase in the parameters Nb and Nt enhances the dimensionless temperature, whereas dimensionless concentration shows the same behavior, as observed in temperature distribution in the case of Nt and the opposite behavior observed in the case of Nb.
The graphical results for Nu and Sh are plotted against Nt for variation of Nb, Pr and Le and shown in Figures 4 and 5. Figure 4 shows that Nu decreases with the increasing Nb while increasing with increasing Pr and Le. Figure 5 reveals that Sh rises with increasing Nb and Le and reduces with the variation of Pr, whereas, an enhancement in this profile is more prominent in the case of Nb and Le. This is due to the reason that the increase in Le is equivalent to a decrease in Brownian diffusion, and as a result, dimensionless concentration reduces and the mass transfer rate ϕ (0) increases.           Figure 6 shows that by increasing the parameters δ 1 and δ 2 , Nu reduces and the change in the values of Nu is more prominent in the case of δ 1 , whereas Figure 7 depicts that concentration of nanofluid particle at the surface increases with the increasing values of δ 2 and opposite behavior is observed in case of δ 1 .      In Cattaneo-Christov heat and flux models, heat flux and mass flux are dependent upon the temperature gradient, concentration gradient, and the fluid velocity.

Concluding Remarks
In this paper, natural convection flow along a circular cone is investigated theoretically in the presence of Brownian motion and thermophoresis parameters of nanoparticles. Implicit finite difference technique is used to solve the transformed ordinary differential equations along with the boundary conditions. Computational results are displayed for essential quantities of physical nature such as velocity, temperature, concentration profiles, Nusselt number and Sherwood number. From this study, the obtained results are concluded here: • With the increase of Brownian motion parameters, Sherwood number increases, whereas it gains reverse behavior against thermophoresis parameter.

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With the increase of thermophoresis and Brownian motion parameters, Nusselt number decreases.  Funding: This research received no external and internal funding.

Conflicts of Interest:
The authors declare no conflict of interest.