1 Introduction

Choi^[1] introduced an innovative technique, which used a mixture of nanoparticles and the base fluid in order to develop advanced heat transfer fluids with substantially higher conductivities. He referred to the resulting mixture as a nanofluid. From an energy saving viewpoint, the heat transfer improvements are necessary in the development and manufacturing of electronic devices. Based on this importance, many researchers in recent years have been interested in studying the nanofluid^[2-6]. Makinde and Aziz^[2] studied heat transfer by mixed convection flow of a nanofluid past a stretching sheet. Cheng^[3] analyzed the free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porous media saturated by a nanofluid. Mansour et al.^[4] presented a numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using a nanofluid. The problem of laminar free convection over a vertical wavy surface embedded in a porous medium saturated with a nanofluid was investigated by Mahdy and Ahmed^[5]. Mansour and Ahmed^[6] studied mixed convection flows in a square lid-driven cavity with heat source at the bottom utilizing a nanofluid. Li et al.^[7] investigated dispersal behaviors of water-based nanofluid droplets using the molecular dynamics simulation. They found that the nanoparticle-tuned dispersal behavior of nanofluid droplets can be significantly used for different applications. Wang and Wu^[8] studied the improved oil droplet objectivity from solid surfaces in charged nanoparticle suspensions. Their results indicated that the surface wettability of the nanoparticles plays an important role in oil elimination processes. The influence of a single nanoparticle on the contact line motion was examined by Li et al.^[9]. They obtained three types of contact line motion including complete slipping, alternate pinning-depinning, and complete pinning, and theoretically explained them.

Micropolar fluids are fluids with microstructure which belong to a class of fluids with nonsymmetric stress tensors. From the physical view, the micropolar fluids may characterize fluids consisting of rigid, randomly spherical (or oriented) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. The theory of micropolar fluids initiated by Eringen^[10] exhibits some microscopic effects arising from the local structure and micro motion of the fluid elements. Furthermore, they can sustain couple stresses and include classical Newtonian fluid as a special case. The common applications of these fluids can be found in the theory of lubrication. El-Aziz^[11] presented an analysis for the unsteady mixed convection flow of a viscous incompressible micropolar fluid adjacent to a heated vertical surface in the presence of viscous dissipation when the buoyancy force assists or opposes the flow. Olajuwon et al.^[12] investigated heat and mass transfer effects on unsteady flow of a viscoelastic micropolar fluid over an infinite moving permeable plate in a saturated porous medium in the presence of a transverse magnetic field with the Hall effect and thermal radiation. Mansour at al.^[13] studied steady axisymmetric flow and heat transfer of micropolar fluid over a vertical permeable slender cylinder in the presence of thermal radiation. Mansour et al.^[14] analyzed thermal stratification and suction/injection effects on flow and heat transfer of micropolar fluid due to stretching cylinder. Recently, Ahmed and Rashad^[15] examined the effects of anisotropic porous medium on the natural convection of micropolar nanofluids inside a rectangular enclosure. They indicated that the increase in the nanoparticles volume fraction enhances the rate of heat transfer.

For the boundary layer flow for cones, Cheng^[16] investigated the natural convection boundary layer flow of a micropolar fluid near a vertical permeable cone with variable wall temperature. Their results showed that the heat transfer rates of the permeable cones with higher suction variables are higher than those with lower suction variables. Hossain and Paul^[17] considered laminar free convection from a vertical permeable circular cone maintained at non-uniform surface temperature. They found that the value of skin friction increases with the increase in the suction parameter ξ near the apex of the cone, the value of skin friction decreases to the asymptotic value as ξ increases, and the local Nusselt number increases due to the increasing values of ξ. The problem of laminar free convection from a vertical permeable circular cone maintained with non-uniform surface heat flux was considered by Hossain and Paul^[18]. The references^[19-24] give a good survey for the present study.

The main objective of the current study is to investigate the natural convection boundary layer flow of a micropolar nanofluid over a vertical permeable cone with variable wall temperatures. Non-similar transformations are used to convert the governing equations to non-similar form. The resulting system is solved numerically using the finite difference method discussed by Blottner^[25]. For each type of the nanoparticles, different experimental correlations for both the nanofluid effective viscosity and the nanofluid thermal conductivity are considered.

2 Basic equations

Consider a two-dimensional boundary layer flow of a micropolar nanofluid near a vertical permeable cone with variable wall temperatures. As we can see from Fig. 1, the origin of the coordinate system is placed at the vertex of the cone, where x is the coordinate along the surface of the cone measured from the origin, and y is the coordinate normal to the surface of the cone. It is assumed that the surface of the permeable cone is held at a variable temperature T_w(x) such that T_w(x)>T_∞.

Using the Boussinesq approximation, the governing equations for the boundary layer flow are

(1)

(2)

(3)

(4)

where r is the radius of the cone, u and v are the velocity components along the x-and y-axes, respectively, N is the angular velocity, T is the temperature, g is the acceleration due to gravity, k is the vortex viscosity, j is the micro-inertia density, γ is the half angle of the cone, ρ_nf is the density, μ_nf is the dynamic viscosity, β_nf is the thermal expansion, γ_nf^* is the spin-gradient viscosity, T_∞ is the temperature outside the boundary layer, and α_nf is the nanofluid thermal diffusivity.

The appropriate boundary conditions are

(5a)

(5b)

where V_w is the suction velocity, α is a constant, and n is the surface temperature exponent.

Introduce the stream function ψ(x, y) which satisfies the continuity equation (Eq.(1)) as

(6)

When the boundary layer thickness is relatively small compared with the radius of the cone, the local radius to a point in the boundary layer can be approximated by

(7)

Furthermore, following Refs.[26] and [27], γ_nf^* can be expressed as

(8)

The nanofluid effective density, the nanofluid effective diffusivity, the nanofluid heat capacitance, and the nanofluid thermal expansion coefficient are assumed to be

(9)

(10)

(11)

(12)

where φ is the nanoparticle volume fraction, c_p is the specific heat, and the subscripts nf, f, and p refer to nanofluid, base fluid, and nanoparticle, respectively.

For each type of nanoparticles, experimental correlations for both the nanofluid effective viscosity μ_nf and the nanofluid thermal conductivity k_nf are used. These correlations can be summarized as

(13)

(14)

where R is the ratio of nanolayer thickness to nanoparticle radius.

Introduce the following dimensionless variables:

(15)

By substituting Eqs.(7) and (15) into Eqs.(1)-(4), the following non-similar equations can be obtained:

(16)

(17)

(18)

The corresponding boundary conditions (Eq.(5)) are converted to

(19a)

(19b)

where and are the material parameters, is the Grashof number, is the Prandtl number, ξ is the suction parameter, and the prime symbol refers to the differentiation with respect to η.

Using the non-similar transformations given in Eq.(15), the velocities in the x-and y-directions may be expressed as

(20)

(21)

The local Nusselt can be defined as

(22)

where q_w is the local heat transfer rate per unit surface area and is defined as

(23)

Using Eqs.(15) and (23), the local Nusselt number can be written as

(24)

3 Results and discussion

The numerical algorithm used to solve the dimensionless governing equations (16)-(18) subjected to the boundary conditions (19) is based on the finite difference method. The three-point central difference formula is used to approximate the first and second derivatives of the dependent variables with respect to η, while the backward formula is used for the derivatives with respect to ξ. The obtained algebraic system is solved using the tri-diagonal matrix algorithm (TDMA). Blottner^[25] discussed this technique in detail. In order to check the accuracy of the present method, the obtained results are compared in special cases with the previously published works. Table 1 shows a comparison of NuGr^-1/4 for different values of ξ at K=0, n=0.5, Pr=0.1, and φ=0. It is observed from this table that the present results are in excellent agreement with the results obtained by Hossain and Paul^[17] and Cheng^[16].

The ranges of the governing parameters considered in this investigation are the nanoparticle volume friction 0≤φ ≤0.2, the vortex-viscosity parameter 0≤K≤2, and the surface temperature exponent 0≤n≤1. In all the obtained results, water is considered as a base fluid with Pr=6.2, and Al₂O₃, TiO₂, Ag, and Cu are considered as nanoparticles. Moreover, it is noted that the thermo-physical properties of water and nanoparticles are presented clearly in Table 2.

Figure 2 shows the velocity profiles for different nanoparticles. It is observed that Cu-nanoparticle gives the largest rate of fluid flow. The Cu-nanoparticle gives a large increase in the velocity profile compared with the other nanoparticles. On the contrary, the Al₂O₃-nanoparticle has the lowest profile of velocity behavior. Also, it can be noted that the Ag-nanoparticle velocity profile is larger than the TiO₂-nanoparticle velocity profile. Like the velocity behavior, the temperature distribution in the case of copper is the largest compared with the other nanoparticles, whereas the Ag-nanoparticle gives the lowest one. This can be observed from Fig. 3 which displays the temperature profiles for different nanoparticles. Also, this figure shows that the temperature distribution in the case of Al₂O₃-nanoparticle is larger than that of the TiO₂-nanoparticle. In fact, the behavior of the local Nusselt number depends mainly on the temperature distributions. Therefore, the Cu-water nanofluid has large values of the local Nusselt number, as shown in Fig. 4. However, the other nanoparticles give very convergent values of the local Nusselt number. The above behavior of velocity, temperature, and local Nusselt number may be due to the nanofluid thermal conductivity ratio k_nf/k_f. As we can see from Eq.(14), this ratio is a function of nanoparticle volume friction only in the cases of Al₂O₃, TiO₂, and Ag, while in the case of Cu-water nanofluid, this ratio is a function of φ, k_p, and k_f. This makes the nanofluid thermal conductivity high in the case of Cu-water compared with the other nanoparticles, which gives a good natural convection.

The effects of the nanoparticle volume friction φ on the velocity profiles, the temperature distribution, and the local Nusselt number for the Cu-water nanofluid at K=1, n=0.5, and Pr=6.2 are displayed in Figs. 5, 6, and 7, respectively. As φ increases, f', θ, and Nu/Gr^1/4 increase. Usually, the thermal conductivity of the nanofluid plays an important role in such kinds of problems. Based on this point, we can explain the behaviors of f', θ, and Nu/Gr^1/4 under the effect of φ. Thus, as φ increases, the nanofluid thermal conductivity increases which results in a good convection, unlike the effect of vortex-viscosity parameter K on the velocity profiles. As we can see from Fig. 8 which displays the effect of K on f', the velocity profiles tend to decrease as K increases. This behavior is due to the total viscosity of the fluid which increases as K increases. As well known, more viscous fluid has small rate of fluid flow. Figure 9 shows that a clear reduction in the rate of heat transfer can be obtained by increasing the vortex-viscosity parameter.

The effects of the surface temperature exponent n on the velocity profiles and the local Nusselt number are presented in Figs. 10 and 11, respectively, at K=1, Pr=6.2, and φ=0.1. It is observed that in the uniform surface temperature case (n=0), the velocity profiles take the highest values. In addition, an increase in the surface temperature exponent n leads to a clear reduction in the velocity profiles. However, the local Nusselt number increases by increasing n. In all the local Nusselt number figures, an increase in the suction parameter ξ results in an increase in the rate of heat transfer.

4 Conclusions

The problem of natural convection boundary layer flow of a micropolar fluid over a vertical permeable cone with non-uniform surface heating is discussed in the present paper. Non-similar solution and an implicit finite difference method are used to solve the governing equations. For each type of fluid, different experimental correlations are used for both the nanofluid effective viscosity and the nanofluid thermal conductivity. It is found that using copper as nanoparticle gives high rates of fluid flow, temperature distribution, and rate of heat transfer. Also, a clear enhancement in the local Nusselt number can be obtained by considering the nanofluids compared with the pure fluid. In addition, as the nanoparticle volume friction increases, both the velocity profiles and the temperature distribution increase. The results also indicate that a clear reduction in the fluid velocity and the local Nusselt number occurs as the vortex-viscosity parameter increases. Finally, as the surface temperature exponent increases, the local Nusselt number increases, whereas the velocity profiles take the opposite behavior.