ARTICLE
J. Cent. South Univ. (2019) 26: 1172-1183
DOI: https://doi.org/10.1007/s11771-019-4079-6
MHD mixed convective stagnation-point flow of Eyring-Powell nanofluid over stretching cylinder with thermal slip conditions
Hammed Abiodun OGUNSEYE, Precious SIBANDA, Hiranmoy MONDAL
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa
Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract:
The optimal design of heating and cooling systems must take into account heat radiation which is a non-linear process. In this study, the mixed convection in a radiative magnetohydrodynamic Eyring-Powell copper- water nanofluid over a stretching cylinder was investigated. The energy balance is modeled, taking into account the non-linear thermal radiation and a thermal slip condition. The effects of the embedded flow parameters on the fluid properties, as well as on the skin friction coefficient and heat transfer rate, are analyzed. Unlike in many existing studies, the recent spectral quasi-linearization method is used to solve the coupled nonlinear boundary-value problem. The computational result shows that increasing the nanoparticle volume fraction, thermal radiation parameter and heat generation parameter enhances temperature profile. We found that the velocity slip parameter and the fluid material parameter enhance the skin friction. A comparison of the current numerical results with existing literature for some limiting cases shows excellent agreement.
Key words:
Cite this article as:
Hammed Abiodun OGUNSEYE, Precious SIBANDA, Hiranmoy MONDAL. MHD mixed convective stagnation-point flow of Eyring-Powell nanofluid over stretching cylinder with thermal slip conditions [J]. Journal of Central South University, 2019, 26(5): 1172–1183.
DOI:https://dx.doi.org/https://doi.org/10.1007/s11771-019-4079-61 Introduction
The term of nanofluid refers to a colloidal suspension of tiny particles having a diameter less than 100 nm. CHOI [1] reported that nanofluids have remarkably enhanced thermal conductivity relative to conventional heat transfer fluids. These fluids now have several applications in engineering and biomedical sciences. SAID et al [2] observed that the heating and cooling of a system using solar energy is enhanced when the collector is a nanofluid. Gold nanoparticles were discovered to have therapeutic properties for cancer treatment either as drug carriers or in photothermal therapy (see JAIN et al [3]). BUONGIORNO [4] proposed a mathematical model for convective transport in a nanofluid. He presented an analysis of the influences of Brownian motion and thermophoretic diffusion in his model. To further understand the thermal behaviour of nanofluids, DAS et al [5] presented another nanofluid model with more emphasis placed on the effective fluid properties.
In recent years, several researchers have given attention to stagnation-point flow due to its relevance in many industrial and engineering processes. Some of the areas of interest include the cooling of electronic devices and nuclear reactors, polymer processes and the flow of ground water. BACHOK et al [6] studied the stagnation-point flow of a nanofluid over a stretching or shrinking plate and observed in their research that the skin friction and heat transfer coefficients are enhanced with nanofluid. RAMZAN et al [7] investigated the stagnation-point flow of a electrically conducting fluid with a generalized slip condition. HAYAT et al [8] investigated the stagnation-point flow of carbon nanotubes over a stretchable cylinder with partial slip. ISHAK et al [9] analyzed the heat transfer of mixed convection stagnation-point flow over a vertical linear stretching sheet. The mixed convection stagnation-point flow of nanofluid over a stretching or shrinking sheet with internal heat generation or absorption in a porous medium was explored by PAL et al [10]. The results of their investigation showed among others, that in a situation of large value of the heat generation or absorption parameter the copper-water temperature increases. ABBAS et al [11] studied stagnation- point flow on a permeable stretching cylinder with heat generation or absorption. The influence of variable viscosity and thermal radiation on stagnation-point flow past a porous stretching sheet was addressed by MUKHOPADHYAY [12]. It was shown that an increase in the thermal parameter led to a significant increase in the thermal boundary layer thickness. Other related studies of nanofluid flow with different geometries can be found in the studies [13–17] and the references therein.
Practically, most industrial and technological real fluids are non-Newtonian. Hence, the Newtonian constitutive relation based on linear shear stress and strain can not be used to study such fluids. For the study of non-Newtonian fluids, several models have been proposed, such as the Casson fluid model, power law model, Maxwell fluid, Jeffrey fluid, Eyring-Powell fluid. In this study, we assume the Eyring-Powell fluid model proposed by EYRING and POWELL [18]. The choice of this model is due to the fact that the constitutive model is derived from the molecular theory of fluid and not based on an empirical relation. JAVED et al [19] studied Eyring-Powell fluid flow over a stretchable plate using the Keller box method to solve the flow equations. They showed that the velocity profiles are enhanced for a non-Newtonian fluid, as against the use of a Newtonian fluid. HAYAT et al [20] obtained a series solution for heat transfer in an Eyring-Powell fluid flow over a continuously moving surface with a convective boundary condition using the homotopy analysis method. AKBAR et al [21] studied the magnetohydrodynamic (MHD) flow of an Eyring-Powell fluid using the implicit finite difference method. It was shown that for large magnetic field intensity, the resistance to flow increases. BABU et al [22] analyzed MHD mass transfer of Eyring-Powell nanofluid over a permeable cone with buoyancy forces and suction or injection effects. HAYAT et al [23] considered the effects of heat generation or absorption on MHD Eyring-Powell nanofluid over an impermeable stretched cylinder. In the work of RAMZAN et al [24] reactive Eyring-Powell nanofluid with variable properties was investigated. MALIK et al [25] reported the mixed convection flow of MHD Eyring-Powell nanofluid past a plate. KHAN et al [26] investigated mixed convection flow of reactive Eyring-Powell nanofluid over a cone and plate.
The thermal properties and flow structure of a copper-water nanofluid using the Eyring-Powell model has not been studied, with most studies limited to the Buongiorno model [4]. The objective of this study is to investigate the thermal properties of a water based Eyring-Powell nanofluid flow past a vertical stretching cylinder under heat generation, velocity and thermal slip boundary conditions. To the best of the authors knowledge, the problem analysis has not been investigated. The transformed equations are solved numerically using the spectral quasi-linearization method proposed by MOTSA [27]. To validate the accuracy and convergence of the numerical method, a comparison of the skin friction coefficient and the Nusselt number for limiting cases with existing literature is presented.
2 Mathematical formulation
Consider the steady two-dimensional, laminar and incompressible mixed convective magnetohydrodynamic flow and heat transfer in a radiative Eyring-Powell nanofluid past a vertical stretching cylinder of radius, a. It is assumed that a uniform magnetic field strength, B0, is applied along the radial direction, r, and the stretching velocity of the cylinder is given by uw(z)=U0z/l, where U0>0 is the stretching constant, z is the co-ordinate measured along the axial direction and l is the characteristic length (see Figure 1).
The nanofluid is composed of copper nanoparticles suspended in water. The base fluid and the suspended nanoparticles are assumed to be in thermal equilibrium. Under the usual boundary layer and Oberbeck-Boussinesq approximations, the equations of conversation of mass, momentum and energy balance describing the Eyring-Powell nanofluid can be written as follows [23]:
(1)
(2)
(3)
subjected to the boundary conditions:
(4)
where u and v are the velocity components in the z and r directions, respectively; b and c are fluid parameters; g is the gravitational acceleration; T is the fluid temperature; qr is the radiative heat flux; ue(z) is the free stream velocity; Q0 denotes the uniform volumetric heat generation or absorption coefficient; vw(z)>0 signifies fluid suction while vw(z)<0 indicates fluid injection; e1 and e2 stand for the velocity and thermal slip coefficients,respectively; 
denotes the prescribed wall temperature; U∞ and T∞ are the reference velocity and temperature, respectively; vnp, ρnp, βnp, σnp, κnp and (ρCp)np denote the kinematic viscosity, density, thermal expansion coefficient, electrical conductivity, thermal conductivity and heat capacity, respectively. For a spherical-shaped nanoparticles, the physical properties of the nanofluid are defined by [5]
(5)
where the subscripts f and s stand for the base fluid and nanoparticles; φ is the solid volume fraction of the nanoparticle; μ, ρ, β, (ρCp), σ and κ respectively represent the viscosity, density, thermal expansion coefficient, heat capacity, electrical conductivity and thermal conductivity, respectively. The thermo- physical properties of water and nanoparticles are given in Table1.
Figure 1 Geometry of problem
Using the Rosseland approximation [28], the radiative heat flux can be expressed as follows:
Table 1 Thermo-physical properties of water and nanoparticles [5]
(6)
where σ* is the Steffan-Boltzman constant and k* is the Rosseland mean absorption coefficient.
Substituting Eqs. (6) into the energy balance Eq. (3) yields
(7)
To transform Eqs. (1), (2) and (7) into ordinary differential equation, we introduce the following stream function and similarity variables (see HAYAT et al [23] and MUKHOPADHYAY [12]):
(8)
where ψ is the stream function such that 
and 
and η is the similarity variable.
Using Eq. (8), Eq. (1) is identically satisfied while Eqs. (2) and (7) reduce to the following coupled nonlinear ordinary differential equations:
(9)
(10)
subjected to the boundary conditions:
(11)
where the prime denotes differentiation with respect to η; γ is the curvature parameter; Γ and δ is the fluid parameter, respectively; Ha is the Hartman number; ε is the velocity ratio of the free stream velocity to that of stretching cylinder wall; λ represents the mixed convection parameter; Rd stands for the thermal radiation parameter; θw is the temperature ratio parameter; Pr is the Prandtl number; Q denotes the heat generation (Q>0) or absorption (Q<0) parameter; fw stands for suction (fw<0) or injection (fw>0); Λ1 and Λ2 are the dimensionless velocity slip and thermal slip parameter, respectively. These parameters are expressed as follows:
(12)
here Gr is the local Grashof number and Rez is the local Reynold number. Furthermore, the skin friction coefficient, Cf and the local Nusselt number, Nuz are defined as follows (see HAYAT et al [23] and JAVED et al [19]):
(13)
The wall shear stress, τw and the wall heat flux, qw at r=a are expressed as:
(14)
(15)
In dimensionless forms, the skin friction coefficient, Cf and the local Nusselt number, Nuz are:
(16)
(17)
3 Spectral quasi-linearization method of solution
The system of coupled, nonlinear differential equations given in Eqs. (9)–(11) are solved numerically using the SQLM. The principle of the SQLM is derived from the pioneering work of Ref. [29]. The nonlinear system is linearized using the Newton-Raphson algorithm. The linearized equations are integrated using Chebyshev spectral collocation method. With the appropriate initial guesses, the SQLM converges rapidly and gives an accurate solution. Studies on the accuracy and convergence of SQLM are reported in Refs. [27] and [30].
Equations (9)–(10) can be rewritten in decomposed form as a sum of both linear and nonlinear components. Linearizing using one term Taylor’s series for multiple variables, gives the iterative scheme:
(18)
(19)
with corresponding boundary conditions:
(20)
where the coefficients 
, are known functions from previous iterations and are given by:
,
(21)
Equations (18)–(20) constitute the SQLM iterative scheme. The equations are solved numerically using the Chebyshev pseudo-spectral technique as described in Ref. [31]. Initializing the algorithm with appropriate initial approximations, the results for fn+1 and θn+1, when n=1, 2, … are computed iteratively.
We discretize Eqs. (18) and (19) using the Chebyshev pseudo-spectral collocation method. Firstly, the semi-infinite domain, 
is truncated by replacing it with 
, where 
.
Secondly, we transform the interval 
using the transformation 
The derivatives of the unknown variables f(η) and θ(η) are computed using the Chebyshev differentiation matrix D (see [32]), at the collocation points as a matrix vector product:
(22)
where 
is the number of collocation points, 
and
is a vector function at the collocation point. The Gauss-Lobatto points are selected to define the nodes in [–1, 1] as:
(23)
Let Θ be a similar vector function representing θ. Higher order derivatives of f and θ are evaluated as powers of D, that is
(24)
Substituting Eqs. (22)–(24) into Eqs. (18) and (19), we obtain the following SQLM scheme in a matrix form:
(25)
where 
(i, j=1, …, 2) are 
matrices and 
and
are 
vectors, defined as:
(26)
subjected to the boundary conditions
(27)
A suitable initial approximation for the SQLM scheme is
(28)
4 Results and discussion
In this section, the computational results showing the effect of flow parameters on the velocity profiles, f′(η), temperature profiles θ(η), skin friction coefficient and Nusselt number are discussed. To validate the correctness of the numerical results obtained from the iterative scheme given by Eqs. (18)–(20), the skin friction coefficient, f″(0) is compared with result of MAHAPATRA et al [33] in Table 2, and in Table 3; the values of the local Nusselt number –θ′(0) are compared with those of ZAIMI et al [34]. Thus, Tables 2 and 3 show the accuracy and convergence of SQLM.
Table 2 Comparison of SQLM results for f ″(0) with MAHAPATRA et al [33] for distinct values of ε when γ=Γ=Ha=fw=λ=0 and Λ1=0
Table 3 Comparison of SQLM results for –θ′(0) with ZAIMI et al [34] for different values of Pr by setting ε=λ=1, φ=γ=Γ=Ha=Ra=Q=Λ1=0 and Λ2=0
The following ranges of values are used; 0≤γ≤1.2, 0≤φ≤0.2, 0≤Γ≤1.0, 0≤Ha≤3.0, 0.8≤ε≤1.2, 0≤fw≤0.7, 0≤Λ1≤0.7, 0≤Rd≤0.9 and 0≤Λ2≤0.6.
The impact of the curvature parameter γ on the velocity profile is shown in Figure 2. It is observed that very close to the surface of the cylinder for 
, the velocity profiles diminish with an increase in the curvature parameter, while the velocity profiles are seen to be enhanced far away from the surface. Physically, higher values of the curvature parameter reduces the radius of the cylinder, thus, the contact area of the nanofluid with the cylinder is reduced. Hence, the momentum boundary layer thickness is improved. Similar outcome was reported by HAYAT et al [8]. In Figure 3, the influence of the nanoparticle volume fraction φ on the velocity distribution is illustrated. The velocity profile and momentum boundary layer thickness retard with an increase in the nanoparticle volume fraction. The influence of the fluid parameter Γ on the velocity profile is presented in Figure 4. It is seen that with the increase in the fluid parameter, the velocity profile and the momentum boundary layer thickness are enhanced. Physically, it is correct since the fluid parameter has an inverse relation with the nanofluid dynamic viscosity, thus, the fluid becomes less viscous with large value of the fluid parameter. Hence, the velocity profile is enhanced. This finding is consistent with JAVED et al [19]. Figure 5 presents the velocity profiles for distinct values of the Hartmann number Ha. A decreasing trend is observed in the velocity profile as the Hartmann number increases. This is physically consistent due to the damping influence of the Lorentzian hydromagnetic drag. The impact of the velocity ratio parameter ε on the velocity profile is presented in Figure 6. The result shows that for ε>1, that is, when the free stream velocity is greater than the stretching velocity, the boundary layer thickness decreases with an increase in the velocity ratio parameter and the opposite phenomenon is observed for ε<1. However, the boundary layer breaks down for ε=1, as the free stream velocity coincides with the stretching velocity. Figure 7 illustrates the effect of the mixed convection parameter λ on the velocity profile. This plot shows that the velocity profile and momentum boundary layer thickness are enhanced for higher values of the mixed convection parameter. Physically, an increase in the mixed convection parameter leads to an increment in the buoyancy force, hence, the velocity profile is improved. Figure 8 depicts the effect of the suction/injection parameter fw on the velocity profile. The thickness of the momentum boundary layer is reduced with an increase in the suction parameter (fw >0). Similar trend was reported by MUKHOPADHYAY [13]. In Figure 9, the effect of the velocity slip parameter Λ1 on the velocity profile is presented. Similar to the effects of the Hartmann number Ha, the velocity profile and momentum boundary layer thickness retard for higher values of the velocity slip parameter. Physically, the adhesive force between the wall and the nanofluid decreases with higher velocity slip parameter, which results in the partial transfer of stretching velocity to the nanofluid. Hence, the velocity profile decreases. This outcome is similar to the report of HAYAT el al [8].
Figure 2 Effect of γ on f ′(η)
Figure 3 Effect of φ on f ′(η)
Figure 4 Effect of Γ on f ′(η)
Figure 5 Effect of Ha on f ′(η)
Figure 6 Effect of ε on f ′(η)
Figure 7 Effect λ on f ′(η)
Figure 8 Effect fw on f ′(η)
Figure 9 Effect of Λ1 on f ′(η)
Figures 10–15 show the temperature profiles in the flow. The response of the temperature to the curvature parameter is presented in Figure 10. The temperature profile and the thermal boundary layer thickness are enhanced with an increase in curvature parameter. Figure 11 shows the temperature profiles for different nanoparticle volume fraction φ. An increase in the nanoparticle fraction is seen to increase the nanofluid temperature. This is physically correct due to the fact that as the nanoparticle volume fraction increases, the thermal conductivity of the nanofluid is enhanced, hence, improving the thermal distribution. This observation is similar to the result of DAS et al [5]. In Figure 12, the influence of Hartmann number on the temperature distribution is displayed. From the figure, it is seen that the thermal boundary layer thickness is enhanced for large Hartmann number. Figure 13 depicts the impact of the thermal radiation parameter Rd on the nanofluid temperature distribution. It is observed that the temperature profile and the thermal boundary layer thickness are enhanced with an increase in the radiation parameter. Physically, higher value of the radiation parameter, implies that, more heat is transfered to the nanofluid since the mean absorption coefficient κ* reduces with an increase in the radiation parameter. This temperature profile is similar to result of HAYAT et al [23]. The influence of the heat generation or absorption parameter Q on the temperature profile is shown in Figure 14. As observed from the figure, the temperature distribution of the nanofluid is seen to be enhanced with an increase in the heat generation parameter (Q>0) whereas a reverse trend is noticed with the heat absorption parameter (Q<0). Figure 15 displays the effect of the thermal slip parameter Λ2 on the temperature profile. From the plot, an increase in the thermal slip parameter is seen to decay the temperature and thermal boundary layer thickness. Physically, for higher values of the thermal slip parameter Λ2, the rate of heat transfer is reduced from the cylinder to the nanofluid, hence, damping the temperature profile.
Figure 10 Effect of γ on θ(η)
Figure 11 Effect of φ on θ(η)
Figure 12 Effect of Ha on θ(η)
Figure 13 Effect of Rd on θ(η)
Figure 14 Effect of Q on θ(η)
Figure 15 Effect of Λ2 on θ(η)
The effect of distinct parameters on the skin friction coefficient is shown in Table 4. From the table, it is observed that, an increase in the parameters γ, φ, Γ, Ha and fw reduces the skin friction coefficient. However, the skin friction coefficient increases with an increase in the parameters δ, λ, ε and Λ1. Table 5 shows the impact of the parameters γ, φ, ε, Rd, Q and Λ2 on the rate of heat transfer. Higher values of the parameters γ, ε and Rd lead to an increase in the Nusselt number. The opposite trend is seen for an increase in the parameters φ, Q and Λ2.
Table 4 Skin friction coefficient for distinct values of φ, Γ, δ, Ha, λ, ε, fw and Λ1 when θw=1.5, Pr=6.2, Rd=Q=0.3 and Λ2=0.2
5 Conclusions
We studied the mixed convective stagnation- point flow of magnetohydrodynamic Eyring-Powell copper-water nanofluid flow over a stretching vertical cylinder with slip effects. The conservation equations have been solved numerically using an iterative spectral quasi-linearization method. The major findings are summarised below:
1) An increase in the curvature parameter and nanoparticle volume fraction significantly increases the nanofluid velocity profiles.
2) Higher values of the mixed convection parameter and velocity slip parameter retard the skin friction coefficient.
3) The thermal radiation parameter and heat generation parameters (Q>0) increase the rate of heat transfer, while the nanoparticle volume fraction, heat absorption parameter (Q<0) and thermal slip parameter decay the heat transfer rate.
4) Increasing the fluid parameter Γ reduces the skin friction coefficient while an increase in the velocity slip parameter increases the skin friction coefficient.
Table 5 Nusselt number for distinct values of φ, ε, Rd, Q and Λ2 when θw=1.5, Pr=6.2, Γ=δ=λ=Λ1=0.3 and Ha=fw=0.5
Nomenclature
a
Radius
u, v
Axial and radial velocity, respectively
v
Kinematic viscosity
ρ
Density
σ
Electrical conductivity
B0
Uniform magnetic field
g
Gravitational acceleration
β
Thermal expansion coefficient
T
Fluid temperature
T∞
Reference temperature
κ
Thermal conductivity
Cp
Specific heat capacity
qr
Radiative heat flux
σ*
Stefan-Boltzman constant
κ*
Coefficient of mean absorption
ue
Free stream velocity
vw
Suction/injection
φ
Solid volume fraction of the nanoparticle
Q0
Uniform volumetric heat generation/
absorption coefficient
e1
Velocity slip coefficient
e2
Thermal slip coefficient
γ
Curvature parameter
Γ, δ
Fluid parameters
Ha
Hartman number
ε
Velocity ratio
λ
Mixed convection parameter
Rd
Thermal radiation parameter
θw
Temperature ratio parameter
Pr
Prandtl number
Q
Heat generation/heat absorption parameter
fw
Suction/injection parameter
Λ1
Velocity slip parameter
Λ2
Thermal slip parameter
Cf
Skin friction coefficient
Nu
Local Nusselt number
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(Edited by YANG Hua)
中文导读
Eyring-Powell磁纳米流体在伸缩圆柱上的热滑移混合对流
摘要：供热冷却系统的优化设计必须考虑非线性过程的热辐射。本文研究了铜-水纳米流体在伸缩圆筒上的Eyring-Powell磁热辐射的混合对流动力学。考虑非线性热辐射和热滑移条件，建立了能量守衡模型，分析了表面摩擦系数和换热速率等流动参数对流体特性的影响。不采用现有的研究方法，应用最近的谱准线性化方法求解耦合非线性边界值问题。计算结果表明，增大纳米粒子的体积分数、热辐射参数和热源参数可增强温度分布。速度滑移参数和流体材料参数增大了表面摩擦力。对比分析了极限情况下的数值结果和文献结果，结果表明：采用本文研究方法计算结果达到同等精度。
关键词：Eyring-Powell模型；伸缩圆筒；纳米流体；热辐射；滑移效应；谱准线性化方法
Received date: 2018-08-30; Accepted date: 2018-10-19
Corresponding author: Hiranmoy MONDAL, PhD; Tel: +27731529463; E-mail: hiranmoymondal@yahoo.co.in; ORCID: 0000-0002- 9153-300X
Abstract: The optimal design of heating and cooling systems must take into account heat radiation which is a non-linear process. In this study, the mixed convection in a radiative magnetohydrodynamic Eyring-Powell copper- water nanofluid over a stretching cylinder was investigated. The energy balance is modeled, taking into account the non-linear thermal radiation and a thermal slip condition. The effects of the embedded flow parameters on the fluid properties, as well as on the skin friction coefficient and heat transfer rate, are analyzed. Unlike in many existing studies, the recent spectral quasi-linearization method is used to solve the coupled nonlinear boundary-value problem. The computational result shows that increasing the nanoparticle volume fraction, thermal radiation parameter and heat generation parameter enhances temperature profile. We found that the velocity slip parameter and the fluid material parameter enhance the skin friction. A comparison of the current numerical results with existing literature for some limiting cases shows excellent agreement.
