ARTICLE

J. Cent. South Univ. (2019) 26: 1294-1305

DOI: https://doi.org/10.1007/s11771-019-4088-5

Flow and natural convection heat transfer characteristics of non-Newtonian nanofluid flow bounded by two infinite vertical flat plates in presence of magnetic field and thermal radiation using Galerkin method

Peyman MAGHSOUDI¹, Gholamreza SHAHRIARI², Hamed RASAM², Sadegh SADEGHI²

1. School of Engineering, University of Tehran, Tehran 1417466191, Iran;

2. School of Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: The main goal of this paper is to investigate natural convective heat transfer and flow characteristics of non-Newtonian nanofluid streaming between two infinite vertical flat plates in the presence of magnetic field and thermal radiation. Initially, a similarity transformation is used to convert momentum and energy conservation equations in partial differential forms into non-linear ordinary differential equations (ODE) applying meaningful boundary conditions. In order to obtain the non-linear ODEs analytically, Galerkin method (GM) is employed. Subsequently, the ODEs are also solved by a reliable numerical solution. In order to test the accuracy, precision and reliability of the analytical method, results of the analytical analysis are compared with the numerical results. With respect to the comparisons, fairly good compatibilities with insignificant errors are observed. Eventually, the impacts of effective parameters including magnetic and radiation parameters and nanofluid volume fraction on the velocity, skin friction coefficient and Nusselt number distributions are comprehensively described. Based on the results, it is revealed that with increasing the role of magnetic force, velocity profile, skin friction coefficient and thermal performance descend. Radiation parameter has insignificant influence on velocity profile while it obviously has augmentative and decreasing effects on skin friction and Nusselt number, respectively.

Key words: non-Newtonian flow; nanofluid flow; Galerkin method; magnetic field; radiation

Cite this article as: Peyman MAGHSOUDI, Gholamreza SHAHRIARI, Hamed RASAM, Sadegh SADEGHI. Flow and natural convection heat transfer characteristics of non-Newtonian nanofluid flow bounded by two infinite vertical flat plates in presence of magnetic field and thermal radiation using Galerkin method [J]. Journal of Central South University, 2019, 26(5): 1294–1305. DOI: https://doi.org/10.1007/s11771-019-4088-5.

1 Introduction

Analysis of heat transfer characteristics of different fluid flows can greatly be beneficial for performance improvement of industrial systems [1–8]. Natural convection can highly be significant especially when moving fluid is minimally influenced by forced convection heat transfer [9]. Natural convection has attracted a great deal of attention among researchers due to its occurrence in many engineering applications such as geothermal systems and heat exchangers. In the aforementioned systems, convection is apparently observed upon creation of microstructures for cooling of molten metals and fluid flows near heat-dissipation fins. Free cooling of solar collectors and small-scale devices, i.e. computer chips, is one of the conventional application of natural convection [9, 10]. Analysis of natural convection is often difficult particularly when a non-Newtonian fluid is flowing in a system. By now, different flows of Newtonian and non-Newtonian fluids bounded by two infinite parallel vertical plates have been investigated by numerous scholars analytically, numerically and experimentally. As expressed earlier, due to variation of density by position, heat can naturally be transported from the vertical plates to the fluid flowing between the plates. It should be noted that different models have so far been developed by many scholars to apply partial differential equations and/or ordinary differential equations for description of flow and heat transfer characteristics of these problems.

Nowadays, much attention is being paid to the application of nanofluids for cooling purposes [10–13]. Nanoparticles can considerably enhance the thermal conductivity of base fluids by changing the thermophysical properties of the base fluids leading to heat transfer enhancement [14–17].

CHAMKHA [18] developed a mathematical model for a steady two-phase non-Newtonian fluid flow over an infinite porous flat plate considering continuum equations. ELLAHI et al [19] used series solutions to analyze the heat transfer characteristics of a fully-developed incompressible non-Newtonian fluid flow in coaxial cylinders considering Reynolds and Vogel models. BRUCE et al [20] examined a natural convection problem to test the performance of several non-Newtonian fluids flowing between vertical flat plates. MAHMOUD [21] numerically investigated the influences of heat generation and surface slip on the flow and heat transfer characteristics of a non-Newtonian fluid streaming on a moving surface using Runge-Kutta method. Their results show that local Nusselt number will decrease as the slip parameter or the heat generation parameter increases. JOHNSTON [22] proposed a solution method to describe the heat transfer characteristics of a non-Newtonian fluid under Dirichlet and Neumann boundary conditions utilizing Sturm- Liouville integral transformation operators. Their results demonstrate that Nusselt number decrease as axial distance rises. CHEN [23] studied a free convection flow influencing a non-Newtonian fluid flow through a porous medium along an isothermal vertical flat plate applying Runge-Kutta method. RAJAKOPAL et al [24] examined natural convection heat transfer in a homogeneous incompressible non-Newtonian fluid which flows between two infinite parallel vertical plates taking into account the friction skin parameter. They showed that Prandtl number decrease will decrease the velocity profile. DOMAIRRY et al [25] analytically studied the natural convection of a non-Newtonian nanofluid flow between two infinite parallel vertical flat plates employing differential transformation method. They showed that as the nanoparticle volume fraction increases, the momentum boundary layer thickness increases, whereas the thermal boundary layer thickness decreases. PITTMAN et al [26] experimentally investigated the natural convection heat transfer of a non-Newtonian fluid moving on an electrically- heated vertical plate under constant surface heat flux conditions and their results show that with increasing the distance, the temperature difference will grow. HATAMI et al [27] utilized analytical and numerical means to investigate the natural convection of a non-Newtonian nanofluid flow between two vertical flat plates. TERNIK et al [28] numerically analyzed the natural convection of non-Newtonian nanofluids in a square differentially heated cavity using Boussinesq approximation.

In the above-reviewed research studies, scholars have mainly investigated the flow and natural convection heat transfer of different kinds of cooling fluids, i.e. nanofluids, streaming between two infinite plates. However, the effects of magnetic field and thermal radiation on the behavior of nanofluids flow between flat plates have not yet been studied. Moreover, reliability of Galerkin method, an efficient and well-known analytical method for solution of ODEs, has not so far been tested to predict the flow and heat transfer characteristics of nanofluid flow between flat plates.

The main aim of this paper is to predict the performance of natural convection through a non-Newtonian nanofluid streaming between two vertical plates by Galerkin method. To carry out a detailed investigation, the effects of magnetic field and thermal radiation are included, for the first time, in the governing equations. Besides, the impact of skin friction coefficient is considered in the current analysis. At the initial stage of this research, a similarity transformation is applied to present the PDF forms of momentum and energy conservation equations in non-linear ordinary differential forms using accurate boundary conditions. Afterwards, Galerkin method is employed to solve the ODEs. Eventually, the effects of magnetic and radiation parameters on the velocity and Nusselt number distributions are comprehensively explained.

2 Governing equations

In this study, as drawn in Figure 1, a non-Newtonian nanofluid flow between two parallel vertical flat plates is considered. The distance between the plates is assumed to be 2b. Walls are sited at x=b and x=-b and held at constant temperatures T₂ and T₁ (T₁>T₂), respectively. This temperature difference pushes the fluid near the walls located at x=-b and x=b upward and downward, respectively.

Figure 1 Schematic diagram of flow considered in this study

Steady-state, incompressibility and absence of chemical reaction are the main assumptions considered for analysis of this problem. In addition, it is presumed that solid nanoparticles and the base fluid are in thermal equilibrium and no slip occurs between the materials. Table 1 lists the properties of the base fluid (water) and the nanoparticle used in this study.

Table 1 Thermo-physical properties of water and nanoparticle [29]

A viscous fluid is conventionally governed by continuity and Navier–Stokes equations. As an incompressible nanofluid flow is considered, conservation equations of momentum and energy are described as below [24, 30]:

                (1)

          (2)

To consider the effect of magnetic field, following formula is used [30]:

                          (3)

Stress in a third-grade non-Newtonian fluid (T^*) can be formulated as below [24]:

       (4)

where β₃, α₁ and α₂ are the material modules which are functions of temperature; –pI represents the spherical stress corresponding to the restraint of incompressibility and kinematical tensors A₁, A₂ presented in Ref. [30].

Heat flux is obtained by aggregation of conduction and radiation terms as presented below:

                            (5)

where radiative heat flux q_rad can be calculated by using Rosseland approximation as below [31]:

                         (6)

where σ^* and K^* are Stephan–Boltzmann constant and mean absorption coefficient, respectively. It is assumed that the temperature differences existing in the flow vary by forth power of T which can be expressed by a linear function of temperature. This can be implemented by expansion of T⁴ based on Taylor series as follows [31]:

       (7)

By neglecting higher-order terms of temperature in Eq. (7) against the first-degree term, following expression is achieved [31]:

                          (8)

Subsequently, by substituting Eq. (8) into Eq. (6), radiative heat flux is rewritten as below [31]:

                        (9)

Conductive heat flux is calculated by the following equation:

                          (10)

Effective density (ρ_nf), heat capacity ((ρC_p)_nf), dynamic viscosity(μ_nf), thermal conductivity (k_nf) and electrical conductivity (σ_nf) of the fluid are defined as below [27,32,33]:

                      (11)

           (12)

                          (13)

              (14)

                (15)

In order to solve the governing equations, following similarity parameters are used [24]:

                 (16)

Considering the aforementioned parameters, the Navier–Stokes and energy equations are transformed into the following ODEs:

  (17)

                   (18)

where A, B, Prandtl number (Pr), Eckert number (Ec), normalized non-Newtonian viscosity (δ), radiation dimensionless parameter (N) and Hartman number (Ha represents the effect of magnetic field) are defined as below:

,

,

      (19)

The boundary conditions considered in this study are presented as below:

                    (20)

                    (21)

Now to solve the coupled non-linear ODEs presented in Eqs. (17) and (18), Galerkin method is employed. It should be mentioned that Galerkin method is an analytical approach operating based on weighted residuals methods (WRMs). In this method, initially, a trial function is approximated. With respect to the approximated function, an error function will subsequently be defined. Finally, the integral of the error function is minimized over the considered domain [34].

In order to scrutinize the thermal performance of the system, Nusselt number and skin friction coefficient are formulated as below [35]:

         (22)

       (23)

         (24)

          (25)

3 Results and discussion

As previously mentioned, the governing equations are solved by Galerkin method. Current calculations and solution of the equations are implemented in Matlab software. Regarding the aforementioned boundary conditions, following approximated functions are used to non- dimensionalize the flow velocity and temperature, respectively:

                  (26)

            (27)

It is notable that above equations are obtained by try-and-error approach. Unknown parameters including c₁, c₂, c₃ and c₄ are calculated using Galerkin method. For validation purposes, results of this study are compared with results obtained by a numerical method, results of a reliable previous study and results calculated by several other analytical methods. For the first case, performance of the method has been compared with results of numerical analysis in which Ec, Hartman number, N and δ are assumed to be unity and φ and Pr are equal to 0.01, 6.2, respectively. By considering the above conditions and applying the Galerkin method, following functions are yielded for the flow velocity and temperature:

        (28)

 (29)

Results of the present numerical and analytical solutions are plotted in Figures 2 and 3, respectively. As it can be observed in these figures, the result of the analytical solution is appropriately compatible with those of the numerical solution obtained from the momentum and energy equations. According to Figures 2 and 3, mean values of error for non-dimensionalized functions of velocity and temperature are found to be 0.081% and 0.0029%, respectively.

Figure 2 Comparison of velocity profiles obtained from current analytical and numerical solutions for Ec=δ=Ha=N=1, φ=0.01, Pr=6.2

Figure 3 Comparison of temperature profiles obtained from current analytical and numerical results for Ec=δ=Ha=N=1, φ=0.01, Pr=6.2

For the second case, temperature and velocity obtained from this analysis are compared with results of a previous work carried out by HATAMI et al [36] when δ, Pr, Ec and φ are equal to 1, 1, 1 and 0.01, respectively. Results of the comparison are depicted in Figures 4 and 5. As it can be seen from the figures, a fairly good agreement exists between the results extracted from the momentum and energy equations.

For the third case, the governing equations are solved by two other analytical methods, namely least square method and collocation method for validation purpose. Table 2 compares the velocity and temperature distributions obtained by the methods when Ec=δ=1, Pr=6.2, φ=0.01, Ha=3 and N=2. As can be observed in Table 2, mean relative error obtained from Galerkin method is less than the errors yielded by the others. Additionally, CPU processing time of the Galerkin and collocation methods is significantly less than that of the least square method.

Figure 4 Comparison of velocity profiles obtained from current analytical analysis and previous work for Ec=δ=Pr=1, φ=0.01

Figure 5 Comparison of temperature profiles obtained from current analytical analysis and previous work for Ec=δ=Pr=1, φ=0.01

Figures 6–8 illustrate the effects of Hartman number (Ha) on the non-dimensionalized velocity, skin friction coefficient and Nusselt number when δ, Pr, Ec, φ and radiation parameter are constant and assumed to be 1, 6.2, 1, 0.01 and 1, respectively. Regarding Figure 6, with an increase in the value of Hartman number, the value of velocity reduces. It is worth mentioning that for the values of Hartman number greater than 20, velocity is approximately equal to zero. In other words, with increasing the value of Hartman number, Lorentz force related to the magnetic field enhances, leading to further resistance against the flow and consequently a reduction in the value of velocity. Figure 7 shows the influence of Ha on skin friction coefficient. As can be observed in Figure 7, with growing Ha, c_f declines. Before Ha=10, c_f drastically declines and after Ha=50, the c_f is almost constant and found to be zero. Variation of Nusselt number with Hartman number is described in Figure 8.

Table 2 Comparison of results obtained by analytical methods

Figure 6 Variation of non-dimensionalized flow velocity with position for several values of Hartman number

Figure 7 Variation of skin fiction coefficient with Hartman number

Figure 8 Variation of flow Nusselt Number with Hartman number

With respect to Figure 8, flow Nusselt number decreases as Hartman number increases, resulting in decrease of velocity. According to this figure, Nusselt number intensively reduces when Hartman number is lower than 20. However, for greater values of Hartman number, flow Nusselt number lessens smoothly.

The effects of non-dimensionalized radiation parameter on the velocity, skin friction coefficient and Nusselt number are delineated in Figures 9–11 when δ, Pr, Ec, φ and Hartman number are constant and considered to be 1, 6.2, 1, 0.01 and 1, respectively. As seen in Figure 9, radiation parameter has negligible effects on the velocity profile. According to Figure 10, c_f drastically grows in the beginning of the considered interval until about N=20 and then the slope of the curve becomes smooth. It can be justified by the fact that radiation parameter has augmentation effect on c_f before N=20 while its impact on c_f is insignificant for the values greater than N=20. As suggested in Figure 11, with enhancing radiation parameter, flow Nusselt number decreases. For radiation parameter values less than 20, variation of flow Nusselt number is intensive while for the values larger than N=20, changes are apparently smooth. Based on Figure 11, Nusselt numbers change slightly over the domain which is related to the flow temperature distribution. As previously explained, the radiation term is obtained by multiplying the dimensionless radiation parameter (N) and second derivative of temperature function (Eq. (9)). According to Figures 3 and 5, temperature profile is almost linear without any curvature. Accordingly, the temperature profile is of small second derivative. Hence, increasing the radiation parameter causes a small reduction in the flow Nusselt number.

Figure 9 Variation of flow non-dimensionalized velocity with position for several values of radiation parameter

Figure 10 Variation of skin friction coefficient with radiation parameter

Figure 11 Variation of Nusselt number with radiation parameter

Figures 12 and 13 show the effects of nanofluid volume fraction (φ) on skin friction coefficient c_f and Nusselt number when δ, Pr, Ec, Hartman number and radiation parameter are constant and assumed to be 1, 6.2, 1, 1 and 1, respectively. Regarding Figure 12, with an increase in the volume fraction, the c_f grows. This growth can be justified by increase of viscosity. As fluid viscosity increases (increase of nanoparticles additives), the skin friction rises. Further, according to Figure 13, with a rise in the nanofluid volume fraction, thermal performance improves, which is due to the increment in thermal conductivity.

Figure 12 Variation of skin friction coefficient with nanofluid volume fraction

Figure 13 Variation of Nusselt number with nanofluid volume fraction

In Figures 14 and 15, the effect of δ on skin friction coefficient c_f and Nusselt number are shown. As seen in Figure 14, increase of δ leads to decline in skin friction. With regard to definition of dimensionless non-Newtonian viscosity in Eq. (19), δ has reverse effect on nanofluid viscosity. With an increment in δ, the viscosity descends and eventually the skin friction decreases. Regarding Figure 15, Nusselt number has an inverse proportionality relationship with dimensionless non-Newtonian viscosity. At the beginning of the considered interval, Nusselt number severely reduces and after δ=20 smoothly goes up to Nu=0.59.

Figures 16 and 17 show the effect of Eckert number on skin friction coefficient and Nusselt number, respectively. With respect to Figure 16,Eckert number increases, leading to a decrement with constant negative slope in skin friction coefficient. According to Figure 17, increment of Eckert number increases the Nusselt number and heat viscous dissipation, leading to a rise in temperature gradient and improvement in thermal performance of the system.

Figure 14 Variation of skin friction coefficient with dimensionless non-Newtonian viscosity

Figure 15 Variation of Nusselt number with dimensionless non-Newtonian viscosity

Figure 16 Variation of skin friction coefficient with Eckert number

Figure 17 Variation of Nusselt number with Eckert number

4 Conclusions

In this research, flow and heat transfer characteristics of non-Newtonian nanofluid flowing between two infinite vertical flat plates in the presence of magnetic field and thermal radiation effect were investigated. Initially, a similarity transformation was applied to convert the governing momentum and energy equations into non-linear ordinary differential equations using reliable and accurate boundary conditions. These ODEs were solved analytically by Galerkin Method. Afterwards, to assess the accuracy and precision of the analytical results, the equations were also solved using least square, collocation methods and a numerical approach. The results were compared with the analytical data obtained by Galerkin method. Fairly good agreements with minute error rates were observed in the validation comparisons. Finally, the effects of magnetic field and radiation parameters and nanofluid volume fraction on the velocity, skin friction coefficient and Nusselt number distributions were examined. According to the results, it is concluded that the magnetic parameter has a reverse relationship with velocity profile, skin friction coefficient and Nusselt number. The impact of thermal radiation parameter on velocity is negligible while flow Nusselt number decreases and skin friction coefficient rises with increasing the radiation parameter. Furthermore, nanofluid volume fraction has a direct relationship with flow Nusselt number and skin friction coefficient. In addition, dimensionless non- Newtonian viscosity has reverse effects on both Nusselt number and skin friction coefficient. Similarly, Eckert number has reverse influence on skin friction coefficient and direct effect on flow Nusselt number.

Nomenclature

x, y

Coordinate directions, m

u

Velocity in y direction, m/s

U

Dimensionless velocity

N

Radiation dimensionless parameter

Ha

Hartman number

Pr

Prandtl number

b

Half of distance between plates, m

q

Heat flux, W/m²

c

Specific heat of the fluid, W/(kg·K)

c_f

Skin friction coefficient

k

Thermal conductivity, W/(m·K)

Ec

Eckert number

g

Gravity, m/s²

Pr

Prandtl number

T

Temperature, K

T₁, T₂

Wall temperature, K

T_m

Mean temperature, K

u₀

Inlet velocity, m/s

K^*

Mean absorption coefficient, m^–1

B

Magnetic field, kg/(s²·A)

Nu

Nusselt number

Greek symbols

δ

Dimensionless viscosity of

non-Newtonian flow

η

Dimensionless form of distance

μ

Dynamic viscosity, kg/(m·s)

θ

Dimensionless temperature

ρ_f

Fluid density, kg/m³

β_T

Thermal expansion coefficient, K^–1

σ

Electrical conductivity, S³·A²/(kg·m³)

σ^*

Stephan–Boltzmann constant, W·K⁴/m²

Subscripts

rad

Radiation

con

Conduction

f

Fluid

nf

Nanofluid

s

Solid

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(Edited by YANG Hua)

中文导读

Galerkin法研究磁场和热辐射作用下的非牛顿纳米流体在垂直平板间的流动和自然对流换热特性

摘要：研究了在磁场和热辐射作用下，非牛顿纳米流体在两个无限大的垂直平板间流动的自然对流换热和流动特性。首先，采用相似变换法，将偏微分形式的动量和能量守恒方程转化为特殊边界条件下的非线性常微分方程组。应用Galerkin方法(GM)，获得非线性常微分方程组的解析解，并给出了非线性常微分方程组的数值解。对比分析解析解和数值解，验证了模型的准确性、精度和可靠性。最后，综合考虑了磁场、辐射参数和纳米流体体积分数等参数对速度、摩擦阻力系数和Nusselt数分布的影响。结果表明，随着磁力作用的增大，速度场、摩擦阻力系数和热性能下降。辐射参数虽然对增强的速度场影响不大，但降低了摩擦阻力系和Nusselt数的影响。

关键词：非牛顿流体；纳米流体流动；Galerkin法；磁场；辐射

Received date: 2018-08-28; Accepted date: 2018-11-10

Corresponding author: Gholamreza SHAHRIARI, Assistant Professor; Tel: +98-2173912979; E-mail: shahriari@iust.ac.ir; ORCID: 0000-0002-9936-5194