Abstract
The optimal homotopy analysis method (OHAM) is employed to investigate the steady laminar incompressible free convective flow of a nanofluid past a chemically reacting upward facing horizontal plate in a porous medium taking into account heat generation/absorption and the thermal slip boundary condition. Using similarity transformations developed by Lie group analysis, the continuity, momentum, energy, and nanoparticle volume fraction equations are transformed into a set of coupled similarity equations. The OHAM solutions are obtained and verified by numerical results using a Runge-Kutta-Fehlberg fourth-fifth order method. The effect of the emerging flow controlling parameters on the dimensionless velocity, temperature, and nanoparticle volume fraction have been presented graphically and discussed. Good agreement is found between analytical and numerical results of the present paper with published results. This close agreement supports our analysis and the accuracy of the numerical computations. This paper also includes a representative set of numerical results for reduced Nusselt and Sherwood numbers in a table for various values of the parameters. It is concluded that the reduced Nusselt number increases with the Lewis number and reaction parameter whist it decreases with the order of the chemical reaction, thermal slip, and generation parameters.
1. Introduction
Research in micro- and nanofluids has become a popular area of research in engineering. At micro- and nanoscale, conventional ideas of classical fluid mechanics do not apply, and traditional approaches to fluid mechanics problems need to be changed to correctly reflect the importance of the interaction between a fluid and a solid boundary. Conventional heat transfer fluids like oil, water, and ethylene glycol mixtures are poor heat transfer fluids because of their poor thermal conductivity. Many attempts have been made by various investigators during the recent years to enhance the thermal conductivity of these fluids by suspending nano/microparticles in liquids [1, 2]. Researchers have observed that the thermal conductivity of a nanofluid is much higher than that of the base fluid even for low solid volume fraction of nanoparticles in the mixture [3–5]. The effect of temperature on thermal conductivity in a model has been considered by Kumar et al. [6]. Patel et al. [7] have improved the model given in [6] by incorporating the effect of microconvection due to particle movement.
Nano- and microfluidics is a new area with significant potential for novel engineering applications, especially for the development of new biomedical devices and procedures [8]. Napoli et al. [9] reviewed applications of nanofluidic phenomena to various nanofabricated devices, in particular ones designed for biomolecule transport and manipulation. There has been significant interest in nanofluids. This interest is due to its diverse applications, ranging from laser-assisted drug delivery to electronic chip cooling. Nanofluids are made of ultrafine nanoparticles (<100 nm) suspended in a base fluid, which can be water or an organic solvent. Nanofluids possess superior thermophysical properties like high thermal conductivity, minimal clogging in flow passages, long term stability and homogeneity. Industrial applications of nanofluid include electronics, automotive and nuclear applications. Nanobiotechnology is also a fast developing field of research with application in many domains such as medicine, pharmacy, cosmetics, and agroindustry. Many of these industrial processes involve nanofluid flow and nanoparticle volume fraction past various geometries. In these applications, the diffusing species can be generated/absorbed due to chemical reaction with the ambient fluid. This can greatly affect the flow and hence the properties and quality of the final product [10, 11].
Different industrial applications of internal heat generation include polymer production and the manufacture of ceramics or glassware, phase change processes, thermal combustion processes, and the development of a metal waste from spent nuclear fuel [12]. A review of convective transport in nanofluids was conducted by Buongiorno [13]. Kuznetsov and Nield [14] presented a similarity solution of natural convective boundary-layer flow of a nanofluid past a vertical plate. They have shown that the reduced Nusselt number is a decreasing function of the buoyancy-ratio number , a Brownian motion number , and a thermophoresis number . Godson et al. [15] presented the recent experimental and theoretical studies on convective heat transfer in nanofluids and their thermophysical properties and applications and clarified the challenges and opportunities for future research.
Convective flow in porous media has received the attention of researchers over the last several decades due to its many applications in mechanical, chemical, and civil engineering. Examples include fibrous insulation, food processing and storage, thermal insulation of buildings, geophysical systems, electrochemistry, metallurgy, the design of pebble bed nuclear reactors, underground disposal of nuclear or nonnuclear waste, and cooling system of electronic devices. Excellent reviews of the fundamental theoretical and experimental works can be found in the books by Nield and Bejan [16], Vadasz [17], Vafai [18]. The Cheng-Minkowycz problem [19] was investigated by Nield and Kuznetsov [20] for nanofluid where the model involves the effect of Brownian motion and thermophoresis. The classical problem of free convective flow in a porous medium near a horizontal flat plate was first investigated by Cheng and Chang [21]. Following him many researchers such as Chang and Cheng [22], Shiunlin and Gebhart [23], Merkin and Zhang [24], and Chaudhary et al. [25] have extended the problem in various aspects. Gorla and Chamkha [26] presented a similarity analysis of free convective flow of nanofluid past a horizontal upward facing plate in a porous medium numerically. Khan and Pop [27] extended this problem for nanofluid. Very recently, Aziz et al. [28] extended the same problem for a water-based nanofluid containing gyrotactic microorganisms.
Lie group analysis has been used by many investigators to analyze various convective phenomena under various flow configurations arising in fluid mechanics, aerodynamics, plasma physics, meteorology, chemical engineering, and other engineering branches [29]. This method has been applied by many investigators to study various transport problems. For example, the symmetrical properties of the turbulent boundary-layer flows and other turbulent flows are investigated by using the Lie group techniques by Avramenko et al. [30]. Kuznetsov et al. [31] investigated a falling bioconvection plume in a deep chamber filled with a fluid saturated porous medium theoretically. Jalil et al. [32] studied mixed convective flow with mass transfer using Lie group analysis. The effect of thermal radiation and convective surface boundary condition on the boundary-layer flow was investigated by Hamad et al. [33]. Aziz et al. [34] studied MHD flow over an inclined radiating plate with the temperature dependent thermal conductivity, variable reactive index and heat generation using scaling group of transformations. Reviews for the fundamental theory and applications of group theory to differential equations can be found in the texts by Hansen [35], Ames [36], Seshadri and Na [37], and Shang [38].
Most scientific problems and phenomena such as the boundary-layer problem occur nonlinearly. For these nonlinear problems we have difficulty in finding their exact analytical solutions. Analytical solutions to these nonlinear equations are of fundamental importance. Where no analytical solutions can be found, researchers have resorted to other approaches. One such approach is a perturbation method [39] that is strongly dependent upon the so-called “small parameters.” The perturbation method cannot provide us with a simple way to adjust and control the convergence region and rate of convergence of a given approximate series.
Another known method is the differential transform method that has been used in recent years [40–44]. In 1992, Liao introduced the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely, homotopy analysis method (HAM) [45] that does not need any small parameter. This method has been successfully applied to solve many types of nonlinear problems by others [46–48]. As this approach is based on the homotopy of topology, the validity of the HAM is independent of whether or not there exists a small parameter in the considered equation. Therefore, the HAM can overcome the foregoing restrictions and limitations of perturbation methods [49]. This method also provides us with great freedom to select proper base functions to approximate solutions of nonlinear problems. Using one interesting property of homotopy, one can transform any nonlinear problem into an infinite number of linear problems.
In this paper, the steady flow of an Ostwald-de Waele power-law fluid induced by a steadily rotating infinite disk to a non-Darcian fluid-saturated porous medium is considered. The coupled governing equations are transformed into ordinary differential equations in the boundary layer. The OHAM is applied to solve the ODEs. The validity of our solutions is verified by the numerical results (by using a fourth-order Runge-Kutta and shooting method).
The aim of the present study is to investigate the effect of higher order chemical reaction, internal heat generation, and the thermal slip boundary condition on the boundary-layer flow of a nanofluid past an upward facing horizontal plate. Lie group analysis is used to develop the similarity transformations and the corresponding similarity representations of the governing equations. The coupled governing equations are transformed into ordinary differential equations in the boundary layer. The OHAM is applied to solve the ODEs. The obtained solutions are verified by the numerical results (obtained by using a Runge-Kutta-Fehlberg fourth-fifth order and shooting method). The effect of relevant parameters on dimensionless fluid velocity, temperature, and nanoparticle volume fraction are investigated and shown graphically and discussed. A table containing data for the reduced Nusselt number and reduced Sherwood number is also provided to show the effects of various parameters on them. To the best of our knowledge, the effects of thermal slip boundary condition with internal heat generation and chemical reaction on the boundary-layer flow of a nanofluid past a horizontal plate in porous media have not been reported in the literature yet.
The paper is divided up as follows. In Section 2, the mathematical formulation is presented. In Section 3, we used the Lie group method to reduce the system of partial differential equations to a system of ordinary differential equations. In Section 4 the basic idea of the HAM is presented. In Section 5, we derived the OHAM solution of the coupled system of nonlinear ordinary differential equations. In Section 6, we compared our results with numerical solutions obtained using a Runge-Kutta-Fehlberg method. In Section 7, we introduced the physical quantities to be considered and compared in this paper. Section 8 contains the results and discussion. The conclusions are summarized in Section 9.
2. Formulation of the Problem
We consider a two-dimensional laminar free convective boundary-layer flow of a nanofluid past an upward facing chemically reacting horizontal plate in a porous media (Figure 1). We assume that a homogeneous isothermal irreversible chemical reaction of order takes place between the plate and nanofluid. There is internal heat generation/absorption within the fluid inside the boundary layer at the volumetric rate . Variation of density of the fluid is taken into account using the Oberbeck-Boussinesq approximation. The conservation of mass, momentum, energy, and nanoparticles describing the flow can be written in dimensional form (see [27]):
The flow is assumed to be slow to ignore an advective term and a Forchheimer quadratic drag term in the momentum equation.
We consider a steady flow where the Oberbeck-Boussinesq approximation is used. In addition, we assume that the nanoparticle concentration is dilute. With a suitable choice for the reference pressure, the momentum equation can be linearized and (2) written as (see [50])
We also consider the effect of temperature-dependent volumetric heat generation/absorption in the flow region that is given by Vajravelu and Hadjinicolaou [51] as where is the heat generation/absorption constant. Also, we consider the case where the reaction rate varies as where is the constant reaction rate.
With these assumptions along with standard boundary-layer approximation, the governing equations can be written in dimensional form as where is the thermal diffusivity of the fluid and is a parameter.
The boundary conditions are taken to be where is the thermal slip factor with dimension . The following new nondimensional variables are introduced to make (8)–(13) dimensionless: where is the Rayleigh number based on the characteristic length . A stream function defined by is introduced into (8)–(13) to reduce the number of dependent variables and the number of equations. Note that (8) is satisfied identically. Hence, we have
The boundary conditions become
The parameters in (16)–(19) are introduced in Nomenclature and defined by
3. Lie Group Analysis
We consider the following scaling group of transformations which is a special form of Lie group analysis [52]:
Here is the parameter of the group and are arbitrary real numbers whose connection will be determined by our analysis. The transformations in (21) can be considered as a point transformation transforming the coordinates to . We now investigate the relationship among the exponents ’s such that
Since this is the requirement that the differential forms , , and be reformed under the transformation group in (19), by using (21), (16)–(18) are transformed to (see [35, 38])
The system will remain invariant (structure of the equations same) under the group transformation , if we have the following relationship among the exponents:
For invariance of the boundary conditions, we have
The set of transformations reduces to
Expanding by the Taylor’s series in powers of and keeping the terms up to the order yields
In terms of differentials, we have
3.1. Similarity Transformations
From (29), , which can be integrated to give
Similarly, yields where is an arbitrary function of . From the equations and , we obtain by integration
From the equation , we obtain by integration where is a constant thermal slip factor.
Thus from (30a)–(30d), we obtain Here is the similarity variable and , , are dependent variables. Note that the similarity transformations in (31) are consistent with the well-known similarity transformations reported in the paper of Cheng and Chang [21] for in their paper. Thus, the dimensionless velocity components , can be expressed as where primes indicate differentiation with respect to .
3.2. Similarity Equations
On substituting the transformations in (31) into the governing (16)–(18), we obtain the following system of ordinary differential equations:
We have to solve the system equations (33)–(35) subject to the boundary conditions where is the thermal slip parameter.
4. Basic Idea of the HAM
Let us consider the following differential equation: where is a nonlinear operator, denotes independent variable, and is an unknown function, respectively. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the traditional homotopy method, Liao [45] constructs the so-called zero-order deformation equation where is the embedding parameter, is a nonzero auxiliary parameter, is an auxiliary function, is an auxiliary linear operator, is an initial guess of , and is an unknown function, respectively. It is important that one has great freedom to choose auxiliary things in the HAM. Obviously, when and , it holds , , respectively. Thus, as increases from 0 to 1, the solution varies from the initial guess to the solution . Expanding in Taylor series with respect to , we have where
If the auxiliary linear operator, the initial guess, the auxiliary parameter , and the auxiliary function are so properly chosen, the series (39) converges at , then we have which must be one of solutions of original nonlinear equation, as proved by Liao [45]. As and , (38) becomes which is used mostly in the homotopy perturbation method, where the solution can be obtained directly without using Taylor series.
According to definition (39), the governing equation can be deduced from the zero-order deformation equation (37). By defining the vector and differentiating equation (37) times with respect to the embedding parameter and then setting and finally dividing them by , we have the so-called th-order deformation equation where
It should be emphasized that for is governed by the linear equation (39) with the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as MAPLE and MATHEMATICA.
5. HAM Solution
In this section, we applied the HAM to obtain approximate analytical solutions of the effect of higher order chemical reaction, internal heat generation, and the thermal slip boundary condition on the boundary-layer flow of a nanofluid past an upward facing horizontal plate (33)–(35). We start with the initial approximation , , and the linear operator
Furthermore, from (33)–(35) we define the nonlinear operators
Using the above definition, with assumption , , and , we construct the th-order deformation equation
Obviously, when and ,
Differentiating the th-order deformation equation (47) times with respect to and finally dividing by , we have the th-order deformation equation subject to boundary condition For (Newtonian fluid), For , Also for , and for ,
Obviously, the solution of the th-order deformation equations (49) for becomes
By using the symbolic software MATHEMATICA to solve the system of linear equations (49) with the boundary conditions (46), it can be obtained finally
The higher orders solutions of , , and were too long to be mentioned here; therefore, they are shown graphically.
5.1. Convergence of the HAM Solution
As mentioned by Liao [45], HAM provides us with great freedom in choosing the solution of a nonlinear problem by different base functions. This has a great effect on the convergence region because the convergence region and rate of a series are chiefly determined by the base functions used to express the solution. We used several terms in evaluating the approximate solution , , ; note that the solution series contains the auxiliary parameter which provides us with a simple way to adjust and control the convergence of the solution series. Generally, by means of the so-called -curve, that is, a curve of a versus . As pointed by Liao [45], the valid region of is a horizontal line segment. Figures 2, 3, and 4 show the -curve with the various order of the HAM for , , , respectively, when , , , , , , , . It can be seen that when the order of series is 12 the segment of the horizontal line is more than the other orders. For example, it can be found that for , the acceptable range of is between and for 12th order of the HAM, but for 8th order of the HAM the acceptable range of is between and or and for 4th order of the HAM, so horizontal line segment of 12th order of the HAM is more than others. Therefore, it is straightforward to choose an appropriate range for which ensures the convergence of the solution series. To choose optimal value of auxiliary parameter, the averaged residual errors (see [53] for more details) are defined as where and . For given order of approximation and the optimal values are given by the minimum of , corresponding to nonlinear algebraic equations
It is noted that the optimal value of is replaced into the equations. Table 1 shows optimal values obtained for the auxiliary parameter , for various quantities of , , and , when , , , , . To see the accuracy of the solutions, the residual errors are defined for the system as (for order approximation)
Figures 5, 6, and 7 show the residual errors for 12th-order deformation solutions when , , , , , , , . For example has the minimum range of residual curve in Figure 5 and so on.
Graphical representation of results is very useful to demonstrate the efficiency and accuracy of the HAM for the above problem.
6. Comparisons and Verification
It is worth citing that for isothermal plate and in the absence of internal heat generation/absorption and chemical reaction our problem reduces to Gorla and Chamkha [26] and Khan and Pop [27]. To verify the accuracy of our results, the present results are compared in Table 2 with Gorla and Chamkha [26] and are found to be in good agreement.
7. Physical Quantities
The parameters of physical interest of the present problem are the local skin friction factor , the local Nusselt number , and the local Sherwood number , respectively. Physically, indicates wall shear stress, indicates the rate of wall heat transfer whilst indicates the rate of wall nanoparticle volume fraction. The following relations are used to find these quantities:
By substituting from (14) and (31) into (37), it can be shown that physical quantities are putted in the following dimensionless form
Here is the local Rayleigh number, is the Prandtl number, and is the reference velocity.
8. Results and Discussion
Graphical representation of results is very useful to discuss the physical features presented by the solutions. This section describes the influence of some interesting parameters on the velocity and temperature fields. Equations (33) to (35) with boundary conditions in (36) were solved analytically by HAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54].
Figures 8, 9, and 10, respectively, for , , and show the comparisons between various order approximation of the optimal HAM and the numerical solutions for the case , , , , , , , . For better presentation, these results are presented in Tables 3, 4, and 5. Also Tables 6, 7, and 8 present the comparisons between various order approximation of the optimal HAM and the numerical solutions for the case , , , , , , , . It is observed that the results of 12th-order approximation of the optimal HAM are very close to the numerical solutions which confirm the validity of these methods.
In the following figures, effects of various physical parameters on the dimensionless velocity, temperature, and temperature profiles will be investigated. These results have been obtained by 12th order of the HAM and have been validated by numerical results. Figures 11–13, respectively, represent that the comparison of solutions of , , and for fix values , , , , , , and different values of buoyancy ratio . In Figure 11, it is clear that velocity of the fluid tremendously decreases in the near of horizontal plate with an increase in the buoyancy ratio. It is observed that the temperature increases slightly but concentration of the fluid does not vary sensibly (Figures 12 and 13). The effects of the Brownian motion are depicted in Figures 14, 15, and 16, when , , , , , , . If increases, and increase lightly and decreases. The dimensionless velocity, dimensionless temperature, and dimensionless concentration profiles for different values of thermophoresis with constant values , , , , , , are presented in Figures 17, 18, and 19. It is observed that the velocity and temperature of the fluid are not impressible from but concentration increases slightly. In Figures 20, 21, and 22, respectively, comparison of solutions of , , and for , , , , , , and different values of heat generation/absorption parameter are shown. It is observed that in presence of heat source, , both increase while heat sink causes that , both decrease extremely. These effects are more visible for temperature distribution. For , the effect of changing is seen to be almost insignificant. This is in agreement with the physical fact. The dimensionless profiles for different values of Lewis number with constant parameters , , , , , , are shown in Figures 23, 24, and 25. It is clear that for these constant parameters, variation of Lewis number has any considerable effect on the velocity and temperature profiles, but concentration profile decreases with the increase of Lewis number. What is similar to these results can be detected in Figures 26, 27, and 28 for chemical reaction parameter . Figures 29, 30, and 31, respectively, present the comparison of solutions of , , and for , , , , , , and different values of order of chemical reaction . It is observed that for higher order of chemical reaction, velocity decreases and temperature and concentration profiles increase extremely. Ultimately, Figures 32, 33, and 34 depict the effect of thermal slip parameter on velocity, temperature and concentration functions when , , , , , , . It is seen that the velocity and temperature of the fluid decrease with the increase of the thermal slip parameter in the near of horizontal plate, while the concentration of the fluid does not vary patently.
Also, for investigation of the parameters of physical interest, Table 9 is presented. In this table, numerical values of reduced Nusselt number and Sherwood number obtained by HAM for different values of the parameters , , , and can be compared, when .
9. Conclusions
In this paper, we studied the steady laminar incompressible free convective flow of a nanofluid past a chemically reacting upward facing horizontal plate in porous medium taking into account heat generation and the thermal slip boundary condition.
The governing partial differential equations have been transformed by similarity transformations into a system of ordinary differential equations which are solved by OHAM and numerical method (fourth-order Runge-Kutta scheme with the shooting method). Dimensionless velocity, temperature, and concentration functions are presented for various values of parameters for the problem, for example, buoyancy ratio , Brownian motion , thermophoresis , heat generation/absorption parameter , Lewis number , chemical reaction parameter , order of chemical reaction , and thermal slip parameter . From the present investigation, the following may be concluded.(i)The velocity increases with increasing and decreases with increasing , , and .(ii)The temperature increases with , , and decreases with .(iii)Parameters of , , and have no effect on velocity and temperature profiles.(iv)The concentration distribution increases with increasing , and decreases with , , and , but , , and are not affected by it.(v)In presence of heat source , , both increase while heat sink causes that , both decrease extremely. These effects are more visible for temperature distribution.
In addition, numerical results for the reduced Nusselt and Sherwood numbers are tabulated. It is concluded that the reduced Nusselt number increases with the Lewis number and reaction parameter whist it decreases with the order of chemical reaction, thermal slip, and generation parameters. It is further concluded that the reduced Sherwood number enhances with the Lewis number, generation and reaction parameters whist it suppresses with thermal slip and order of chemical reaction.
The convergence of the solution series and the power of HAM in controlling and adjusting the convergence region and rate of solution series were discussed. The proper range of auxiliary parameter to ensure the convergence of the solution series was obtained through the so called -curve. The HAM provides us with a convenient way to control the convergence of approximation series, which is a fundamental qualitative difference in analysis between the HAM and other methods. The numerical results of the above problems display a fast convergence, with minimal calculations. This shows that the HAM is a very efficient method. Finally, the agreement between analytical and numerical results of the present study with previous published results is excellent.
Nomenclature
| Thermal slip parameter | |
| : | Dimensional concentration |
| : | Skin friction factor |
| : | Thermal slip factor |
| : | Brownian diffusion coefficient |
| : | Thermophoretic diffusion coefficient |
| : | Dimensionless velocity functions |
| : | Gravitation acceleration |
| : | Chemical reaction parameter |
| : | Permeability of the porous media |
| : | The constant reaction rate |
| : | Effective thermal conductivity of the porous medium |
| : | Variable reaction rate |
| : | Linear operator of the HAM |
| : | Length of horizontal plate |
| : | Lewis number |
| : | Nonlinear operator of the HAM |
| : | Order of chemical reaction |
| : | Brownian motion |
| : | Buoyancy ratio |
| : | Thermophoresis |
| : | Nusselt number |
| : | Pressure |
| : | Prandtl number |
| : | Heat generation/absorption parameter |
| : | Heat generation/absorption constant |
| : | Internal heat generation rate |
| : | Rayleigh number |
| : | Sherwood number |
| : | Temperature |
| : | Time |
| : | Velocity in -direction |
| : | Reference velocity |
| : | Velocity vector |
| : | Velocity in -direction |
| : | Distance along the surface |
| : | Distance normal to the surface. |
| : | Thermal diffusivity |
| : | Volumetric expansion coefficient of nanofluid |
| : | Porosity of the porous media |
| : | Dimensionless concentration |
| : | Similarity variable |
| : | Dynamic viscosity |
| : | Dimensionless temperature |
| : | Density of the base fluid |
| : | Density of the nanoparticles |
| : | Effective heat capacity of the fluid |
| : | Effective heat capacity of the nanoparticle material |
| : | Stream function. |
| : | Conditions far away from the surface |
| ′: | Differentiation with respect to . |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
M. M. Rashidi wishes to thank the Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, for the hospitality during his visit. E. Momoniat thanks the National Research Foundation of South Africa for their support.