An Analytical Study on Effects of adding Nanoparticles to Water and Enhancement in Thermal Properties Based on Falkner-Skan Model

Document Type: Research Paper

Authors

1 Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

2 Department of Electrical and Energy Engineering, University of Vaasa, Vaasa 65200, Finland

Abstract

In the age of technology, it is vital to cool down different parts of a device to use it more beneficially. Using nanofluids is one of the most common methods which has shown very effective results. In this paper, we have rephrased a classic equation in fluid mechanics, i.e. the Falkner-Skan boundary layer equation, in order to be used for nanofluid. This nonlinear equation, which was presented by Liao, has been solved by Homotopy Analysis Method (HAM). This method is very capable to solve a wide range of nonlinear equations. The physical interpretation of results which are velocity and temperature profiles are explained in details and they are parallel with experimental outcomes of previous researchers.

Keywords


An Analytical Study on Effects of adding Nanoparticles to Water and Enhancement in Thermal Properties Based on Falkner-Skan Model

Y. Rostamiyana*, M. Abbasia, F. Aghajania, F. Hedayatia, S.M. Hamidib

a Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

bDepartment of Electrical and Energy Engineering, University of Vaasa, Vaasa 65200, Finland

 

paper INFO

 

ABSTRACT

Paper history:

Received 6 August 2014

Received in revised form 15 September 2014

Accepted 22 Semptember 2014

 In the age of technology, it is vital to cool down different parts of a device to use it more beneficially. Using nanofluids is one of the most common methods which has shown very effective results. In this paper, we have rephrased a classic equation in fluid mechanics, i.e. the Falkner-Skan boundary layer equation, in order to be used for nanofluid. This nonlinear equation, which was presented by Liao, has been solved by Homotopy Analysis Method (HAM). This method is very capable to solve a wide range of nonlinear equations. The physical interpretation of results which are velocity and temperature profiles are explained in details and they are parallel with experimental outcomes of previous researchers.

 

Keywords:

Falkner-skan

Boundary layer

Nanofluids

HAM

 

© 2014 Published by Semnan University Press. All rights reserved.

 

 

  1. 1.      Introduction

As technology improves, it was realized that devices have to cool down in a more effective way and the conventional fluids such as water are not appropriate anymore, so the idea of adding particles to a fluid was presented. These tiny particles have high thermal conductivity, so the mixed fluids have better thermal properties [1-3]. The material of these nano scale particles is aluminum oxide (Al2O3), copper (Cu), copper oxide (CuO), gold (Au), silver (Ag) etc, which are suspending in base fluids such as water, oil, acetone and ethylene glycol, etc. Al2O3 and CuO are the most well-known nanoparticles used by many researchers [4-8] in their experimental researches. They claimed different results due to the size and shape and so the contact surface of the particles. In addition the base fluid characteristics were important as well. The main obstacle in this field was how to keep the particles suspended in static fluid which is discussed in [5-9]. Fortunately, the result was in a same trace that the thermal conductivity of the nanofluids is higher than the conventional fluids and this term is modeled mathematically in [4, 5], [10-15]. In this paper we will rephrase the Falkner-Skan equation [16-18] for a nanofluid with a semi analytical method, i. e. Homotopy Analysis Method, which was presented by Liao [19-21] in 1992. The HAM is one of the well-known methods to solve nonlinear equations. This method has been used by many authors in a wide range of engineering problems [22-25]. Falkner and Skan considered two-dimensional wedge flows. They developed a similarity solution method in which the partial differential boundary-layer equation was reduced to a nonlinear third-order ordinary differential equation which does not have an exact solution ;besides, the numerical solution for this equation is time consuming and difficult.

 

  1. 2.      Governing Equations

   In this paper, we consider an incompressible viscous fluid which flows over a wedge, as is shown in Fig. 1.

 

Figure 1.Velocity and thermal boundary layers for the Falkner–Skan wedge flow.

 

The wall temperature, i.e. , is uniformed and constant and is greater than the free stream temperature, . It is assumed that the free stream velocity, , is also uniformed and constant as well. Further, assuming that the flow in the laminar boundary layer is two-dimensional, and that the temperature gradients resulting from viscous dissipation are small, the continuity, momentum and energy equations can be expressed as:

 

(1)

 

 

(2)

 

(3)

where  and  are the respective velocity components in the  and  direction of the fluid flow, is the kinematic viscosity of the fluid, and is the reference velocity at the edge of the boundary layer and is a function of . is the thermal diffusivity of the fluid, is the temperature in the vicinity of the wedge, and the boundary conditions are given by:

 

(4)

 

(5)

 

(6)

where  is the mean stream velocity,  is the length of the wedge,  is the Falkner–Skan power-law parameter, and  is measured from the tip of the wedge. A stream function, is introduced such as this:

 

(7)

By Substituting Eq.  into Eq. , the momentum equation will be obtained as follows:

 

(8)

By using these similarity variable yields, we obtain the following ordinary differential equation:

 

(9)

 

(10)

in which  is the kinematic viscosity of the fluid. Substituting Eqs.  and  into Eq. , gives:

 

(11)

this is known as the Falkner–Skan boundary-layer equation [13-15].The boundary conditions of  are:

 

(12)

 

(13)

Note that in the equations above, parameters  and  are related through the expression. β is the Hartree pressure gradient parameter which corresponds to β=Ω/π for a total angle Ω of the wedge and m=0(β=0) represents the boundary layer flow past a horizontal flat plate, while m=1(β=1) corresponds to the boundary layer flow near the stagnation point of a vertical flat plate. m is a dimensionless constant. In the Blasius solution, m = 0 corresponds to an angle of attack of zero radians. Thus we can write:

   .

A dimensionless temperature is defined as follows:

 

(14)

If Eq.  is substituted into Eq. , then the boundary-layer energy equation becomes:

 

(15)

where  /  is the Prandtl number.

With the following boundary conditions:

 

 

(16)

 

(17)

The following equations can be obtained by changing these parameters to rephrase this classic equation and use it for nanofluids:

 

 

(18)

 

 

(19)

 

 

(20)

 

 

(21)

 

 

(22)

 

(23)

Finally:

 

(24)

With these boundary conditions:

 

 

(25)

 

(26)

And also:

 

(27)

With these boundary conditions:

 

 

(28)

 

(29)

The physical properties are also mentioned in the Table 1.

Table 1. Thermophysical properties of the base fluid and the nanoparticles (Oztop and Abu-Nada [36]).

Physical properties

Fluid phase (water)

Cu

Al2O3

 (J/kg K)

4179

385

765

 (kg /m3)

997.1

8933

3970

 (W/m K)

0.613

400

40

 (m2/s)

1.47

1163.1

131.7

  1. 3.      Basic Idea of HAM

Let us assume the following nonlinear differential equation in the form of:

 

(30)

where  is a nonlinear operator,  is an independent variable and  is the solution of equation. We define the function,  as follows:

 

(31)

where,  and  is the initial guess which satisfies the initial or boundary conditions and if

 

(32)

And by using the generalized homotopy method, Liao’s so-called zero-order deformation Eq.  will be:

 

(33)

where  is the auxiliary parameter which helps us increase the results convergence,  is the auxiliary function and  is the linear operator. It should be noted that there is a great freedom to choose the auxiliary parameter  , the auxiliary function  , the initial guess  and the auxiliary linear operator

Thus, when  increases from  to  the solution  changes between the initial guess  and the solution . The Taylor series expansion of  with respect to  is:

 

(34)

And

 

(35)

where  for brevity is called the order of deformation derivation which reads:

 

(36)

According to the definition in Eq. , the governing equation and the corresponding initial conditions of  can be deduced from zero-order deformation Eq. . Differentiating Eq.  for  times with respect to the embedding parameter  and setting  and finally dividing by , we will have the so-called  order deformation equation in the following form:

 

(37)

where:

 

(38)

And

 

(39)

So by applying an inverse linear operator to both sides of the linear equation, Eq. , we can easily solve the equation and compute the generation constant by applying the initial or boundary condition.

  1. 4.      Application of HAM to Falkner-Skan Problem

       As mentioned by Liao [19-21], a solution may be expressed with different base functions, among which some converge to the exact solution of the problem faster than others. Such base functions are obviously better suited for the final solution to be expressed. Noting these facts, we have decided to express  and  by a set of base functions in the following form:

 

 

(40)

 

(41)

     

The rule of solution expression provides us with a starting point. It is under the rule of solution expression that the initial approximations, the auxiliary linear operators, and the auxiliary functions are determined. So, according to the rule of solution expression, we choose the initial guess and auxiliary linear operator in the following form

 

 

(42)

 

 

(43)

 

 

(44)

 

 

(45)

 

 

(46)

 

(47)

where  are constants. Let  denotes the embedding parameter and  indicates non –zero auxiliary parameters. We construct the following equations:

4.1    Zeroth–Order Deformation Equations

 

 

(48)

 

 

(49)

 

 

 

(50)

 

(51)

 

 

 

 

 

 

 

 

 

 

 

 

 

         (52)

 

 

 

 

 

         (53)

For  and  we have:

 

 

(54)

 

(55)

When p increases from 0 to 1 then  and  vary from  and  to  and . Using Taylor's theorem for Eqs.  and ,  and  can be expanded in a power series of p as follows:

 

 

(56)

 

(57)

in which  is chosen in such a way that these two series are convergent at , therefore we have through Eqs.  and  that

 

(58)

4.2    mth –order deformation equations

 

 

(59)

 

 

(60)

 

 

(61)

 

 

(62)

 

 

(63)

 

 

(64)

 

(65)

     

The general solutions will be:

 

 

(66)

 

(67)

     

where  to  are constants that can be obtained by applying the boundary conditions in Eqs.  and .

As discussed by Liao the rule of coefficient ergodicity and the rule of solution existence play important roles in determining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutions. In many cases, by means of the rule of solution expression and the rule of coefficient ergodicity, auxiliary functions can be uniquely determined. So we define the auxiliary functions  and in the following form:

 

 

(68)

 

(69)

  1. 5.      Convergence of HAM Solution

    HAM provides us with great freedom in choosing the solution of a nonlinear problem by different base funtions. This has a great effect on the convergence region because the convergence region and the rate of a series are chiefly determined by the base functions used to express the solution. Therefore, a more accurate approximation of a nonlinear problem can be obtained by choosing a proper set of base functions and ensuring about its convergency.

On the other hand, as pointed out by Liao, the convergence and the rate of approximation for the HAM solution strongly depend on the value of auxiliary parameters . Even if the initial approximations  and , the auxiliary linear operator , and the auxiliary functions   and are given, we still have great freedom to choose the value of the auxiliary parameters  and . So, the auxiliary parameters provide us with an additional way to conveniently adjust and control the convergence region and the rate of solution series. By means of the so-called -curves it is easy to find out the so-called valid regions of auxiliary parameters to gain a convergent solution series. When the valid region of auxiliary parameters is a horizontal line segment, then the solution is converged.

 

 

Figure 2. The -validity for CuO for .

In our case study, suitable range of  and  for CuO and Al2O3 can be obtained from Figures 2 to 5.

 

  1. 6.      Results and Discussion

  Fig. 6 shows the relation between  and  which represents the velocity of the fluid. Adding nano particles to a base fluid will cause a rise in fluid viscosity, therefore, the momentum exchange between the nanofluid layers is higher. It can be clearly seen that the pure fluid reaches to  approximately in , however, nanofluids approach to  in . Considering Eq.  we have expected this trend. Fig. 7 shows temprature curve variation with  which represents the particle volume fraction of the suspension. When  climbs up from  to , as we expected, there is a  noticeable enhancement in heat transfer rate. This can be justified by torbulace which the nano-scale particles cause. In addition, with the increase in the amount of  the effects of the nanoparticles conductivity appear in the heat transfer rate as well.

 

 

 

Figure 3. The -validity for CuO for .

 

 

 

 

 

Figure 4. The -validity for Al2O3 for .

 

Figure 5. The -validity for Al2O3 for .

As we can see at about  the nanofluid curve with , has dropped to  but when  this happens at .

This trend was observed before in exprimental studies by others [3].

Fig. 8 shows the bulk temprature as a function of the fluids types.

 

 

Figure 6. Velocity profile for different types of fluids for .

 

Figure 7.Temperature profile for CuO for different values of   and .

 

Figure 8. Temperature profile for different types of fluids for  and

In agreement with the previous experimental studies results, our solution shows that nanofluids have better thermal properties and the temprature diffrence between the fluid and the wedge disappears sooner.

Due to adding the nano particles in the base fluid, we can see a noticeable enhancement in heat transfer rate.

In other words, the nano fluids tempratures are equal with the wedge surface at  for “Cuo/water” and  for “Al2O3/water”, however, water experiences this at . To see this effect better, we have magnified the region between  in the related figure.

Fig. 9 shows the relation between fluid’s bulk temperature, i.e.  and . We can observe a marked change in the  at which profiles drop to . As  grows, the effects of the stagnation point will be more important and the conductivity of the fluid will play a more important role. Because of high heat conduction in nanofluids the curve approaches to  sooner with the increase in .

 

 

 

  1. 7.      Conclusion

     To make it brief, in this paper, we derived the Falkner-Skan equation for a nanofluid and solved it with the Homotopy Analysis Method. The convergence region of HAM is discussed, i.e. the auxiliary parameters and .

 

Figure 9.Temperature  profile for CuO for different values of   and .

The physical interpretation of the results was explained.

Temperature and velocity diagrams have been discussed in details which has shown that Cuo as a nanoparticle has more efficiency in comparison with Al2O3. Both of these nanoparticles are suspended in water as the base fluid. In addition we observed that with adding nano-scale particles to a fluid, the viscosity of the fluid will increase and this will affect the momentum exchange in the fluid layers. 

References

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[2]    Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000;21:58–64.

[3]   Das SK, Putra N, Roetzel W. Pool boiling characteristics of nano-fluids. Int J Heat Mass Transfer 2003; 46: 851–62.

[4]   Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.

[5]    Xuan Y, Li Q.: Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000; 21: 58–64.

[6]   Das SK, Putra N, Roetzel W. Pool boiling characteristics of nano-fluids. Int J Heat Mass Transfer 2003; 46: 851–62.

[7]   Das SK, Putra N, Roetzel W. Pool boiling of nano-fluids on horizontal narrow tubes. Int J Multiphase Flow 2003; 29: 1237–47.

[8]   Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J Heat Transfer 2003; 125: 567–74.

[9]   Wang X, Xu X, Choi SUS. Thermal conductivity of nanoparticle–fluid mixture. J Therm Phys Heat Transfer 1999; 13: 474–80.

[10]    Eastman JA, Choi SUS, Li S, Yu W, Thomson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 2001; 78: 718–20.

[11]    Xie H, Wang J, Xi T, Liu Y, Ai F, Wu Q. Thermal conductivity enhancement of suspensions containing nanosized alumina particles. Appl Phys 2002; 91: 4568–72.

[12]    Dongsheng Wen, Guiping Lin, SaeidVafaei, Kai Zhang. Particuology   2009; 7: 141–150.

[13]    Wang BX, Zhou LP, Peng XF. A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Int J Heat Mass Transfer 2003; 46: 2665–72.

[14]    Xue QZ. Model for effective thermal conductivity of nanofluids.PhysLett A 2003; 307: 313–7.

[15]    Yu W, Choi SUS. The role of interfacial layers in the enhanced thermal conductivity of nanofluids: arenovated Maxwell model. Nanoparticle Res 2003; 5: 167–71.

[16]    Falkner, V. M, and Skan, S. W. Solutions of the boundary-layer equations. Philosophical Magazine, 1931; 7(12): 865-896.

[17]    S. Abbasbandy and T. Hayat. Solution of the MHD Falkner-Skan flow by homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; (14): 3591–3598.

[18]    Pade. On the solution of  Falkner–Skan equations, J Math Anal Appl 2003; (285): 264–274.

[19]    Liao, S. J. On the Homotopy Analysis Method for Nonlinear Problems. Appl Math Comput 2004; 47 (2)

[20]    Liao, S. J. An Explicit, Totally Analytic Approximation of Blasius’s Viscous Flow Problems. Int J Non-Linear Mech 1999; 34 (4)

[21]    Liao, S. J. Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. Ph.D. Dissertation, Shanghai Liao Tong University, China 1992.

[22]    Domairry, G. and N. Nadim. Assessment of Homotopy Analysis Method and Homotopy Perturbation Method in Non-Linear Heat Transfer Equation. Int Commun Heat Mass Transfer 2008; 35 (1):

[23]    Domairry, G. and M. Fazeli. Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity. Commun Non-linear Sci Numer Simul 2009; 14 (2)

[24]    Domairry, G., A. Mohsenzadeh, and M. Famouri. The Application of Homotopy Analysis Method to Solve Nonlinear Differential Equation Governing Jeffery–Hamel Flow. Commun Non-linear Sci Numer Simul 2009; 14 (1): 85

[25]    Fakhari A, Domairry G, Ebrahimpour. Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys lett A  2007; 368: 64-68.

[26]    Sohouli AR, Domairry D, Famouri M, Mohsenzadeh A. Analytical solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by means of HAM, Int Commun Heat and Mass Transfer 2008; 35(10): 1380-1384.

[27]    Ziabakhsh Z, Domairry G. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(5): 1868-1880.

[28]    Sohouli AR, Famouri M, Kimiaeifar A, Domairry G. Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux, Commun Nonlinear Sci Numer Simul 2010; 15(7): 1691-1699

[29]     Ziabakhsh Z, Domairry G, Ghazizadeh HR. Analytical solution of the stagnation-point flow in a porous medium by using the homotopy analysis method, Taiwan Institute Chem Eng 2009; 40(1): 91-97.

[30]    Rashidi MM, Domairry G, Dinarvand S. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(3): 708-717.

[31]    Rashidi MM, Mohimanianpour SA, Abbasbandy S. Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation, Commun Nonlinear Sci Numer Simul 2011; 16(4): 1874-1889.

[32]    Abdoul R. Ghotbi, H. Bararnia, G. Domairry, A. Barari: Investigation of a powerful analytical method into natural convection boundary layer flow, Commun Nonlinear Sci Numer Simul 2009; 14(5): 2222-2228

[33]    Bararnia H, Ghasemi E, Domairry G, Soleimani S. Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction, Adv Eng Softw 2010; 41(6): 893-897.

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[35]    Kimiaeifar A, Saidi AR, Bagheri GH, Rahimpour M, Domairry DG. Analytical solution for Van der Pol–Duffing oscillators. Chaos Solitons & Fractals 2009; 42(5): 2660-2666.

[36]    Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int J Heat Fluid Flow 2008; 29: 1326–1336.

 

An Analytical Study on Effects of adding Nanoparticles to Water and Enhancement in Thermal Properties Based on Falkner-Skan Model

Y. Rostamiyana*, M. Abbasia, F. Aghajania, F. Hedayatia, S.M. Hamidib

a Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

bDepartment of Electrical and Energy Engineering, University of Vaasa, Vaasa 65200, Finland

 

paper INFO

 

ABSTRACT

Paper history:

Received 6 August 2014

Received in revised form 15 September 2014

Accepted 22 Semptember 2014

 In the age of technology, it is vital to cool down different parts of a device to use it more beneficially. Using nanofluids is one of the most common methods which has shown very effective results. In this paper, we have rephrased a classic equation in fluid mechanics, i.e. the Falkner-Skan boundary layer equation, in order to be used for nanofluid. This nonlinear equation, which was presented by Liao, has been solved by Homotopy Analysis Method (HAM). This method is very capable to solve a wide range of nonlinear equations. The physical interpretation of results which are velocity and temperature profiles are explained in details and they are parallel with experimental outcomes of previous researchers.

 

Keywords:

Falkner-skan

Boundary layer

Nanofluids

HAM

 

© 2014 Published by Semnan University Press. All rights reserved.

 

 

  1. 1.      Introduction

As technology improves, it was realized that devices have to cool down in a more effective way and the conventional fluids such as water are not appropriate anymore, so the idea of adding particles to a fluid was presented. These tiny particles have high thermal conductivity, so the mixed fluids have better thermal properties [1-3]. The material of these nano scale particles is aluminum oxide (Al2O3), copper (Cu), copper oxide (CuO), gold (Au), silver (Ag) etc, which are suspending in base fluids such as water, oil, acetone and ethylene glycol, etc. Al2O3 and CuO are the most well-known nanoparticles used by many researchers [4-8] in their experimental researches. They claimed different results due to the size and shape and so the contact surface of the particles. In addition the base fluid characteristics were important as well. The main obstacle in this field was how to keep the particles suspended in static fluid which is discussed in [5-9]. Fortunately, the result was in a same trace that the thermal conductivity of the nanofluids is higher than the conventional fluids and this term is modeled mathematically in [4, 5], [10-15]. In this paper we will rephrase the Falkner-Skan equation [16-18] for a nanofluid with a semi analytical method, i. e. Homotopy Analysis Method, which was presented by Liao [19-21] in 1992. The HAM is one of the well-known methods to solve nonlinear equations. This method has been used by many authors in a wide range of engineering problems [22-25]. Falkner and Skan considered two-dimensional wedge flows. They developed a similarity solution method in which the partial differential boundary-layer equation was reduced to a nonlinear third-order ordinary differential equation which does not have an exact solution ;besides, the numerical solution for this equation is time consuming and difficult.

 

  1. 2.      Governing Equations

   In this paper, we consider an incompressible viscous fluid which flows over a wedge, as is shown in Fig. 1.

 

Figure 1.Velocity and thermal boundary layers for the Falkner–Skan wedge flow.

 

The wall temperature, i.e. , is uniformed and constant and is greater than the free stream temperature, . It is assumed that the free stream velocity, , is also uniformed and constant as well. Further, assuming that the flow in the laminar boundary layer is two-dimensional, and that the temperature gradients resulting from viscous dissipation are small, the continuity, momentum and energy equations can be expressed as:

 

(1)

 

 

(2)

 

(3)

where  and  are the respective velocity components in the  and  direction of the fluid flow, is the kinematic viscosity of the fluid, and is the reference velocity at the edge of the boundary layer and is a function of . is the thermal diffusivity of the fluid, is the temperature in the vicinity of the wedge, and the boundary conditions are given by:

 

(4)

 

(5)

 

(6)

where  is the mean stream velocity,  is the length of the wedge,  is the Falkner–Skan power-law parameter, and  is measured from the tip of the wedge. A stream function, is introduced such as this:

 

(7)

By Substituting Eq.  into Eq. , the momentum equation will be obtained as follows:

 

(8)

By using these similarity variable yields, we obtain the following ordinary differential equation:

 

(9)

 

(10)

in which  is the kinematic viscosity of the fluid. Substituting Eqs.  and  into Eq. , gives:

 

(11)

this is known as the Falkner–Skan boundary-layer equation [13-15].The boundary conditions of  are:

 

(12)

 

(13)

Note that in the equations above, parameters  and  are related through the expression. β is the Hartree pressure gradient parameter which corresponds to β=Ω/π for a total angle Ω of the wedge and m=0(β=0) represents the boundary layer flow past a horizontal flat plate, while m=1(β=1) corresponds to the boundary layer flow near the stagnation point of a vertical flat plate. m is a dimensionless constant. In the Blasius solution, m = 0 corresponds to an angle of attack of zero radians. Thus we can write:

   .

A dimensionless temperature is defined as follows:

 

(14)

If Eq.  is substituted into Eq. , then the boundary-layer energy equation becomes:

 

(15)

where  /  is the Prandtl number.

With the following boundary conditions:

 

 

(16)

 

(17)

The following equations can be obtained by changing these parameters to rephrase this classic equation and use it for nanofluids:

 

 

(18)

 

 

(19)

 

 

(20)

 

 

(21)

 

 

(22)

 

(23)

Finally:

 

(24)

With these boundary conditions:

 

 

(25)

 

(26)

And also:

 

(27)

With these boundary conditions:

 

 

(28)

 

(29)

The physical properties are also mentioned in the Table 1.

Table 1. Thermophysical properties of the base fluid and the nanoparticles (Oztop and Abu-Nada [36]).

Physical properties

Fluid phase (water)

Cu

Al2O3

 (J/kg K)

4179

385

765

 (kg /m3)

997.1

8933

3970

 (W/m K)

0.613

400

40

 (m2/s)

1.47

1163.1

131.7

  1. 3.      Basic Idea of HAM

Let us assume the following nonlinear differential equation in the form of:

 

(30)

where  is a nonlinear operator,  is an independent variable and  is the solution of equation. We define the function,  as follows:

 

(31)

where,  and  is the initial guess which satisfies the initial or boundary conditions and if

 

(32)

And by using the generalized homotopy method, Liao’s so-called zero-order deformation Eq.  will be:

 

(33)

where  is the auxiliary parameter which helps us increase the results convergence,  is the auxiliary function and  is the linear operator. It should be noted that there is a great freedom to choose the auxiliary parameter  , the auxiliary function  , the initial guess  and the auxiliary linear operator

Thus, when  increases from  to  the solution  changes between the initial guess  and the solution . The Taylor series expansion of  with respect to  is:

 

(34)

And

 

(35)

where  for brevity is called the order of deformation derivation which reads:

 

(36)

According to the definition in Eq. , the governing equation and the corresponding initial conditions of  can be deduced from zero-order deformation Eq. . Differentiating Eq.  for  times with respect to the embedding parameter  and setting  and finally dividing by , we will have the so-called  order deformation equation in the following form:

 

(37)

where:

 

(38)

And

 

(39)

So by applying an inverse linear operator to both sides of the linear equation, Eq. , we can easily solve the equation and compute the generation constant by applying the initial or boundary condition.

  1. 4.      Application of HAM to Falkner-Skan Problem

       As mentioned by Liao [19-21], a solution may be expressed with different base functions, among which some converge to the exact solution of the problem faster than others. Such base functions are obviously better suited for the final solution to be expressed. Noting these facts, we have decided to express  and  by a set of base functions in the following form:

 

 

(40)

 

(41)

     

The rule of solution expression provides us with a starting point. It is under the rule of solution expression that the initial approximations, the auxiliary linear operators, and the auxiliary functions are determined. So, according to the rule of solution expression, we choose the initial guess and auxiliary linear operator in the following form

 

 

(42)

 

 

(43)

 

 

(44)

 

 

(45)

 

 

(46)

 

(47)

where  are constants. Let  denotes the embedding parameter and  indicates non –zero auxiliary parameters. We construct the following equations:

4.1    Zeroth–Order Deformation Equations

 

 

(48)

 

 

(49)

 

 

 

(50)

 

(51)

 

 

 

 

 

 

 

 

 

 

 

 

 

         (52)

 

 

 

 

 

         (53)

For  and  we have:

 

 

(54)

 

(55)

When p increases from 0 to 1 then  and  vary from  and  to  and . Using Taylor's theorem for Eqs.  and ,  and  can be expanded in a power series of p as follows:

 

 

(56)

 

(57)

in which  is chosen in such a way that these two series are convergent at , therefore we have through Eqs.  and  that

 

(58)

4.2    mth –order deformation equations

 

 

(59)

 

 

(60)

 

 

(61)

 

 

(62)

 

 

(63)

 

 

(64)

 

(65)

     

The general solutions will be:

 

 

(66)

 

(67)

     

where  to  are constants that can be obtained by applying the boundary conditions in Eqs.  and .

As discussed by Liao the rule of coefficient ergodicity and the rule of solution existence play important roles in determining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutions. In many cases, by means of the rule of solution expression and the rule of coefficient ergodicity, auxiliary functions can be uniquely determined. So we define the auxiliary functions  and in the following form:

 

 

(68)

 

(69)

  1. 5.      Convergence of HAM Solution

    HAM provides us with great freedom in choosing the solution of a nonlinear problem by different base funtions. This has a great effect on the convergence region because the convergence region and the rate of a series are chiefly determined by the base functions used to express the solution. Therefore, a more accurate approximation of a nonlinear problem can be obtained by choosing a proper set of base functions and ensuring about its convergency.

On the other hand, as pointed out by Liao, the convergence and the rate of approximation for the HAM solution strongly depend on the value of auxiliary parameters . Even if the initial approximations  and , the auxiliary linear operator , and the auxiliary functions   and are given, we still have great freedom to choose the value of the auxiliary parameters  and . So, the auxiliary parameters provide us with an additional way to conveniently adjust and control the convergence region and the rate of solution series. By means of the so-called -curves it is easy to find out the so-called valid regions of auxiliary parameters to gain a convergent solution series. When the valid region of auxiliary parameters is a horizontal line segment, then the solution is converged.

 

 

Figure 2. The -validity for CuO for .

In our case study, suitable range of  and  for CuO and Al2O3 can be obtained from Figures 2 to 5.

 

  1. 6.      Results and Discussion

  Fig. 6 shows the relation between  and  which represents the velocity of the fluid. Adding nano particles to a base fluid will cause a rise in fluid viscosity, therefore, the momentum exchange between the nanofluid layers is higher. It can be clearly seen that the pure fluid reaches to  approximately in , however, nanofluids approach to  in . Considering Eq.  we have expected this trend. Fig. 7 shows temprature curve variation with  which represents the particle volume fraction of the suspension. When  climbs up from  to , as we expected, there is a  noticeable enhancement in heat transfer rate. This can be justified by torbulace which the nano-scale particles cause. In addition, with the increase in the amount of  the effects of the nanoparticles conductivity appear in the heat transfer rate as well.

 

 

 

Figure 3. The -validity for CuO for .

 

 

 

 

 

Figure 4. The -validity for Al2O3 for .

 

Figure 5. The -validity for Al2O3 for .

As we can see at about  the nanofluid curve with , has dropped to  but when  this happens at .

This trend was observed before in exprimental studies by others [3].

Fig. 8 shows the bulk temprature as a function of the fluids types.

 

 

Figure 6. Velocity profile for different types of fluids for .

 

Figure 7.Temperature profile for CuO for different values of   and .

 

Figure 8. Temperature profile for different types of fluids for  and

In agreement with the previous experimental studies results, our solution shows that nanofluids have better thermal properties and the temprature diffrence between the fluid and the wedge disappears sooner.

Due to adding the nano particles in the base fluid, we can see a noticeable enhancement in heat transfer rate.

In other words, the nano fluids tempratures are equal with the wedge surface at  for “Cuo/water” and  for “Al2O3/water”, however, water experiences this at . To see this effect better, we have magnified the region between  in the related figure.

Fig. 9 shows the relation between fluid’s bulk temperature, i.e.  and . We can observe a marked change in the  at which profiles drop to . As  grows, the effects of the stagnation point will be more important and the conductivity of the fluid will play a more important role. Because of high heat conduction in nanofluids the curve approaches to  sooner with the increase in .

 

 

 

  1. 7.      Conclusion

     To make it brief, in this paper, we derived the Falkner-Skan equation for a nanofluid and solved it with the Homotopy Analysis Method. The convergence region of HAM is discussed, i.e. the auxiliary parameters and .

 

Figure 9.Temperature  profile for CuO for different values of   and .

The physical interpretation of the results was explained.

Temperature and velocity diagrams have been discussed in details which has shown that Cuo as a nanoparticle has more efficiency in comparison with Al2O3. Both of these nanoparticles are suspended in water as the base fluid. In addition we observed that with adding nano-scale particles to a fluid, the viscosity of the fluid will increase and this will affect the momentum exchange in the fluid layers. 

References

[1]   Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.

[2]    Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000;21:58–64.

[3]   Das SK, Putra N, Roetzel W. Pool boiling characteristics of nano-fluids. Int J Heat Mass Transfer 2003; 46: 851–62.

[4]   Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.

[5]    Xuan Y, Li Q.: Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000; 21: 58–64.

[6]   Das SK, Putra N, Roetzel W. Pool boiling characteristics of nano-fluids. Int J Heat Mass Transfer 2003; 46: 851–62.

[7]   Das SK, Putra N, Roetzel W. Pool boiling of nano-fluids on horizontal narrow tubes. Int J Multiphase Flow 2003; 29: 1237–47.

[8]   Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J Heat Transfer 2003; 125: 567–74.

[9]   Wang X, Xu X, Choi SUS. Thermal conductivity of nanoparticle–fluid mixture. J Therm Phys Heat Transfer 1999; 13: 474–80.

[10]    Eastman JA, Choi SUS, Li S, Yu W, Thomson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 2001; 78: 718–20.

[11]    Xie H, Wang J, Xi T, Liu Y, Ai F, Wu Q. Thermal conductivity enhancement of suspensions containing nanosized alumina particles. Appl Phys 2002; 91: 4568–72.

[12]    Dongsheng Wen, Guiping Lin, SaeidVafaei, Kai Zhang. Particuology   2009; 7: 141–150.

[13]    Wang BX, Zhou LP, Peng XF. A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Int J Heat Mass Transfer 2003; 46: 2665–72.

[14]    Xue QZ. Model for effective thermal conductivity of nanofluids.PhysLett A 2003; 307: 313–7.

[15]    Yu W, Choi SUS. The role of interfacial layers in the enhanced thermal conductivity of nanofluids: arenovated Maxwell model. Nanoparticle Res 2003; 5: 167–71.

[16]    Falkner, V. M, and Skan, S. W. Solutions of the boundary-layer equations. Philosophical Magazine, 1931; 7(12): 865-896.

[17]    S. Abbasbandy and T. Hayat. Solution of the MHD Falkner-Skan flow by homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; (14): 3591–3598.

[18]    Pade. On the solution of  Falkner–Skan equations, J Math Anal Appl 2003; (285): 264–274.

[19]    Liao, S. J. On the Homotopy Analysis Method for Nonlinear Problems. Appl Math Comput 2004; 47 (2)

[20]    Liao, S. J. An Explicit, Totally Analytic Approximation of Blasius’s Viscous Flow Problems. Int J Non-Linear Mech 1999; 34 (4)

[21]    Liao, S. J. Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. Ph.D. Dissertation, Shanghai Liao Tong University, China 1992.

[22]    Domairry, G. and N. Nadim. Assessment of Homotopy Analysis Method and Homotopy Perturbation Method in Non-Linear Heat Transfer Equation. Int Commun Heat Mass Transfer 2008; 35 (1):

[23]    Domairry, G. and M. Fazeli. Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity. Commun Non-linear Sci Numer Simul 2009; 14 (2)

[24]    Domairry, G., A. Mohsenzadeh, and M. Famouri. The Application of Homotopy Analysis Method to Solve Nonlinear Differential Equation Governing Jeffery–Hamel Flow. Commun Non-linear Sci Numer Simul 2009; 14 (1): 85

[25]    Fakhari A, Domairry G, Ebrahimpour. Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys lett A  2007; 368: 64-68.

[26]    Sohouli AR, Domairry D, Famouri M, Mohsenzadeh A. Analytical solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by means of HAM, Int Commun Heat and Mass Transfer 2008; 35(10): 1380-1384.

[27]    Ziabakhsh Z, Domairry G. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(5): 1868-1880.

[28]    Sohouli AR, Famouri M, Kimiaeifar A, Domairry G. Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux, Commun Nonlinear Sci Numer Simul 2010; 15(7): 1691-1699

[29]     Ziabakhsh Z, Domairry G, Ghazizadeh HR. Analytical solution of the stagnation-point flow in a porous medium by using the homotopy analysis method, Taiwan Institute Chem Eng 2009; 40(1): 91-97.

[30]    Rashidi MM, Domairry G, Dinarvand S. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(3): 708-717.

[31]    Rashidi MM, Mohimanianpour SA, Abbasbandy S. Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation, Commun Nonlinear Sci Numer Simul 2011; 16(4): 1874-1889.

[32]    Abdoul R. Ghotbi, H. Bararnia, G. Domairry, A. Barari: Investigation of a powerful analytical method into natural convection boundary layer flow, Commun Nonlinear Sci Numer Simul 2009; 14(5): 2222-2228

[33]    Bararnia H, Ghasemi E, Domairry G, Soleimani S. Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction, Adv Eng Softw 2010; 41(6): 893-897.

[34]    Jalaal M, Bararnia H, Domairry G. A series exact solution for one-dimensional non-linear particle equation of motion, Powder Technol 2011; 207 (1-3): 461-464.

[35]    Kimiaeifar A, Saidi AR, Bagheri GH, Rahimpour M, Domairry DG. Analytical solution for Van der Pol–Duffing oscillators. Chaos Solitons & Fractals 2009; 42(5): 2660-2666.

[36]    Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int J Heat Fluid Flow 2008; 29: 1326–1336.

 

[1]          Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.

[2]           Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000;21:58–64.

[3]          Das SK, Putra N, Roetzel W. Pool boiling characteristics of nano-fluids. Int J Heat Mass Transfer 2003; 46: 851–62.

[4]          Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxidenanoparticles. ASME J Heat Transfer 1999; 121: 280–9.

[5]           Xuan Y, Li Q.: Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000; 21: 58–64.

[6]          Das SK, Putra N, Roetzel W. Pool boiling characteristics of nano-fluids. Int J Heat Mass Transfer 2003; 46: 851–62.

[7]          Das SK, Putra N, Roetzel W. Pool boiling of nano-fluids on horizontal narrow tubes. Int J Multiphase Flow 2003; 29: 1237–47.

[8]          Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J Heat Transfer 2003; 125: 567–74.

[9]          Wang X, Xu X, Choi SUS. Thermal conductivity of nanoparticle–fluid mixture. J Therm Phys Heat Transfer 1999; 13: 474–80.

[10]        Eastman JA, Choi SUS, Li S, Yu W, Thomson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 2001; 78: 718–20.

[11]        Xie H, Wang J, Xi T, Liu Y, Ai F, Wu Q. Thermal conductivity enhancement of suspensions containing nanosized alumina particles. Appl Phys 2002; 91: 4568–72.

[12]        Dongsheng Wen, Guiping Lin, SaeidVafaei, Kai Zhang. Particuology   2009; 7: 141–150.

[13]        Wang BX, Zhou LP, Peng XF. A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Int J Heat Mass Transfer 2003; 46: 2665–72.

[14]        Xue QZ. Model for effective thermal conductivity of nanofluids.PhysLett A 2003; 307: 313–7.

[15]        Yu W, Choi SUS. The role of interfacial layers in the enhanced thermal conductivity of nanofluids: arenovated Maxwell model. Nanoparticle Res 2003; 5: 167–71.

[16]        Falkner, V. M, and Skan, S. W. Solutions of the boundary-layer equations. Philosophical Magazine, 1931; 7(12): 865-896.

[17]        S. Abbasbandy and T. Hayat. Solution of the MHD Falkner-Skan flow by homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; (14): 3591–3598.

[18]        Pade. On the solution of  Falkner–Skan equations, J Math Anal Appl 2003; (285): 264–274.

[19]        Liao, S. J. On the Homotopy Analysis Method for Nonlinear Problems. Appl Math Comput 2004; 47 (2)

[20]        Liao, S. J. An Explicit, Totally Analytic Approximation of Blasius’s Viscous Flow Problems. Int J Non-Linear Mech 1999; 34 (4)

[21]        Liao, S. J. Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. Ph.D. Dissertation, Shanghai Liao Tong University, China 1992.

[22]        Domairry, G. and N. Nadim. Assessment of Homotopy Analysis Method and Homotopy Perturbation Method in Non-Linear Heat Transfer Equation. Int Commun Heat Mass Transfer 2008; 35 (1):

[23]        Domairry, G. and M. Fazeli. Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity. Commun Non-linear Sci Numer Simul 2009; 14 (2)

[24]        Domairry, G., A. Mohsenzadeh, and M. Famouri. The Application of Homotopy Analysis Method to Solve Nonlinear Differential Equation Governing Jeffery–Hamel Flow. Commun Non-linear Sci Numer Simul 2009; 14 (1): 85

[25]        Fakhari A, Domairry G, Ebrahimpour. Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys lett A  2007; 368: 64-68.

[26]        Sohouli AR, Domairry D, Famouri M, Mohsenzadeh A. Analytical solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by means of HAM, Int Commun Heat and Mass Transfer 2008; 35(10): 1380-1384.

[27]        Ziabakhsh Z, Domairry G. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(5): 1868-1880.

[28]        Sohouli AR, Famouri M, Kimiaeifar A, Domairry G. Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux, Commun Nonlinear Sci Numer Simul 2010; 15(7): 1691-1699

[29]         Ziabakhsh Z, Domairry G, Ghazizadeh HR. Analytical solution of the stagnation-point flow in a porous medium by using the homotopy analysis method, Taiwan Institute Chem Eng 2009; 40(1): 91-97.

[30]        Rashidi MM, Domairry G, Dinarvand S. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Commun Nonlinear Sci Numer Simul 2009; 14(3): 708-717.

[31]        Rashidi MM, Mohimanianpour SA, Abbasbandy S. Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation, Commun Nonlinear Sci Numer Simul 2011; 16(4): 1874-1889.

[32]        Abdoul R. Ghotbi, H. Bararnia, G. Domairry, A. Barari: Investigation of a powerful analytical method into natural convection boundary layer flow, Commun Nonlinear Sci Numer Simul 2009; 14(5): 2222-2228

[33]        Bararnia H, Ghasemi E, Domairry G, Soleimani S. Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction, Adv Eng Softw 2010; 41(6): 893-897.

[34]        Jalaal M, Bararnia H, Domairry G. A series exact solution for one-dimensional non-linear particle equation of motion, Powder Technol 2011; 207 (1-3): 461-464.

[35]        Kimiaeifar A, Saidi AR, Bagheri GH, Rahimpour M, Domairry DG. Analytical solution for Van der Pol–Duffing oscillators. Chaos Solitons & Fractals 2009; 42(5): 2660-2666.

[36]        Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int J Heat Fluid Flow 2008; 29: 1326–1336.