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Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel
Computers and Mathematics with Applications (
)
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel Abdelhalim Ebaid Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
article
info
Article history: Received 8 December 2013 Received in revised form 4 May 2014 Accepted 14 May 2014 Available online xxxx Keywords: Peristaltic flow Nanofluid Exact solution Homotopy perturbation method
abstract In applied science, the exact solution (when available) for any physical model is of great importance. Such exact solution not only leads to the correct physical interpretation, but also very useful in validating the approximate analytical or numerical methods. However, the exact solution is not always available for the reason that many authors resort to the approximate solutions by using any of the analytical or the numerical methods. To ensure the accuracy of these approximate solutions, the convergence issue should be addressed, otherwise, such approximate solutions inevitably lead to incorrect interpretations for the considered model. Recently, several peristaltic flow problems have been solved via the homotopy perturbation method, which is an approximate analytical method. One of these problems is selected in this paper to show that the solutions obtained by the homotopy perturbation method were inaccurate, especially, when compared with the exact solutions provided currently and also when compared with a well known accurate numerical method. The comparisons reveal that great remarkable differences have been detected between the exact current results and those approximately obtained in the literatures for the temperature distribution and the nano-particle concentration. Hence, many similar problems that have been approximately solved by using the homotopy perturbation method should be re-investigated by taking the convergence issue into consideration, otherwise, the published results were really incorrect. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The homotopy perturbation method (HPM) was proposed by He [1–7] as an analytical technique to solve nonlinear differential equations. This method has been widely used by many authors to investigate various models [8–11]. Unlike, the requirement for the regular perturbation techniques [12], the homotopy perturbation method is always valid no matter whether there exists a small physical parameter or not. It combines the traditional perturbation method and the homotopy technique to deform a nonlinear problem into a simple solving one. The solution using this method is expressed as the summation of an infinite series in terms of an artificial parameter. Moreover, this method depends on some initial guesses for the differential operators to generating the solutions. Here, it should be noted that if such initial guesses are chosen by inappropriate ways, the resulting solutions may be inaccurate. However, the method gives accurate solutions in many cases in which successful initial guesses have been chosen as pointed out by [8,9,11]. Very recently, many authors have used the homotopy perturbation method to analyze several peristaltic flow problems of nanofluids [13–17]. The concept of nanofluids is put into practice particularly after the tremendous development of E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.camwa.2014.05.008 0898-1221/© 2014 Elsevier Ltd. All rights reserved.
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nanotechnologies in the last decade, as they are of great importance in many industrial and engineering applications, especially in heat transfer enhancement [18,19]. This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu [20]). Nanofluids are produced by dispersing nanometer-scale solid particles into base liquids such as water, ethylene glycol, and oils. Normally, if the particle sizes are in the 1100 nm ranges, they are generally called nano-particles. Recently, the peristaltic flow of nanofluids has attracted much attention [13–17,21,22] due to its important applications in engineering, medicine, and biology. The flow field of such problems is governed by systems of linear and nonlinear partial differential equations. Due to the difficulties arise in obtaining the exact solutions for the most of these systems, several authors have resorted to approximate analytical or approximate numerical methods. Sometimes, these approximate analytical solutions do not lead to accurate numerical results, especially, when compared with the exact solutions. In this regard, we aim in this paper to show that the approximate solutions obtained by using the HPM in [17] were not accurate enough to be presented to the scientific community. This shall be done by first achieving the exact solutions for the temperature distribution and the nano-particle concentration for the current physical model. Then several plots shall be presented for the comparisons between the approximate solutions using HPM with the current exact solutions along with the numerical solutions derived from MATHEMATICA. 2. The mathematical model Akram et al. [17] considered the peristaltic flow of an incompressible Jeffrey fluid in an asymmetric channel of width d1 + d2 . The asymmetric channel flow is produced due to different amplitudes and phases of the peristaltic waves on the channel. Heat transfer along with nano-particle phenomena has been taken into account. The wall surfaces are chosen in the following forms: H1 (X , t ) = d1 + a1 cos
H2 (X , t ) = −d2 − b1 cos
2π
λ
(X − ct ) ,
2π
λ
Upper wall,
(X − ct ) + φ ,
Lower wall,
(1)
(2)
where a1 , b1 are the amplitudes of the upper and lower waves, λ is the wave length, φ is the phase difference which varies in the range 0 ≤ φ ≤ π . In addition, a1 , a2 , b1 , b2 , and φ should satisfy the following condition [17]: a21 + b21 + 2a1 b1 cos φ ≤ (a1 + a2 )2 .
(3)
The flow is assumed to be steady in the wave frame (x, y) moving with velocity c away from the fixed frame (X , Y ). The transformation between these two frames is given by x = X − ct ,
y = Y,
u = U − c,
p(x) = P (X , t )
(4)
where u and v are the velocity components in the wave frame (x, y), p and P are pressure in wave and fixed frame of reference, respectively. Akram et al. [17] found that under the assumptions of long wavelength and low Reynolds number approximation the flow is governed by the following system of partial differential equations in non-dimensional form:
∂p = 0, ∂y ∂p ∂ 1 ∂ 2ψ = + Gr θ + Br Φ , ∂x ∂ y (1 + λ1 ) ∂ y2 2 1 ∂ 2θ ∂θ ∂ Φ ∂θ + N + N = 0, b t 2 Pr ∂ y ∂y ∂y ∂y
(5)
(6)
(7)
∂ 2Φ Nt ∂ 2 θ + = 0, (8) 2 ∂y Nb ∂ y2 ∂θ ∂2 1 ∂ 2ψ ∂Φ + Gr + Br = 0, (9) ∂ y2 (1 + λ1 ) ∂ y2 ∂y ∂y where λ1 , Nb , Nt , Gr, and Br are the ratio of relaxation to retardation times, the Brownian motion parameter, thermophoresis parameter, local temperature Grashof number and nano-particle Grashof number, respectively. In addition, a, b, d, and φ should satisfy the following condition [17]: a2 + b2 + 2ab cos φ ≤ (1 + d)2 .
(10)
Eq. (5) shows that p is independent of y. Accordingly, Eq. (6) can be written as dp dx
=
∂ 1 ∂ 2ψ + Gr θ + Br Φ . ∂ y (1 + λ1 ) ∂ y2
(11)
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The flow is governed by the following boundary conditions:
ψ=
q 2
∂ψ = −1, ∂y
,
at y = h1 (x),
∂ψ = −1, ∂y
q
ψ =− , 2
(12)
at y = h2 (x),
(13)
θ = 0,
φ = 0,
at y = h1 (x),
(14)
θ = 1,
φ = 1,
at y = h2 (x),
(15)
where h1 (x) = 1 + a cos(2π x),
h1 (x) = −d − b cos(2π x + φ).
(16)
3. Analysis and discussion Although it seems to be difficult to obtain the analytical solutions for the current system of linear and nonlinear partial differential equations, it is discussed below an effective procedure to achieve this goal. From Eq. (8), we obtain
Φ=−
Nt Nb
θ + f1 (x)y + f2 (x),
(17)
where f1 (x) and f2 (x) are two unknown functions to be determined. By inserting Eq. (17) into (7) results
∂ 2θ ∂θ + PrNb f1 (x) = 0, ∂ y2 ∂y
(18)
which can be integrated once w.r.t. y to give
∂θ + PrNb f1 (x)θ = f3 (x), ∂y
(19)
where f3 (x) is an additional unknown function. Eq. (19) is solved exactly to give the temperature distribution as
θ (x, y) =
f3 (x)
+ f4 (x) e−Nb Prf1 (x)y ,
Nb Prf1 (x)
(20)
where f4 (x) is also unknown function. Therefore the nano-particle concentration φ is given from Eq. (17) as
Φ (x, y) = f1 (x)y + f2 (x) −
Nt
f3 (x)
Nb
Nb Prf1 (x)
+ f4 (x) e−Nb Prf1 (x)y .
(21)
It is our task now to find the unknown functions f1 (x), f2 (x), f3 (x), and f4 (x) through applying the given boundary conditions. First of all, f1 (x) and f2 (x) can be determined by applying the boundary conditions for θ and φ on Eq. (17) and this gives f 1 ( x) =
1+
Nt Nb
h2 − h1
f2 (x) = −h1
,
(22)
1+
Nt Nb
h2 − h1
.
(23)
Further, we can get f3 (x) and f4 (x) by applying the boundary conditions for φ on Eq. (21), this leads to the following system: f1 (x)h1 + f2 (x) − f1 (x)h2 + f2 (x) −
Nt
Nb Prf1 (x)
Nb Nt
f 3 ( x)
Nb
f 3 ( x) Nb Prf1 (x)
+ f4 (x) e
−Nb Prf1 (x)h1
= 0,
(24)
+ f4 (x) e−Nb Prf1 (x)h2 = 1.
(25)
On solving the system (24)–(25), we obtain f3 (x) = −Nb Prf1 (x)
e−Nb Prf1 (x)h1 e−Nb Prf1 (x)h2 − e−Nb Prf1 (x)h1
1 f4 (x) = −N Prf (x)h . e b 1 2 − e−Nb Prf1 (x)h1
,
(26) (27)
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Fig. 1. Temperature profile for different values of Nt for fixed values of a = 0.7; b = 1.2; d = 2; Nb = 0.6; x = 0; Pr = 2; φ = Pi/2.
These values given in Eqs. (26) and (27) for f3 (x) and f4 (x), respectively, can be also obtained by a simpler way through applying the boundary conditions θ (x, h1 ) = 0 and θ (x, h2 ) = 1 on Eq. (20). Hence, the exact expressions for the temperature distribution θ and the nano-particle concentration Φ are given by
θ (x, y) =
e−Nb Prf1 (x)y − e−Nb Prf1 (x)h1 e−Nb Prf1 (x)h2 − e−Nb Prf1 (x)h1
,
(28)
and
Φ (x, y) = 1 +
Nt
y − h1 h2 − h1
Nb
−
Nt Nb
e−Nb Prf1 (x)y − e−Nb Prf1 (x)h1
e−Nb Prf1 (x)h2 − e−Nb Prf1 (x)h1
.
(29)
The approximate solutions obtained by the homotopy perturbation method for the temperature distribution θ and the nanoparticle concentration Φ are given by
θ (HPM) =
h1 − y
+
PrNt
(1 + Nb ) h21 − y2 +
PrNt
(1 + Nb ) h21 − h22 (y − h1 )
2(h1 − h2 ) 2(h1 − h2 ) 2 3 (PrNt )2 Nb ( PrN ) N t b + (1 + Nb ) h1 − y3 + (1 + Nb )(h1 + h2 ) y2 − h21 3 3 6(h1 − h2 ) 4(h1 − h2 ) h1 − h2
+
2
3
(PrNt )2 Nb (PrNt )2 Nb (1 + Nb )(h1 + h2 )2 (h1 − y) + (1 + Nb ) h32 − h31 (h1 − y), 3 4 4(h1 − h2 ) 4(h1 − h2 )
(30)
and
Φ (HPM) =
h1 − y h1 − h2
+
Pr (Nt )2 (1 + Nb ) 2 y − h21 + (h1 + h2 )(h1 − y) . 2Nb (h1 − h2 )2
(31)
Here, it should be noted that after achieving the exact solutions for the temperature distribution θ and the nano-particle concentration Φ we can easily obtain the exact stream function and hence the exact expression for the pressure gradient. The exact stream function is obtained by solving a very simple partial differential equation, Eq. (9), under the boundary conditions (12) and (13). However, we focus in this paper on the temperature distribution θ and the nano-particle concentration Φ for declaring that the approximate homotopy perturbation solutions for these quantities were not correct. No doubt, the exact solutions given by (28) and (29) are of great importance in validating any of the approximate solutions that have been previously obtained to analyze the physical model. In the present section, the obtained exact solutions are used to explore the actual effects of various parameters on the temperature distribution and the nano-particle concentration. Figs. 1–3 show the variation of temperature profile for different values of the physical parameters Nt , Nb and Pr. These figures reveal that the temperature profile increases with an increase in Nt , Nb and Pr. It is observed from these figures that the temperature profile remains between 0 (at the upper wall) and 1 (at the lower wall). However, the corresponding results depicted by Figures 15 to 17 in [17] contradict with the current exact results, where all the temperature curves obtained in [17] reached values higher than 1. This refers to that the approximate solutions derived from the homotopy perturbation method in [17] were not really effective. Unfortunately, these differences in results can be also observed through comparing Figs. 4–6 with Figures 18–20 in [17] for the nano-particle concentration profile. In addition, the current exact results in Fig. 4 declare that the concentration profile decreases with an increase in Nt . It is also clear from Fig. 5 that the concentration profile increases with an increase in Nb . Besides, it is observed from Fig. 6 that the concentration profile decreases with an increase in Pr. Further comparisons between the current exact solutions and those approximately obtained by the HPM
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Fig. 2. Temperature profile for different values of Nb for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.2; x = 0; Pr = 2; φ = Pi/2.
Fig. 3. Temperature profile for different values of Pr for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.5; x = 0; Nb = 1.2; φ = Pi/2.
Fig. 4. Concentration profile for different values of Nt for fixed values of a = 0.7; b = 1.2; d = 2; Nb = 0.6; x = 0; Pr = 2; φ = Pi/2.
in [17] and also with MATHEMATICA numerical solution have been also depicted in Figs. 7–14. It is clear from Figs. 7 and 8 that the approximate solutions for the temperature distribution θ using the HPM are not accurate, especially, when both the thermophoresis parameter Nt and the Brownian motion parameter Nb are greater than one, i.e., Nb , Nt > 1. However, the HPM solutions approach to the exact solution when Nb and Nt have values smaller than one, this can be seen in Figs. 9 and 10. Regarding the nano-particle concentration Φ , it is shown from Figs. 11 and 12 that the approximate solutions for Φ using the HPM are not accurate when either the thermophoresis parameter Nt or the Brownian motion parameter Nb or both take values greater than one. In the case of Nb , Nt ≪ 1, the approximate solutions become more accurate as shown in
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Fig. 5. Concentration profile for different values of Nb for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.2; x = 0; Pr = 2; φ = Pi/2.
Fig. 6. Concentration profile for different values of Pr for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.5; x = 0; Nb = 1.2; φ = Pi/2.
Fig. 7. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 0.6; Nt = 1.7; x = 0; Pr = 2; φ = Pi/2.
Figs. 13 and 14. It may be concluded here that the HPM used in [17] gives acceptable approximate numerical solutions under the restriction that both Nb and Nt take very small values, i.e., Nb , Nt ≪ 0.2, otherwise, these approximate solutions are not acceptable to analyze the peristaltic flow of nanofluid. One way to obtain accurate solutions by the HPM at wider ranges for Nb and Nt is to impose more terms in the homotopy series solutions. This conclusion may be very useful for researchers who are analyzing the peristaltic flow of nanofluid by using the HPM. Accordingly, many published results in this field should be revised and also should be re-investigated.
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Fig. 8. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 2; Nt = 2; x = 0; Pr = 2; φ = Pi/2.
Fig. 9. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 0.5; Nt = 0.5; x = 0; Pr = 2; φ = Pi/2.
Fig. 10. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 0.2; Nt = 0.2; x = 0; Pr = 2; φ = Pi/2.
4. Conclusion The exact solutions for the temperature and the nano-particle concentration profiles have been obtained in this paper for the physical model describing the peristaltic flow of a nanofluid. The obtained exact solutions have been used to study the effects of the thermophoresis parameter Nt , the Brownian motion parameter Nb and Prandtl number Pr on the temperature
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Fig. 11. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 0.6; Nt = 1.7; x = 0; Pr = 2; φ = Pi/2.
Fig. 12. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 2; Nt = 2; x = 0; Pr = 2; φ = Pi/2.
Fig. 13. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 0.5; Nt = 0.5; x = 0; Pr = 2; φ = Pi/2.
and the nano-particle concentration profiles. On comparing our exact results with those obtained in [17] and also with the MATHEMATICA numerical solutions, remarkable differences in the behavior of the physical quantities have been detected. This refers to that the approximate solutions obtained by using the homotopy perturbation method in [17] were not sufficient to obtain the correct physical interpretations of the quantities just mentioned when either/both Nt > 1 or/and Nb > 1. The inaccurate numerical results obtained in [17] come back to the convergence issue which was not addressed by the authors. In addition, the authors in [17] used a few terms of the homotopy series solution. In order to obtain more
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Fig. 14. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 0.2; Nt = 0.2; x = 0; Pr = 2; φ = Pi/2.
accurate solution, especially at Nt > 1 or/and Nb > 1, additional terms should be added to the series solution. A final note on the current comparative study is that when it is difficult to achieve the exact solutions of the considered physical problem we instead search for the approximate solutions taking into account the convergence of such solutions. References [1] J.-H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. 178 (3–4) (1999) 257–262. [2] J.-H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35 (1) (2000) 37–43. [3] J.-H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (1) (2003) 73–79. [4] J.-H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput. 156 (2) (2004) 527–539. [5] J.-H. He, Asymptotology by homotopy perturbation method, Appl. Math. Comput. 156 (3) (2004) 591–596. [6] J.-H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (1–2) (2006) 87–88. [7] J.-H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons Fractals 26 (3) (2005) 695–700. [8] P.D. Ariel, The three-dimensional flow past a stretching sheet and the homotopy perturbation method, Comput. Math. Appl. 54 (7–8) (2007) 920–925. [9] S. Pamuk, N. Pamuk, He’s homotopy perturbation method for continuous population models for single and interacting species, Comput. Math. Appl. 59 (2) (2010) 612–621. [10] H. Aminikhah, The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs, Appl. Math. Model. 36 (8) (2012) 3638–3644. [11] E.H. Aly, A. Ebaid, New analytical and numerical solutions for mixed convection boundary-layer nanofluid flow along an inclined plate embedded in a porous medium, J. Appl. Math. 2013 (2013) http://dx.doi.org/10.1155/2013/219486. Article ID 219486, 7 pages. [12] A.H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, NY, USA, 1973. [13] N.S. Akbar, S. Nadeem, Endoscopic effects on peristaltic flow of a nanofluid, Commun. Theor. Phys. 56 (4) (2011) 761–768. [14] N.S. Akbar, S. Nadeem, T. Hayat, A.A. Hendi, Peristaltic flow of a nanofluid in a non-uniform tube, Heat Mass Transfer/Waerme-und Stoffuebertragung 48 (2) (2012) 451–459. [15] N.S. Akbar, S. Nadeem, Peristaltic flow of a Phan–Thien–Tanner nanofluid in a diverging tube, Heat Transf. 41 (1) (2012) 10–22. [16] N.S. Akbar, S. Nadeem, T. Hayat, A.A. Hendi, Peristaltic flow of a nanofluid with slip effects, Meccanica 47 (2012) 1283–1294. [17] Safia Akram, S. Nadeem, Abdul Ghafoor, Changhoon Lee, Consequences of nanofluid on peristaltic flow in an asymmetric channel, Int. J. Basic Appl. Sci. IJBAS-IJENS 12 (5) (2012) 75–96. [18] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD, 66 (1995) 99. [19] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspension, Appl. Phys. Lett. 79 (2001) 2252. [20] J. Buongiorno, W. Hu, Nanofluid coolants for advanced nuclear power plants, Paper no. 5705, in: Proceedings of ICAPP ’05, Seoul, May 15–19, 2005. [21] M. Mustafa, S. Hina, T. Hayat, A. Alsaedi, Influence of wall properties on the peristaltic flow of a nanofluid: analytic and numerical solutions, Int. J. Heat Mass Transfer 55 (2012) 4871. [22] A. Ebaid, E.H. Aly, Exact analytical solution of the peristaltic nanofluids flow in an asymmetric channel with flexible walls and slip condition: application to the cancer treatment, Comput. Math. Meth. Med. 2013 (2013) http://dx.doi.org/10.1155/2013/825376. Article ID 825376, 8 pages.
)
–
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel Abdelhalim Ebaid Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
article
info
Article history: Received 8 December 2013 Received in revised form 4 May 2014 Accepted 14 May 2014 Available online xxxx Keywords: Peristaltic flow Nanofluid Exact solution Homotopy perturbation method
abstract In applied science, the exact solution (when available) for any physical model is of great importance. Such exact solution not only leads to the correct physical interpretation, but also very useful in validating the approximate analytical or numerical methods. However, the exact solution is not always available for the reason that many authors resort to the approximate solutions by using any of the analytical or the numerical methods. To ensure the accuracy of these approximate solutions, the convergence issue should be addressed, otherwise, such approximate solutions inevitably lead to incorrect interpretations for the considered model. Recently, several peristaltic flow problems have been solved via the homotopy perturbation method, which is an approximate analytical method. One of these problems is selected in this paper to show that the solutions obtained by the homotopy perturbation method were inaccurate, especially, when compared with the exact solutions provided currently and also when compared with a well known accurate numerical method. The comparisons reveal that great remarkable differences have been detected between the exact current results and those approximately obtained in the literatures for the temperature distribution and the nano-particle concentration. Hence, many similar problems that have been approximately solved by using the homotopy perturbation method should be re-investigated by taking the convergence issue into consideration, otherwise, the published results were really incorrect. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The homotopy perturbation method (HPM) was proposed by He [1–7] as an analytical technique to solve nonlinear differential equations. This method has been widely used by many authors to investigate various models [8–11]. Unlike, the requirement for the regular perturbation techniques [12], the homotopy perturbation method is always valid no matter whether there exists a small physical parameter or not. It combines the traditional perturbation method and the homotopy technique to deform a nonlinear problem into a simple solving one. The solution using this method is expressed as the summation of an infinite series in terms of an artificial parameter. Moreover, this method depends on some initial guesses for the differential operators to generating the solutions. Here, it should be noted that if such initial guesses are chosen by inappropriate ways, the resulting solutions may be inaccurate. However, the method gives accurate solutions in many cases in which successful initial guesses have been chosen as pointed out by [8,9,11]. Very recently, many authors have used the homotopy perturbation method to analyze several peristaltic flow problems of nanofluids [13–17]. The concept of nanofluids is put into practice particularly after the tremendous development of E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.camwa.2014.05.008 0898-1221/© 2014 Elsevier Ltd. All rights reserved.
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)
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nanotechnologies in the last decade, as they are of great importance in many industrial and engineering applications, especially in heat transfer enhancement [18,19]. This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu [20]). Nanofluids are produced by dispersing nanometer-scale solid particles into base liquids such as water, ethylene glycol, and oils. Normally, if the particle sizes are in the 1100 nm ranges, they are generally called nano-particles. Recently, the peristaltic flow of nanofluids has attracted much attention [13–17,21,22] due to its important applications in engineering, medicine, and biology. The flow field of such problems is governed by systems of linear and nonlinear partial differential equations. Due to the difficulties arise in obtaining the exact solutions for the most of these systems, several authors have resorted to approximate analytical or approximate numerical methods. Sometimes, these approximate analytical solutions do not lead to accurate numerical results, especially, when compared with the exact solutions. In this regard, we aim in this paper to show that the approximate solutions obtained by using the HPM in [17] were not accurate enough to be presented to the scientific community. This shall be done by first achieving the exact solutions for the temperature distribution and the nano-particle concentration for the current physical model. Then several plots shall be presented for the comparisons between the approximate solutions using HPM with the current exact solutions along with the numerical solutions derived from MATHEMATICA. 2. The mathematical model Akram et al. [17] considered the peristaltic flow of an incompressible Jeffrey fluid in an asymmetric channel of width d1 + d2 . The asymmetric channel flow is produced due to different amplitudes and phases of the peristaltic waves on the channel. Heat transfer along with nano-particle phenomena has been taken into account. The wall surfaces are chosen in the following forms: H1 (X , t ) = d1 + a1 cos
H2 (X , t ) = −d2 − b1 cos
2π
λ
(X − ct ) ,
2π
λ
Upper wall,
(X − ct ) + φ ,
Lower wall,
(1)
(2)
where a1 , b1 are the amplitudes of the upper and lower waves, λ is the wave length, φ is the phase difference which varies in the range 0 ≤ φ ≤ π . In addition, a1 , a2 , b1 , b2 , and φ should satisfy the following condition [17]: a21 + b21 + 2a1 b1 cos φ ≤ (a1 + a2 )2 .
(3)
The flow is assumed to be steady in the wave frame (x, y) moving with velocity c away from the fixed frame (X , Y ). The transformation between these two frames is given by x = X − ct ,
y = Y,
u = U − c,
p(x) = P (X , t )
(4)
where u and v are the velocity components in the wave frame (x, y), p and P are pressure in wave and fixed frame of reference, respectively. Akram et al. [17] found that under the assumptions of long wavelength and low Reynolds number approximation the flow is governed by the following system of partial differential equations in non-dimensional form:
∂p = 0, ∂y ∂p ∂ 1 ∂ 2ψ = + Gr θ + Br Φ , ∂x ∂ y (1 + λ1 ) ∂ y2 2 1 ∂ 2θ ∂θ ∂ Φ ∂θ + N + N = 0, b t 2 Pr ∂ y ∂y ∂y ∂y
(5)
(6)
(7)
∂ 2Φ Nt ∂ 2 θ + = 0, (8) 2 ∂y Nb ∂ y2 ∂θ ∂2 1 ∂ 2ψ ∂Φ + Gr + Br = 0, (9) ∂ y2 (1 + λ1 ) ∂ y2 ∂y ∂y where λ1 , Nb , Nt , Gr, and Br are the ratio of relaxation to retardation times, the Brownian motion parameter, thermophoresis parameter, local temperature Grashof number and nano-particle Grashof number, respectively. In addition, a, b, d, and φ should satisfy the following condition [17]: a2 + b2 + 2ab cos φ ≤ (1 + d)2 .
(10)
Eq. (5) shows that p is independent of y. Accordingly, Eq. (6) can be written as dp dx
=
∂ 1 ∂ 2ψ + Gr θ + Br Φ . ∂ y (1 + λ1 ) ∂ y2
(11)
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The flow is governed by the following boundary conditions:
ψ=
q 2
∂ψ = −1, ∂y
,
at y = h1 (x),
∂ψ = −1, ∂y
q
ψ =− , 2
(12)
at y = h2 (x),
(13)
θ = 0,
φ = 0,
at y = h1 (x),
(14)
θ = 1,
φ = 1,
at y = h2 (x),
(15)
where h1 (x) = 1 + a cos(2π x),
h1 (x) = −d − b cos(2π x + φ).
(16)
3. Analysis and discussion Although it seems to be difficult to obtain the analytical solutions for the current system of linear and nonlinear partial differential equations, it is discussed below an effective procedure to achieve this goal. From Eq. (8), we obtain
Φ=−
Nt Nb
θ + f1 (x)y + f2 (x),
(17)
where f1 (x) and f2 (x) are two unknown functions to be determined. By inserting Eq. (17) into (7) results
∂ 2θ ∂θ + PrNb f1 (x) = 0, ∂ y2 ∂y
(18)
which can be integrated once w.r.t. y to give
∂θ + PrNb f1 (x)θ = f3 (x), ∂y
(19)
where f3 (x) is an additional unknown function. Eq. (19) is solved exactly to give the temperature distribution as
θ (x, y) =
f3 (x)
+ f4 (x) e−Nb Prf1 (x)y ,
Nb Prf1 (x)
(20)
where f4 (x) is also unknown function. Therefore the nano-particle concentration φ is given from Eq. (17) as
Φ (x, y) = f1 (x)y + f2 (x) −
Nt
f3 (x)
Nb
Nb Prf1 (x)
+ f4 (x) e−Nb Prf1 (x)y .
(21)
It is our task now to find the unknown functions f1 (x), f2 (x), f3 (x), and f4 (x) through applying the given boundary conditions. First of all, f1 (x) and f2 (x) can be determined by applying the boundary conditions for θ and φ on Eq. (17) and this gives f 1 ( x) =
1+
Nt Nb
h2 − h1
f2 (x) = −h1
,
(22)
1+
Nt Nb
h2 − h1
.
(23)
Further, we can get f3 (x) and f4 (x) by applying the boundary conditions for φ on Eq. (21), this leads to the following system: f1 (x)h1 + f2 (x) − f1 (x)h2 + f2 (x) −
Nt
Nb Prf1 (x)
Nb Nt
f 3 ( x)
Nb
f 3 ( x) Nb Prf1 (x)
+ f4 (x) e
−Nb Prf1 (x)h1
= 0,
(24)
+ f4 (x) e−Nb Prf1 (x)h2 = 1.
(25)
On solving the system (24)–(25), we obtain f3 (x) = −Nb Prf1 (x)
e−Nb Prf1 (x)h1 e−Nb Prf1 (x)h2 − e−Nb Prf1 (x)h1
1 f4 (x) = −N Prf (x)h . e b 1 2 − e−Nb Prf1 (x)h1
,
(26) (27)
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Fig. 1. Temperature profile for different values of Nt for fixed values of a = 0.7; b = 1.2; d = 2; Nb = 0.6; x = 0; Pr = 2; φ = Pi/2.
These values given in Eqs. (26) and (27) for f3 (x) and f4 (x), respectively, can be also obtained by a simpler way through applying the boundary conditions θ (x, h1 ) = 0 and θ (x, h2 ) = 1 on Eq. (20). Hence, the exact expressions for the temperature distribution θ and the nano-particle concentration Φ are given by
θ (x, y) =
e−Nb Prf1 (x)y − e−Nb Prf1 (x)h1 e−Nb Prf1 (x)h2 − e−Nb Prf1 (x)h1
,
(28)
and
Φ (x, y) = 1 +
Nt
y − h1 h2 − h1
Nb
−
Nt Nb
e−Nb Prf1 (x)y − e−Nb Prf1 (x)h1
e−Nb Prf1 (x)h2 − e−Nb Prf1 (x)h1
.
(29)
The approximate solutions obtained by the homotopy perturbation method for the temperature distribution θ and the nanoparticle concentration Φ are given by
θ (HPM) =
h1 − y
+
PrNt
(1 + Nb ) h21 − y2 +
PrNt
(1 + Nb ) h21 − h22 (y − h1 )
2(h1 − h2 ) 2(h1 − h2 ) 2 3 (PrNt )2 Nb ( PrN ) N t b + (1 + Nb ) h1 − y3 + (1 + Nb )(h1 + h2 ) y2 − h21 3 3 6(h1 − h2 ) 4(h1 − h2 ) h1 − h2
+
2
3
(PrNt )2 Nb (PrNt )2 Nb (1 + Nb )(h1 + h2 )2 (h1 − y) + (1 + Nb ) h32 − h31 (h1 − y), 3 4 4(h1 − h2 ) 4(h1 − h2 )
(30)
and
Φ (HPM) =
h1 − y h1 − h2
+
Pr (Nt )2 (1 + Nb ) 2 y − h21 + (h1 + h2 )(h1 − y) . 2Nb (h1 − h2 )2
(31)
Here, it should be noted that after achieving the exact solutions for the temperature distribution θ and the nano-particle concentration Φ we can easily obtain the exact stream function and hence the exact expression for the pressure gradient. The exact stream function is obtained by solving a very simple partial differential equation, Eq. (9), under the boundary conditions (12) and (13). However, we focus in this paper on the temperature distribution θ and the nano-particle concentration Φ for declaring that the approximate homotopy perturbation solutions for these quantities were not correct. No doubt, the exact solutions given by (28) and (29) are of great importance in validating any of the approximate solutions that have been previously obtained to analyze the physical model. In the present section, the obtained exact solutions are used to explore the actual effects of various parameters on the temperature distribution and the nano-particle concentration. Figs. 1–3 show the variation of temperature profile for different values of the physical parameters Nt , Nb and Pr. These figures reveal that the temperature profile increases with an increase in Nt , Nb and Pr. It is observed from these figures that the temperature profile remains between 0 (at the upper wall) and 1 (at the lower wall). However, the corresponding results depicted by Figures 15 to 17 in [17] contradict with the current exact results, where all the temperature curves obtained in [17] reached values higher than 1. This refers to that the approximate solutions derived from the homotopy perturbation method in [17] were not really effective. Unfortunately, these differences in results can be also observed through comparing Figs. 4–6 with Figures 18–20 in [17] for the nano-particle concentration profile. In addition, the current exact results in Fig. 4 declare that the concentration profile decreases with an increase in Nt . It is also clear from Fig. 5 that the concentration profile increases with an increase in Nb . Besides, it is observed from Fig. 6 that the concentration profile decreases with an increase in Pr. Further comparisons between the current exact solutions and those approximately obtained by the HPM
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Fig. 2. Temperature profile for different values of Nb for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.2; x = 0; Pr = 2; φ = Pi/2.
Fig. 3. Temperature profile for different values of Pr for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.5; x = 0; Nb = 1.2; φ = Pi/2.
Fig. 4. Concentration profile for different values of Nt for fixed values of a = 0.7; b = 1.2; d = 2; Nb = 0.6; x = 0; Pr = 2; φ = Pi/2.
in [17] and also with MATHEMATICA numerical solution have been also depicted in Figs. 7–14. It is clear from Figs. 7 and 8 that the approximate solutions for the temperature distribution θ using the HPM are not accurate, especially, when both the thermophoresis parameter Nt and the Brownian motion parameter Nb are greater than one, i.e., Nb , Nt > 1. However, the HPM solutions approach to the exact solution when Nb and Nt have values smaller than one, this can be seen in Figs. 9 and 10. Regarding the nano-particle concentration Φ , it is shown from Figs. 11 and 12 that the approximate solutions for Φ using the HPM are not accurate when either the thermophoresis parameter Nt or the Brownian motion parameter Nb or both take values greater than one. In the case of Nb , Nt ≪ 1, the approximate solutions become more accurate as shown in
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Fig. 5. Concentration profile for different values of Nb for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.2; x = 0; Pr = 2; φ = Pi/2.
Fig. 6. Concentration profile for different values of Pr for fixed values of a = 0.7; b = 1.2; d = 2; Nt = 1.5; x = 0; Nb = 1.2; φ = Pi/2.
Fig. 7. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 0.6; Nt = 1.7; x = 0; Pr = 2; φ = Pi/2.
Figs. 13 and 14. It may be concluded here that the HPM used in [17] gives acceptable approximate numerical solutions under the restriction that both Nb and Nt take very small values, i.e., Nb , Nt ≪ 0.2, otherwise, these approximate solutions are not acceptable to analyze the peristaltic flow of nanofluid. One way to obtain accurate solutions by the HPM at wider ranges for Nb and Nt is to impose more terms in the homotopy series solutions. This conclusion may be very useful for researchers who are analyzing the peristaltic flow of nanofluid by using the HPM. Accordingly, many published results in this field should be revised and also should be re-investigated.
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Fig. 8. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 2; Nt = 2; x = 0; Pr = 2; φ = Pi/2.
Fig. 9. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 0.5; Nt = 0.5; x = 0; Pr = 2; φ = Pi/2.
Fig. 10. Comparison between the HPM, the numerical solution and the exact solution for θ at a = 0.7; b = 1.2; d = 2; Nb = 0.2; Nt = 0.2; x = 0; Pr = 2; φ = Pi/2.
4. Conclusion The exact solutions for the temperature and the nano-particle concentration profiles have been obtained in this paper for the physical model describing the peristaltic flow of a nanofluid. The obtained exact solutions have been used to study the effects of the thermophoresis parameter Nt , the Brownian motion parameter Nb and Prandtl number Pr on the temperature
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Fig. 11. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 0.6; Nt = 1.7; x = 0; Pr = 2; φ = Pi/2.
Fig. 12. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 2; Nt = 2; x = 0; Pr = 2; φ = Pi/2.
Fig. 13. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 0.5; Nt = 0.5; x = 0; Pr = 2; φ = Pi/2.
and the nano-particle concentration profiles. On comparing our exact results with those obtained in [17] and also with the MATHEMATICA numerical solutions, remarkable differences in the behavior of the physical quantities have been detected. This refers to that the approximate solutions obtained by using the homotopy perturbation method in [17] were not sufficient to obtain the correct physical interpretations of the quantities just mentioned when either/both Nt > 1 or/and Nb > 1. The inaccurate numerical results obtained in [17] come back to the convergence issue which was not addressed by the authors. In addition, the authors in [17] used a few terms of the homotopy series solution. In order to obtain more
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Fig. 14. Comparison between the HPM, the numerical solution and the exact solution for Φ at a = 0.7; b = 1.2; d = 2; Nb = 0.2; Nt = 0.2; x = 0; Pr = 2; φ = Pi/2.
accurate solution, especially at Nt > 1 or/and Nb > 1, additional terms should be added to the series solution. A final note on the current comparative study is that when it is difficult to achieve the exact solutions of the considered physical problem we instead search for the approximate solutions taking into account the convergence of such solutions. References [1] J.-H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. 178 (3–4) (1999) 257–262. [2] J.-H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35 (1) (2000) 37–43. [3] J.-H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (1) (2003) 73–79. [4] J.-H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput. 156 (2) (2004) 527–539. [5] J.-H. He, Asymptotology by homotopy perturbation method, Appl. Math. Comput. 156 (3) (2004) 591–596. [6] J.-H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (1–2) (2006) 87–88. [7] J.-H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons Fractals 26 (3) (2005) 695–700. [8] P.D. Ariel, The three-dimensional flow past a stretching sheet and the homotopy perturbation method, Comput. Math. Appl. 54 (7–8) (2007) 920–925. [9] S. Pamuk, N. Pamuk, He’s homotopy perturbation method for continuous population models for single and interacting species, Comput. Math. Appl. 59 (2) (2010) 612–621. [10] H. Aminikhah, The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs, Appl. Math. Model. 36 (8) (2012) 3638–3644. [11] E.H. Aly, A. Ebaid, New analytical and numerical solutions for mixed convection boundary-layer nanofluid flow along an inclined plate embedded in a porous medium, J. Appl. Math. 2013 (2013) http://dx.doi.org/10.1155/2013/219486. Article ID 219486, 7 pages. [12] A.H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, NY, USA, 1973. [13] N.S. Akbar, S. Nadeem, Endoscopic effects on peristaltic flow of a nanofluid, Commun. Theor. Phys. 56 (4) (2011) 761–768. [14] N.S. Akbar, S. Nadeem, T. Hayat, A.A. Hendi, Peristaltic flow of a nanofluid in a non-uniform tube, Heat Mass Transfer/Waerme-und Stoffuebertragung 48 (2) (2012) 451–459. [15] N.S. Akbar, S. Nadeem, Peristaltic flow of a Phan–Thien–Tanner nanofluid in a diverging tube, Heat Transf. 41 (1) (2012) 10–22. [16] N.S. Akbar, S. Nadeem, T. Hayat, A.A. Hendi, Peristaltic flow of a nanofluid with slip effects, Meccanica 47 (2012) 1283–1294. [17] Safia Akram, S. Nadeem, Abdul Ghafoor, Changhoon Lee, Consequences of nanofluid on peristaltic flow in an asymmetric channel, Int. J. Basic Appl. Sci. IJBAS-IJENS 12 (5) (2012) 75–96. [18] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD, 66 (1995) 99. [19] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspension, Appl. Phys. Lett. 79 (2001) 2252. [20] J. Buongiorno, W. Hu, Nanofluid coolants for advanced nuclear power plants, Paper no. 5705, in: Proceedings of ICAPP ’05, Seoul, May 15–19, 2005. [21] M. Mustafa, S. Hina, T. Hayat, A. Alsaedi, Influence of wall properties on the peristaltic flow of a nanofluid: analytic and numerical solutions, Int. J. Heat Mass Transfer 55 (2012) 4871. [22] A. Ebaid, E.H. Aly, Exact analytical solution of the peristaltic nanofluids flow in an asymmetric channel with flexible walls and slip condition: application to the cancer treatment, Comput. Math. Meth. Med. 2013 (2013) http://dx.doi.org/10.1155/2013/825376. Article ID 825376, 8 pages.