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Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls
Ain Shams Engineering Journal (2015) xxx, xxx–xxx
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
MECHANICAL ENGINEERING
Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls E. Rahimi a, A. Rahimifar b, R. Mohammadyari c, I. Rahimipetroudi d, M. Rahimi-Esbo c,* a
Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Arak, Iran Department of Mechanical Engineering, Babol University of Technology, Babol, Iran c Department of Mathematics, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran d Young Researchers Club, Sari Branch, Islamic Azad University, Sari, Iran b
Received 20 February 2015; revised 21 June 2015; accepted 23 July 2015
KEYWORDS Homotopy Analysis Method; Homotopy Perturbation Method; Viscous flow; Weak permeability
Abstract In this article, an analysis has been performed to study the two dimensional viscous flow between slowly expanding and contracting walls with weak permeability. The governing equations for the base fluid of this problem are described by dimensionless parameters wall dilation rate ðaÞ and permeation Reynolds number (Re). The nonlinear differential equation is solved using two different analytically approaches by Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM). Then, the results are compared with numerical solution by fourth order Runge– Kutta–Fehlberg technique. Furthermore, the effects of dimensionless parameters on the velocity distributions are investigated in this research. As an important outcome, it is observed that, great agreement was found between the obtained results from the analytical and the numerical models. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The flow of Newtonian and non-Newtonian fluids in a porous surface channel has attracted the interest of many investigators * Corresponding author. Tel.: +98 9116277073. E-mail address: [email protected] (M. Rahimi-Esbo). Peer review under responsibility of Ain Shams University.
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in view of its applications in engineering practice, particularly in chemical industries. Examples of these are the cases of boundary layer control, transpiration cooling and gaseous diffusion. Theoretical research on steady flow of this type was initiated by Berman [1] who found a series solution for the two-dimensional laminar flow of a viscous incompressible fluid in a parallel-walled channel for the case of a very low cross-flow Reynolds number. After his work, this problem has been studied by many researchers considering various variations in the problem, e.g., Choi et al. [2] and references cited therein. For the case of a converging or diverging channel with a permeable wall, if the Reynolds number is large and if
http://dx.doi.org/10.1016/j.asej.2015.07.013 2090-4479 Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
2
E. Rahimi et al.
there is suction or injection at the walls whose magnitude is inversely proportional to the distance along the wall from the origin of the channel, a solution for laminar boundary layer equations can be obtained [3]. An interesting subject has been carried out by different authors [4–9]. Most of problems and scientific phenomena such as heat transfer are inherently of nonlinearity. We know that except a limited number of these problems, most of them do not have exact solutions. Therefore, these nonlinear equations should be solved approximately either numerically or analytically. In the numerical method, stability and convergence should be considered so as to avoid divergence or inappropriate results. Time consuming is another problem of numerical techniques. Analytical solutions often fit under classical perturbation methods. Perturbation method [10] provides the most versatile tools available in nonlinear analysis of engineering problem, but its limitations restrict its application [11,12]: Perturbation method is based on assuming a small parameter. The majority of nonlinear problems, especially those having strong nonlinearity, have no small parameters at all. The approximate solutions obtained by the perturbation methods, in most cases, are valid only for small values of the small parameter. Generally, the perturbation solutions are uniformly valid as long as a scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion on which the small parameter should exist. Thus, it is essential to check the validity of the approximations numerically and/or experimentally. To overcome these difficulties, some new methods have been proposed such as VIM, HPM, and ADM. Disappointingly, the majority of nonlinear problems have no small parameter at all. Recently, several new techniques have been presented to overcome the mentioned difficulties. Some of these techniques include Variational Iteration Method (VIM) [13,14], decomposition method [15], Homotopy Perturbation Method (HPM) [16,17] and Homotopy Analysis Method [18–26]. In this letter, analytical solutions of nonlinear equations arising of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability have been studied by three analytical methods. These methods called HAM and HPM that which does not small parameter. Obtaining the analytical solution of the models and comparing with numerical result reveal the capability, effectiveness and convenience of HAM and HPM. These methods give successive approximations of high accuracy solution. 2. Governing equations Consider the laminar, isothermal, and incompressible flow in a rectangular domain bounded by two permeable surfaces that enable the fluid to enter or exit during successive expansions or contractions [27]. A schematic diagram of the problem is shown in Fig. 1. Both walls are assumed to have equal perme_ ability and to expand uniformly at a time dependent rate a. Furthermore, the origin x ¼ 0 is assumed to be the center of the classic squeeze film problem. This enables us to assume flow symmetry about x ¼ 0. Under these assumptions, the equations for continuity and motion become @u @v þ ¼ 0; @x @y
ð1Þ
Figure 1 Two-dimensional domain with expanding or contracting porous walls.
@u @u @u 1 @P þ u þ v ¼ þ tr2 u ; q @x @t @x @y
ð2Þ
_
@v @v @v 1 @p þ u þ v ¼ þ tr2 v : q @y @t @x @y
ð3Þ
In the above equations u and v indicate the velocity components in x and y directions, P denotes the dimensional pressure, q; t and t are the density, kinematic viscosity, and time. Auxiliary conditions can be specified as follows: y ¼ aðtÞ : y ¼ 0 : x ¼ 0 :
a_ v ¼ vw ¼ ; c
u ¼ 0; @u ¼ 0; @y u ¼ 0;
v ¼ 0;
ð4Þ
ð5Þ
_ w Þ is the wall permeance or injection/suction where c ðc a=v coefficient, that is a measure of wall permeability. At this point, the stream function and mean flow vorticity can be introduced by putting [27]: u ¼
@w ; @y
n ¼
@v @u ; @x @y
v ¼
@w ; @x
ð6Þ ð7Þ
2 @n @n @n @ n @ 2 n þ u þ v ¼ t : þ @t @x @y @x2 @y2
ð8Þ
Substituting Eq. (7) into Eq. (8) yields vxt uyt þ u ðvxx uyx Þ þ v ðvxy uyy Þ ¼ tðvxxx uyxx þ vxyy uyyy ÞÞ:
ð9Þ
Due to mass conservation, a similar solution can be developed with respect to x starting with [28]: tx f ðy; tÞ ; u ¼ tx a2 fy ; a y @f : y ¼ ; fy ¼ a @y
w ¼
v ¼ ta1 f ðy; tÞ; ð10Þ
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow
3
From Eqs. (9) and (10), we have uyt þ u uyx þ v uyy ¼ tðuyxx þ uyyy ÞÞ:
ð11Þ
0000 P1 : f1 ðyÞ þ Ref0 ðyÞ f0000 ðyÞ þ 3a f000 ðyÞ þ ay f0000 ðyÞ Re f00 ðyÞ f000 ðyÞ ¼ 0; f1 ð0Þ ¼ 0; f100 ð0Þ ¼ 0; f1 ð1Þ ¼ 0; f10 ð1Þ ¼ 0;
ð22Þ
In order to solve Eq. (8), one uses the chain rule to obtain the following: fyyyy þ a yfyyy þ 3fyy þ f fyyy fy fyy a2 t1 fyyt ¼0 ð12Þ
0000 p2 : f2 ðyÞ þ ay f1000 ðxÞ Re f00 ðyÞ f100 ðyÞ þ 3a f100 ðyÞ 000 000 0 00 þ Ref0 ðyÞ f1 ðyÞ þ Ref1 ðyÞ f0 ðyÞ Re f1 ðyÞ f0 ðyÞ ¼ 0;
with the following boundary conditions:
f2 ð0Þ ¼ 0; f200 ð0Þ ¼ 0; f2 ð1Þ ¼ 0; f20 ð1Þ ¼ 0:
y ¼ 0 ! f ¼ 0;
ð23Þ
fyy ¼ 0;
y ¼ 1 ! f ¼ Re;
ð13Þ
fy ¼ 0;
_ is the non-dimensional wall dilation rate where aðtÞ ¼ aa=t defined positive for expansion and negative for contraction. Furthermore, Re ¼ aVw =t is the permeation Reynolds number defined positive for injection and negative for suction through the walls. Eqs. (10), (12) and (13) can be normalized by putting [29]: w w¼ ; aa_
u u¼ ; a_
v v¼ ; a_
f : f¼ Re
Solving Eqs. (21)–(23) with boundary conditions, we have 1 3 f0 ðyÞ ¼ y3 þ y; 2 2 f1 ðyÞ ¼
1 1 1 9 6 Re y7 þ ay5 þ Re a y3 280 10 6 140 5 1 1 Re þ a y; þ 140 10
ð24Þ
ð25Þ
ð14Þ
And so w¼
xf ; c
u¼
xf 0 ; c
v¼
f ; c
c¼
a ; Re
ð15Þ
0000
f ðyÞ þ a½yf 000 ðyÞ þ 3f 00 ðyÞ þ Re½fðyÞf 000 ðyÞ f 0 ðyÞf 00 ðyÞ:
ð16Þ
In which f 0 is an axial velocity. With the boundary conditions: y¼0:
f 00 ¼ 0;
f ¼ 0;
y¼1:
f 0 ¼ 0;
f ¼ 1;
ð17Þ
where a prime denotes differentiation with respect to y. Note that Berman’s [1] well-known ODE can be viewed as a special case of Eq. (14) with a ¼ 0. 3. Application of Homotopy Perturbation Method In this section, we employ HPM to solve Eq. (16) subject to boundary conditions Eq. (17). We can construct Homotopy function of Eq. (16) as described in [16]: h 0000 i Hðf; pÞ ¼ð1 PÞ f ðyÞ g0 ðyÞ n 0000 þ p f ðyÞ þ a½yf 000 ðyÞ þ 3f 00 ðyÞ þRe½ðfðyÞf 000 ðyÞ f0 ðyÞf00 ðyÞÞg ¼ 0;
ð18Þ
where p 2 ½0; 1 is an embedding parameter. For p ¼ 0 and p ¼ 1 we have fðy; 0Þ ¼ f0 ðyÞ;
fðy; 1Þ ¼ fðyÞ:
ð19Þ
Note that when p increases from 0 to 1, fðy; pÞ varies from f0 ðyÞ to fðyÞ. By substituting fðyÞ ¼ f0 ðyÞ þ pf1 ðyÞ þ p2 f2 ðyÞ þ ¼
n X pi fi ðyÞ; i¼0
g0 ¼ 0
ð20Þ
into Eq. (18) and rearranging the result based on powers of p-terms, we have P0 :
0000
f0 ðyÞ ¼ 0; f0 ð0Þ ¼ 0; f000 ð0Þ ¼ 0;
f0 ð1Þ ¼ 1;
f00 ð1Þ ¼ 0;
ð21Þ
Figure 2 The h-validity of f 000 ð0Þ given by the 9th-order approximate solution for a ¼ 0:1; 1 and different value of Re.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
4 f2 ðyÞ ¼
E. Rahimi et al. 1 1 Re2 y11 þ 21Re2 56aRe y9 92; 400 70; 560 1 2 112a þ 6aRe y5 þ 2800 1 280a2 þ 3Re2 84aRe y7 þ 19; 600 1 219 37 39 2 3 þ Re2 þ aRe a y 6 53; 900 525 175 703 37 2 2 2 Re aRe þ a y: þ 1; 293; 600 4200 175
The solution of this equation, when p ! 1, will be as follows: 2 X fðyÞ ¼ Lim pi fi ðyÞ: ð27Þ i¼0
p!1
4. Application of Homotopy Analysis Method
ð26Þ
For HAM solutions, we choose the initial guess and auxiliary linear operator in the following form: 1 3 ð28Þ f0 ðyÞ ¼ y3 þ y; 2 2 0000
LðfÞ ¼ f ;
ð29Þ
1 3 1 2 L c1 y þ c2 y þ c3 y þ c4 ¼ 0; 6 2
ð30Þ
where ci ði ¼ 1; 2; 3; 4Þ are constants. Let P 2 ½0; 1 denotes the embedding parameter and h indicates non-zero auxiliary parameters. We then construct the following equations: 4.1. Zeroth-order deformation equations hHðyÞN½Fðy; pÞ; ð1 PÞL½Fðy; pÞ f0 ðyÞ ¼ p
ð31Þ
Fð0; pÞ ¼ 0; F00 ð0; pÞ ¼ 0; Fð1; pÞ ¼ 1; F0 ð1; pÞ ¼ 0;
ð32Þ
3 d4 Fðy;pÞ d Fðy;pÞ d2 Fðy;pÞ þa y þ3 N½Fðy;pÞ ¼ dy4 dy3 dy2 d3 Fðy;pÞ dFðy;pÞ d2 Fðy;pÞ : þ Re Fðy;pÞ dy dy3 dy2 For p ¼ 0 and p ¼ 1 we have Fðy; 0Þ ¼ f0 ðyÞ Fðy; 1Þ ¼ fðyÞ:
ð33Þ
ð34Þ
When p increases from 0 to 1 then Fðy; pÞ varies from f0 ðyÞ to fðyÞ. By Taylor’s theorem and using Eq. (34), Fðy; pÞ can be expanded in a power series of p as follows: Fðy; pÞ ¼ f0 ðyÞ þ
1 X fm ðyÞpm ;
m1
1 @ m ðFðy; pÞÞ
fm ðyÞ ¼
: m! @pm p¼0
Figure 3 The comparison between the HAM, HPM and numerical solutions for fðyÞ; f 0 ðyÞ when a ¼ 0:1; Re ¼ 5; 7.
ð35Þ
Figure 4 The comparison of the errors in answer results by HAM and HPM when a ¼ 0:1; Re ¼ 1.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow
5
In which h is chosen in such a way that this series is convergent at p ¼ 1, and therefore we have through Eq. (35) that fðyÞ ¼ f0 ðyÞ þ
1 X
fm ðyÞ:
ð36Þ
m1
Fm ð0; pÞ ¼ 0; F0m ð1; pÞ ¼ 0;
Fm00 ð0; pÞ ¼ 0;
þ
m1 X 0 ½Re fm1k fk000 fm1k fk00 :
ð39Þ
k¼0
Now we determine the convergency of the result, the differential equation, and the auxiliary function according to the solution expression. So let us assume
4.2. mth-order deformation equations L½fm ðyÞ vm fm1 ðyÞ ¼ hHðyÞRm ðyÞ;
000 0000 00 Rm ðyÞ ¼ fm1 þ a yfm1 þ 3fm1
ð37Þ
HðyÞ ¼ 1:
ð40Þ
Fm ð1; pÞ ¼ 0; ð38Þ
Figure 5 The effect of non-dimensional wall dilation rate and Reynolds number on the velocity profiles of expanding wall ða > 0Þ at Re = 7 and 6 when h ¼ 0:9.
Figure 6 The effect of non-dimensional wall dilation rate and Reynolds number on the velocity profiles of contracting wall (a < 0) at Re = 7 and 6 when h ¼ 0:9.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
6
E. Rahimi et al. Table 1
The results of HAM, HPM and numerical methods for fðyÞ when a ¼ 0:1; Re ¼ 1.
x
HAM
HPM
NUM
Error of HAM
Error of HPM
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00
0.000000000 0.075104609 0.149826628 0.223783841 0.296594781 0.367879065 0.437257701 0.504353337 0.568790454 0.630195468 0.688196753 0.792510687 0.878791323 0.944107887 0.985512872 1.000000000
0.000000000 0.075104330 0.149826076 0.223783029 0.296593728 0.367877796 0.437256246 0.504351733 0.568788740 0.630193690 0.688194959 0.792509019 0.878789990 0.944107058 0.985512584 1.000000000
0.000000000 0.075104609 0.149826629 0.223783842 0.296594783 0.367879067 0.437257704 0.504353340 0.568790457 0.630195472 0.688196757 0.792510690 0.878791325 0.944107888 0.985512871 1.000000000
0.0000000000 0.0000000001 0.0000000011 0.0000000016 0.0000000019 0.0000000024 0.0000000028 0.0000000031 0.0000000034 0.0000000034 0.0000000034 0.0000000031 0.0000000026 0.0000000010 0.0000000011 0.0000000000
0.0000000000 0.0000002787 0.0000005526 0.0000008129 0.0000010543 0.0000012711 0.0000014574 0.0000016078 0.0000017174 0.0000017818 0.0000017975 0.0000016715 0.0000013353 0.0000008298 0.0000002867 0.0000000000
We have found the answer by maple analytic solution device. For first deformation of the solution are presented below 1 1 3 1 hRe y7 hay5 þ hRe þ ha y3 f1 ðyÞ ¼ 280 10 280 5 1 1 hRe ha y: ð41Þ þ 140 10
compared to the numerical method NUM is shown in Fig. 4 and Table 1. In this table, the %Error is defined as
%Error ¼ fðyÞNUM fðyÞAnalytical :
the auxiliary parameter h , the h -curve of f000 ð0Þ is illustrated for various quantities of Re in Fig. 2. As seen clearly in Fig. 2(a) and (b), it can be concluded that h ¼ 0:9 is suitable value for different quantities of a ¼ 0:1; a ¼ 1:0 in ranges of 7 < Re w < 7 and 7 < Re w < 7 respectively.
From the achieved results, it can be deduced that this method is precise and accurate in solving a wide range of mathematical and engineering problems. This accuracy gives high confidence to us about validity of this problem and reveals an excellent agreement of engineering accuracy. This study is completed by depicting the effects of the permeation Reynolds number and non-dimensional wall dilation rate on the dimensionless axial velocity near the center and the lower axial velocity near the wall. The behavior of axial velocity for different permeation Reynolds number, over a range of non-dimensional wall dilation rate a ¼ 1 and a ¼ 9, was plotted in Figs. 5 and 6. As can be seen clearly in Fig. 5, for every level of injection or suction, in the case of expanding wall that is, the expansion ratio of the wall, as a increases from 0.1 to 0.9, the higher will be the axial velocity near the center, and the lower near the wall. That is because the flow toward the center becomes greater to make up for the space caused by the expansion of the wall and as a result the axial velocity also becomes greater near the center. Similarly, for the case of contracting wall as shown in Fig. 6, increasing contraction ratio leads to lower axial velocity near the center, and the higher near the wall because the flow toward the wall becomes greater and as a result the axial velocity near the wall becomes greater.
6. Results and discussion
7. Conclusion
The objective of the present study was to apply HAM and HPM to obtain an explicit analytic solution of laminar, isothermal, incompressible viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions. In order to demonstrate the efficiency and accuracy of this method, the obtained results for the temperature distribution for different permeation Reynolds number and non-dimensional wall dilation rate are compared with the numerical procedure as shown in Fig. 3 respectively. Validity of HAM and HPM
In this letter, the analytical approaches called Homotopy Analysis Method and Homotopy Perturbation Methods are successfully applied into the governing differential equation of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. Also, the transformed ODE equation is numerically solved with Runge–Ku tta–Fehlberg. Moreover, the effects of wall dilation rate ðaÞ and permeation Reynolds number (Re) on the dimensionless axial velocity distributions are represented. The following conclusions are drawn from the present research.
The solutions fðyÞ were too long to be mentioned here; therefore, they are shown graphically. 5. Convergence of the HAM solution As pointed out by Liao [18–20], the convergence region and rate of solution series can be adjusted and controlled by means ndash
of the auxiliary parameter h . In order to check the converndash
gence of the present solution, the so-called
h -curve of ndash
f000 ð0Þ is shown in Fig. 2. The solutions converge for h values ndash
which are corresponding to the horizontal line segment in h curve. In order to investigate the range of admissible values of ndash
ndash
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow The results of the present study show that for every level of injection or suction, in the case of expanding wall, increasing a leads to higher axial velocity near the center and the lower axial velocity near the wall. Also, it is found that these methods are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering specially some heat transfer equations. References [1] Berman AS. Laminar, flow in channels with porous walls. J Appl Phys 1953;24:1232–5. [2] Choi JJ, Rusak Z, Tichy JA. Maxwell fluid suction flow in a channel. J Non-Newton Fluid Mech 1999;85:165–87. [3] Rosenhead L. Laminar boundary layers. Oxford: Clerendon Press; 1963. [4] Yahyazadeh A, Yahyazadeh H, Khalili M, Malekzadeh M. Analytical solution of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J Math Comput Sci 2012;5:331–6. [5] Sushila, Singh J, Shishodia YS. A reliable approach for twodimensional viscous flow between slowly expanding or contracting walls with weak permeability using sumudu transform. Ain Shams Eng J 2014;5:237–42. [6] Freidoonimehr N, Rostami B, Rashidi MM. Predictor homotopy analysis method for nano-fluid flow through expanding or contracting gaps with permeable walls. Int J Biomath 2015. http://dx.doi.org/10.1142/S1793524515500503. [7] Rashidi MM, Shahmohamadi H, Dinarvand S. Analytic approximate solutions for unsteady two-dimensional and axisymmetric squeezing flows between parallel plates. Math Probl Eng 2008. http://dx.doi.org/10.1155/2008/93509. Article ID 935095. [8] Rashidi MM, Siddiqui AM, Asadi M. Application of homotopy analysis method to the unsteady squeezing flow of a second grade fluid between circular plates. Math Probl Eng 2010. http://dx.doi. org/10.1155/2010/ 706840. Article ID 706840. [9] Rashidi MM, Siddiqui AM, Rastegari MT. Analytical solution of squeezing flow between two circular plates. Int J Comput Methods Eng Sci Mech 2012;13(5):342–9. [10] Aziz A, Na TY. Perturbation method in heat transfer. Washington, DC: Hemisphere Publishing Corporation; 1984. [11] Bildik N, Konuralp A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 2006;7:65–70. [12] He JH. Non-perturbative methods for strongly nonlinear problems. Dissertation. de-Verlag im Internet GmbH, Berlin; 2006. [13] Ganji DD, Sadighi A. Application of homotopy-perturbation and variational iteration methods to non-linear heat transfer and porous media equations. J Comput Appl Math 2007;207: 24–34. [14] He JH. Variational iteration method – a kind of non-linear analytical technique: some examples. Int J Non-Linear Mech 1999;34:699–708. [15] Adomian G. Nonlinear stochastic operator equations. New York: Academic Press Inc; 1986. [16] He JH. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999;178:257–62.
7 [17] Rostamiyan Y, Ganji DD, Rahimipetroudi I, Khazayinejadabaei M. Analytical investigation of nonlinear model arising in heat transfer through the porous fin. Therm Sci 2014;18:409–17. [18] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003. [19] Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;47(2):499–513. [20] Liao SJ. Homotopy analysis method in nonlinear differential equation. Berlin & Beijing: Springer & Higher Education Press; 2012. [21] Ghasempour M, Rokni E, kimiaeifar A, Rahimpour M. Assessment of HAM and PEM to find analytical solution for calculating displacement functions of geometrically nonlinear prestressed cable structures with concentrated mass. Int J World Appl Sci 2009;9:2264–71. [22] Abbasi M, Hamzehnava GH, Rahimipetroudi I. Analytic solution of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid with convective surface boundary condition. Indian J Sci Res 2014;1:15–9. [23] Abbasi M, Ahmadian CHashmi A, Rahimipetroudi I, Hosseinzadeh KH. Analysis of a fourth grade fluid flow in a channel by application of VIM and HAM. Indian J Sci Res 2014;1:389–95. [24] 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;15:1691–9. [25] Fooladi M, Abaspour SR, Kimiaeifar A, Rahimpour M. on the analytical solution of nonlinear normal mode for continuous systems by means of HAM. World Appl Sci J 2009;6:297–302. [26] Shahbabaei M, Saedodin S, Soleymanibeshei M, Rahimipetroudi I. MHD effect on thermal performance of cylindrical spin porous fin with temperature dependent heat transfer coefficient and emissivity. Int J Energy Technol 2014;6:1–10. [27] Majdalani J, Zhou C, Dawson CA. Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J Biomech 2002;35:1399–403. [28] Ganji ZZ, Ganji DD, Janalizadeh A. Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. Math Comput Appl 2010;15:957–61. [29] Jafaryar M, Iman Pourmousavi S, Hosseini M, Mohammadian E. Application of DTM for 2D viscous flow through expanding or contracting gaps with permeable walls. New Trends Math Sci 2014;2(3):145–58.
Ehsan Rahimi received his M.S degree of mechanical engineering at Faculty of Mechanical Engineering in Science and Research Branch of Azad University, located at Arak, Iran in 2013. His research interests include Thermal Hydraulic Analysis of PWR Nuclear Power Plant, Magneto-hydrodynamic and analytical solutions. The author can be contacted through Tel./fax: +989112543474 and at ehsan.rahimipetroudi@ gmail.com. The author has published the following articles: Transverse magnetic field on Jeffery–Hamel problem with Cu–water Nano fluid between two nonparallel plane walls by using collocation method, Case Studies in Thermal Engineering (Elsevier); Semi-analytical method for solving non-linear equation arising of natural convection porous fin, Thermal science.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
8 Amir Rahimifar is a M.S student of mechanical engineering at Faculty of Mechanical Engineering in Babol University of Technology located at PO Box 484, Babol, Iran. The author can be contacted through Tel./fax: +989119496092 and at amirrahimifar@ yahoo.com. His research interests include renewable energy, direct methanol fuel cell, numerical simulation, and analytical solutions. The author has published the following article: Transverse magnetic field on Jeffery– Hamel problem with Cu–water Nano fluid between two non-parallel plane walls by using collocation method, Case Studies in Thermal Engineering (Elsevier). Reza Mohammadyari received his Ph.D degree of mathematics at Azad University, located at Tehran, Iran in 2012. At present, he is a assistant professor of mathematics at Azad University, located at Buinzahra. The author can be contacted through Tel./fax: +98 9125240975 and at r.mohammadyari@ buiniau.ac.ir. The author has published the following articles in ISI: Topological indices of the double odd graph 2O_K, Creative mathematics and informatics, 20 (2011), No. 2, 163–170, Topological indices of the Kneser graph KG_n,k, Filomat 26:4 (2012), 665–672, The PI index of polyomino chains of 2k-cycles, Creative mathematics and informatics, creative mathematics and informatics, 22(2013), No.1, 89–94, Differential Transformation method to determine magneto hydrodynamics flow of compressible fluid in a channel with porous walls, Bol. Soc. Paran. Mat. V. 32 2 (2014):249–261, On recognition of simple group L_2(r) by the number of sylow subgroups, Maringa, v. 36, n. 3, p. 487–489, july–sept., 2014, Analytical and Numerical investigation of natural convection in a heated cylinder using Homotopy Perturbation Method. Maringa´, v.36, n. 4, p. 669–677, Oct.–Dec., 2014, Invettigation of flow and heat transfer of nanofluid in a diverging sinusoidal channel, int. j. nano dimensions. 2014 6(3): 241–253, Numerical simulation of flow and heat transfer of turbulent nanofluid flow in a triangular rod bundle, Indian journal of fundamental and applied life sciences issn:2231–6345.2014 v.4(s4) 1448–14462, Flow behavior of boundary layer flow of nanoflouid over a flat plate using ham, advances in environmental biology, 8(24) December 2014, 324–331, Topological indices of the bipartite kneser graphs h_n,k, filomat18:10(2014), 1989–1996, On sylow numbers of some finite groups. Siberian electronic mathematical reports, tam 12, cmp. 309–317 (2015), Analytical solution of settling behavior of a particle in incompressible Newtonian fluid by using Parameterized Perturbation Method. Bol. Soc. Paran. Mat. V. 33 2 (2015):143–158, Analytical solution of settling behavior of a particle in incompressible Newtonian fluid by using parameterized method, bol soc. paran. Mat. (3s) v. 33 2 (2015):143–158, Implementation of homotopy analysis
E. Rahimi et al. method on circular permeable slider containing of incompressible Newtonian fluid, bol. Soc. Paran. Mat. (3s) v. 34 1(2016):21–31, Homotopy analysis method to determine magneto hydrodynamics flow of compressible fluid in a channel with porous walls. Bol. Soc. Paran. Mat. (3s) v. 34 1(2016):173–186, Thermo-hydraulic investigation of nanofluid as a coolant in vver-440 fuel rod bundle, transport phenomena in nano and micro scales,3 (2015) 77–88 and Recognizing alternating groups by their order and one conjugacy class length, Journal of algebra and its applications, Vol.15, No. 2 (2015). Iman Rahimipetroudi received his M.S degree of mechanical engineering at Faculty of Mechanical Engineering in Azad University, located at Sari, Iran in 2013. At present, he is a PhD student at Faculty of Mechanical Engineering at Sogang University, Seoul, South Korea. His research interests include Multiphase flow, Magneto-hydrodynamic, numerical, and analytical solutions. The author can be contacted through Tel./fax: +989113526500 and at iman.rahimipetroudi@ gmail.com. The author has published the following articles: Transverse magnetic field on Jeffery–Hamel problem with Cu–water Nano fluid between two non-parallel plane walls by using collocation method, Case Studies in Thermal Engineering (Elsevier); On the MHD squeeze flow between two parallel disks with suction or injection via HAM and HPM, Frontiers of Mechanical Engineering (Springer); Analytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces, Central European Journal of Engineering (Springer); Application of Galerkin and Collocation method to the Electro-hydrodynamic flow analysis in a circular cylindrical conduit, Journal of the Brazilian Society of Mechanical Sciences and Engineering (Springer), Analysis of MHD flow characteristics of an UCM viscoelastic flow in a permeable channel under slip conditions, Journal of the Brazilian Society of Mechanical Sciences and Engineering (Springer). Analytic solution of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid with convective surface boundary condition, Indian J. Sci. Res; Analysis of a fourth grade fluid flow in a channel by application of VIM and HAM, Indian J.Sci.Res; Analytical investigation of nonlinear model arising in heat transfer through the porous fin, Thermal science; Prediction of an Semi-Exact Analytic Solution of a Convective Porous Fin with Variable Cross Section by Different Types of Methods, Walailak Journal for science and Technology; Comparative Analysis of MHD Boundary-Layer Flow of Viscoelastic Fluid in Permeable Channel with Slip Boundaries by using HAM, VIM, HPM, Walailak Journal for science and Technology; MHD effect on thermal performance of cylindrical spin porous fin with temperature dependent heat transfer coefficient and emissivity, International Journal of Energy & Technology; Semi-analytical investigation of a transverse magnetic field on Viscous Flow over a Stretching Sheet, MAGNT Research Report.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow M. Rahimi-Esbo is a PhD student of mechanical engineering at Faculty of Mechanical Engineering in Babol University of Technology located at PO Box 484, Babol, Iran. The author can be contacted through Tel./fax: +98 9116277073 and at rahimi.mazaher@ gmail.com. The author has published the following articles in ISI: Numerical study of turbulent forced convection jet flow in a convergence sinusoidal channel, International journal of Thermal Science (Elsevier); Numerical study of turbulent forced convection jet flow of nanofluids in a converging duct, Numerical Heat Transfer Part A (Taylor & Francis); Numerical simulation of forced convection of nanofluid in a confined jet, Journal of Heat and Mass Transfer; Analytical and Numerical Investigation of fin Efficiency and Temperature Distribution of Conductive, Convective and Radiative Straight fins, International journal of heat transfer Asian research (Wiley), Analytical and Numerical
9 Investigation of Natural Convection in a Heated Cylinder using Homotopy Perturbation Method, journal of Acta Scientiarum Technology; and Determining the Fin Efficiency of Convective Straight Fins with Temperature Dependent Thermal Conductivity by using Homotopy Perturbation Method, International Journal of Numerical Methods for Heat and Fluid Flow (Emerald). The author has also published the following articles: Numerical Investigation of Fin Efficiency and Temperature Distribution of Conductive, Convective and Radiative Straight fins using Gallerkin method, International journal of research and review in applied science; An Analytical Solution For Laminar Flow through a Tube with Porous Walls, International journal of nonlinear dynamics in engineering science; Homotopy Perturbation Method and Variational Iteration Method for solving Burger’s equation, International journal of nonlinear dynamics in engineering science; and Differential Transformation Method to determine Magneto Hydrodynamics flow of compressible fluid in a channel with porous walls, Boletim da Sociedade Paranaense de Matema´tica.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
MECHANICAL ENGINEERING
Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls E. Rahimi a, A. Rahimifar b, R. Mohammadyari c, I. Rahimipetroudi d, M. Rahimi-Esbo c,* a
Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Arak, Iran Department of Mechanical Engineering, Babol University of Technology, Babol, Iran c Department of Mathematics, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran d Young Researchers Club, Sari Branch, Islamic Azad University, Sari, Iran b
Received 20 February 2015; revised 21 June 2015; accepted 23 July 2015
KEYWORDS Homotopy Analysis Method; Homotopy Perturbation Method; Viscous flow; Weak permeability
Abstract In this article, an analysis has been performed to study the two dimensional viscous flow between slowly expanding and contracting walls with weak permeability. The governing equations for the base fluid of this problem are described by dimensionless parameters wall dilation rate ðaÞ and permeation Reynolds number (Re). The nonlinear differential equation is solved using two different analytically approaches by Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM). Then, the results are compared with numerical solution by fourth order Runge– Kutta–Fehlberg technique. Furthermore, the effects of dimensionless parameters on the velocity distributions are investigated in this research. As an important outcome, it is observed that, great agreement was found between the obtained results from the analytical and the numerical models. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The flow of Newtonian and non-Newtonian fluids in a porous surface channel has attracted the interest of many investigators * Corresponding author. Tel.: +98 9116277073. E-mail address: [email protected] (M. Rahimi-Esbo). Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
in view of its applications in engineering practice, particularly in chemical industries. Examples of these are the cases of boundary layer control, transpiration cooling and gaseous diffusion. Theoretical research on steady flow of this type was initiated by Berman [1] who found a series solution for the two-dimensional laminar flow of a viscous incompressible fluid in a parallel-walled channel for the case of a very low cross-flow Reynolds number. After his work, this problem has been studied by many researchers considering various variations in the problem, e.g., Choi et al. [2] and references cited therein. For the case of a converging or diverging channel with a permeable wall, if the Reynolds number is large and if
http://dx.doi.org/10.1016/j.asej.2015.07.013 2090-4479 Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
2
E. Rahimi et al.
there is suction or injection at the walls whose magnitude is inversely proportional to the distance along the wall from the origin of the channel, a solution for laminar boundary layer equations can be obtained [3]. An interesting subject has been carried out by different authors [4–9]. Most of problems and scientific phenomena such as heat transfer are inherently of nonlinearity. We know that except a limited number of these problems, most of them do not have exact solutions. Therefore, these nonlinear equations should be solved approximately either numerically or analytically. In the numerical method, stability and convergence should be considered so as to avoid divergence or inappropriate results. Time consuming is another problem of numerical techniques. Analytical solutions often fit under classical perturbation methods. Perturbation method [10] provides the most versatile tools available in nonlinear analysis of engineering problem, but its limitations restrict its application [11,12]: Perturbation method is based on assuming a small parameter. The majority of nonlinear problems, especially those having strong nonlinearity, have no small parameters at all. The approximate solutions obtained by the perturbation methods, in most cases, are valid only for small values of the small parameter. Generally, the perturbation solutions are uniformly valid as long as a scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion on which the small parameter should exist. Thus, it is essential to check the validity of the approximations numerically and/or experimentally. To overcome these difficulties, some new methods have been proposed such as VIM, HPM, and ADM. Disappointingly, the majority of nonlinear problems have no small parameter at all. Recently, several new techniques have been presented to overcome the mentioned difficulties. Some of these techniques include Variational Iteration Method (VIM) [13,14], decomposition method [15], Homotopy Perturbation Method (HPM) [16,17] and Homotopy Analysis Method [18–26]. In this letter, analytical solutions of nonlinear equations arising of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability have been studied by three analytical methods. These methods called HAM and HPM that which does not small parameter. Obtaining the analytical solution of the models and comparing with numerical result reveal the capability, effectiveness and convenience of HAM and HPM. These methods give successive approximations of high accuracy solution. 2. Governing equations Consider the laminar, isothermal, and incompressible flow in a rectangular domain bounded by two permeable surfaces that enable the fluid to enter or exit during successive expansions or contractions [27]. A schematic diagram of the problem is shown in Fig. 1. Both walls are assumed to have equal perme_ ability and to expand uniformly at a time dependent rate a. Furthermore, the origin x ¼ 0 is assumed to be the center of the classic squeeze film problem. This enables us to assume flow symmetry about x ¼ 0. Under these assumptions, the equations for continuity and motion become @u @v þ ¼ 0; @x @y
ð1Þ
Figure 1 Two-dimensional domain with expanding or contracting porous walls.
@u @u @u 1 @P þ u þ v ¼ þ tr2 u ; q @x @t @x @y
ð2Þ
_
@v @v @v 1 @p þ u þ v ¼ þ tr2 v : q @y @t @x @y
ð3Þ
In the above equations u and v indicate the velocity components in x and y directions, P denotes the dimensional pressure, q; t and t are the density, kinematic viscosity, and time. Auxiliary conditions can be specified as follows: y ¼ aðtÞ : y ¼ 0 : x ¼ 0 :
a_ v ¼ vw ¼ ; c
u ¼ 0; @u ¼ 0; @y u ¼ 0;
v ¼ 0;
ð4Þ
ð5Þ
_ w Þ is the wall permeance or injection/suction where c ðc a=v coefficient, that is a measure of wall permeability. At this point, the stream function and mean flow vorticity can be introduced by putting [27]: u ¼
@w ; @y
n ¼
@v @u ; @x @y
v ¼
@w ; @x
ð6Þ ð7Þ
2 @n @n @n @ n @ 2 n þ u þ v ¼ t : þ @t @x @y @x2 @y2
ð8Þ
Substituting Eq. (7) into Eq. (8) yields vxt uyt þ u ðvxx uyx Þ þ v ðvxy uyy Þ ¼ tðvxxx uyxx þ vxyy uyyy ÞÞ:
ð9Þ
Due to mass conservation, a similar solution can be developed with respect to x starting with [28]: tx f ðy; tÞ ; u ¼ tx a2 fy ; a y @f : y ¼ ; fy ¼ a @y
w ¼
v ¼ ta1 f ðy; tÞ; ð10Þ
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow
3
From Eqs. (9) and (10), we have uyt þ u uyx þ v uyy ¼ tðuyxx þ uyyy ÞÞ:
ð11Þ
0000 P1 : f1 ðyÞ þ Ref0 ðyÞ f0000 ðyÞ þ 3a f000 ðyÞ þ ay f0000 ðyÞ Re f00 ðyÞ f000 ðyÞ ¼ 0; f1 ð0Þ ¼ 0; f100 ð0Þ ¼ 0; f1 ð1Þ ¼ 0; f10 ð1Þ ¼ 0;
ð22Þ
In order to solve Eq. (8), one uses the chain rule to obtain the following: fyyyy þ a yfyyy þ 3fyy þ f fyyy fy fyy a2 t1 fyyt ¼0 ð12Þ
0000 p2 : f2 ðyÞ þ ay f1000 ðxÞ Re f00 ðyÞ f100 ðyÞ þ 3a f100 ðyÞ 000 000 0 00 þ Ref0 ðyÞ f1 ðyÞ þ Ref1 ðyÞ f0 ðyÞ Re f1 ðyÞ f0 ðyÞ ¼ 0;
with the following boundary conditions:
f2 ð0Þ ¼ 0; f200 ð0Þ ¼ 0; f2 ð1Þ ¼ 0; f20 ð1Þ ¼ 0:
y ¼ 0 ! f ¼ 0;
ð23Þ
fyy ¼ 0;
y ¼ 1 ! f ¼ Re;
ð13Þ
fy ¼ 0;
_ is the non-dimensional wall dilation rate where aðtÞ ¼ aa=t defined positive for expansion and negative for contraction. Furthermore, Re ¼ aVw =t is the permeation Reynolds number defined positive for injection and negative for suction through the walls. Eqs. (10), (12) and (13) can be normalized by putting [29]: w w¼ ; aa_
u u¼ ; a_
v v¼ ; a_
f : f¼ Re
Solving Eqs. (21)–(23) with boundary conditions, we have 1 3 f0 ðyÞ ¼ y3 þ y; 2 2 f1 ðyÞ ¼
1 1 1 9 6 Re y7 þ ay5 þ Re a y3 280 10 6 140 5 1 1 Re þ a y; þ 140 10
ð24Þ
ð25Þ
ð14Þ
And so w¼
xf ; c
u¼
xf 0 ; c
v¼
f ; c
c¼
a ; Re
ð15Þ
0000
f ðyÞ þ a½yf 000 ðyÞ þ 3f 00 ðyÞ þ Re½fðyÞf 000 ðyÞ f 0 ðyÞf 00 ðyÞ:
ð16Þ
In which f 0 is an axial velocity. With the boundary conditions: y¼0:
f 00 ¼ 0;
f ¼ 0;
y¼1:
f 0 ¼ 0;
f ¼ 1;
ð17Þ
where a prime denotes differentiation with respect to y. Note that Berman’s [1] well-known ODE can be viewed as a special case of Eq. (14) with a ¼ 0. 3. Application of Homotopy Perturbation Method In this section, we employ HPM to solve Eq. (16) subject to boundary conditions Eq. (17). We can construct Homotopy function of Eq. (16) as described in [16]: h 0000 i Hðf; pÞ ¼ð1 PÞ f ðyÞ g0 ðyÞ n 0000 þ p f ðyÞ þ a½yf 000 ðyÞ þ 3f 00 ðyÞ þRe½ðfðyÞf 000 ðyÞ f0 ðyÞf00 ðyÞÞg ¼ 0;
ð18Þ
where p 2 ½0; 1 is an embedding parameter. For p ¼ 0 and p ¼ 1 we have fðy; 0Þ ¼ f0 ðyÞ;
fðy; 1Þ ¼ fðyÞ:
ð19Þ
Note that when p increases from 0 to 1, fðy; pÞ varies from f0 ðyÞ to fðyÞ. By substituting fðyÞ ¼ f0 ðyÞ þ pf1 ðyÞ þ p2 f2 ðyÞ þ ¼
n X pi fi ðyÞ; i¼0
g0 ¼ 0
ð20Þ
into Eq. (18) and rearranging the result based on powers of p-terms, we have P0 :
0000
f0 ðyÞ ¼ 0; f0 ð0Þ ¼ 0; f000 ð0Þ ¼ 0;
f0 ð1Þ ¼ 1;
f00 ð1Þ ¼ 0;
ð21Þ
Figure 2 The h-validity of f 000 ð0Þ given by the 9th-order approximate solution for a ¼ 0:1; 1 and different value of Re.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
4 f2 ðyÞ ¼
E. Rahimi et al. 1 1 Re2 y11 þ 21Re2 56aRe y9 92; 400 70; 560 1 2 112a þ 6aRe y5 þ 2800 1 280a2 þ 3Re2 84aRe y7 þ 19; 600 1 219 37 39 2 3 þ Re2 þ aRe a y 6 53; 900 525 175 703 37 2 2 2 Re aRe þ a y: þ 1; 293; 600 4200 175
The solution of this equation, when p ! 1, will be as follows: 2 X fðyÞ ¼ Lim pi fi ðyÞ: ð27Þ i¼0
p!1
4. Application of Homotopy Analysis Method
ð26Þ
For HAM solutions, we choose the initial guess and auxiliary linear operator in the following form: 1 3 ð28Þ f0 ðyÞ ¼ y3 þ y; 2 2 0000
LðfÞ ¼ f ;
ð29Þ
1 3 1 2 L c1 y þ c2 y þ c3 y þ c4 ¼ 0; 6 2
ð30Þ
where ci ði ¼ 1; 2; 3; 4Þ are constants. Let P 2 ½0; 1 denotes the embedding parameter and h indicates non-zero auxiliary parameters. We then construct the following equations: 4.1. Zeroth-order deformation equations hHðyÞN½Fðy; pÞ; ð1 PÞL½Fðy; pÞ f0 ðyÞ ¼ p
ð31Þ
Fð0; pÞ ¼ 0; F00 ð0; pÞ ¼ 0; Fð1; pÞ ¼ 1; F0 ð1; pÞ ¼ 0;
ð32Þ
3 d4 Fðy;pÞ d Fðy;pÞ d2 Fðy;pÞ þa y þ3 N½Fðy;pÞ ¼ dy4 dy3 dy2 d3 Fðy;pÞ dFðy;pÞ d2 Fðy;pÞ : þ Re Fðy;pÞ dy dy3 dy2 For p ¼ 0 and p ¼ 1 we have Fðy; 0Þ ¼ f0 ðyÞ Fðy; 1Þ ¼ fðyÞ:
ð33Þ
ð34Þ
When p increases from 0 to 1 then Fðy; pÞ varies from f0 ðyÞ to fðyÞ. By Taylor’s theorem and using Eq. (34), Fðy; pÞ can be expanded in a power series of p as follows: Fðy; pÞ ¼ f0 ðyÞ þ
1 X fm ðyÞpm ;
m1
1 @ m ðFðy; pÞÞ
fm ðyÞ ¼
: m! @pm p¼0
Figure 3 The comparison between the HAM, HPM and numerical solutions for fðyÞ; f 0 ðyÞ when a ¼ 0:1; Re ¼ 5; 7.
ð35Þ
Figure 4 The comparison of the errors in answer results by HAM and HPM when a ¼ 0:1; Re ¼ 1.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow
5
In which h is chosen in such a way that this series is convergent at p ¼ 1, and therefore we have through Eq. (35) that fðyÞ ¼ f0 ðyÞ þ
1 X
fm ðyÞ:
ð36Þ
m1
Fm ð0; pÞ ¼ 0; F0m ð1; pÞ ¼ 0;
Fm00 ð0; pÞ ¼ 0;
þ
m1 X 0 ½Re fm1k fk000 fm1k fk00 :
ð39Þ
k¼0
Now we determine the convergency of the result, the differential equation, and the auxiliary function according to the solution expression. So let us assume
4.2. mth-order deformation equations L½fm ðyÞ vm fm1 ðyÞ ¼ hHðyÞRm ðyÞ;
000 0000 00 Rm ðyÞ ¼ fm1 þ a yfm1 þ 3fm1
ð37Þ
HðyÞ ¼ 1:
ð40Þ
Fm ð1; pÞ ¼ 0; ð38Þ
Figure 5 The effect of non-dimensional wall dilation rate and Reynolds number on the velocity profiles of expanding wall ða > 0Þ at Re = 7 and 6 when h ¼ 0:9.
Figure 6 The effect of non-dimensional wall dilation rate and Reynolds number on the velocity profiles of contracting wall (a < 0) at Re = 7 and 6 when h ¼ 0:9.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
6
E. Rahimi et al. Table 1
The results of HAM, HPM and numerical methods for fðyÞ when a ¼ 0:1; Re ¼ 1.
x
HAM
HPM
NUM
Error of HAM
Error of HPM
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00
0.000000000 0.075104609 0.149826628 0.223783841 0.296594781 0.367879065 0.437257701 0.504353337 0.568790454 0.630195468 0.688196753 0.792510687 0.878791323 0.944107887 0.985512872 1.000000000
0.000000000 0.075104330 0.149826076 0.223783029 0.296593728 0.367877796 0.437256246 0.504351733 0.568788740 0.630193690 0.688194959 0.792509019 0.878789990 0.944107058 0.985512584 1.000000000
0.000000000 0.075104609 0.149826629 0.223783842 0.296594783 0.367879067 0.437257704 0.504353340 0.568790457 0.630195472 0.688196757 0.792510690 0.878791325 0.944107888 0.985512871 1.000000000
0.0000000000 0.0000000001 0.0000000011 0.0000000016 0.0000000019 0.0000000024 0.0000000028 0.0000000031 0.0000000034 0.0000000034 0.0000000034 0.0000000031 0.0000000026 0.0000000010 0.0000000011 0.0000000000
0.0000000000 0.0000002787 0.0000005526 0.0000008129 0.0000010543 0.0000012711 0.0000014574 0.0000016078 0.0000017174 0.0000017818 0.0000017975 0.0000016715 0.0000013353 0.0000008298 0.0000002867 0.0000000000
We have found the answer by maple analytic solution device. For first deformation of the solution are presented below 1 1 3 1 hRe y7 hay5 þ hRe þ ha y3 f1 ðyÞ ¼ 280 10 280 5 1 1 hRe ha y: ð41Þ þ 140 10
compared to the numerical method NUM is shown in Fig. 4 and Table 1. In this table, the %Error is defined as
%Error ¼ fðyÞNUM fðyÞAnalytical :
the auxiliary parameter h , the h -curve of f000 ð0Þ is illustrated for various quantities of Re in Fig. 2. As seen clearly in Fig. 2(a) and (b), it can be concluded that h ¼ 0:9 is suitable value for different quantities of a ¼ 0:1; a ¼ 1:0 in ranges of 7 < Re w < 7 and 7 < Re w < 7 respectively.
From the achieved results, it can be deduced that this method is precise and accurate in solving a wide range of mathematical and engineering problems. This accuracy gives high confidence to us about validity of this problem and reveals an excellent agreement of engineering accuracy. This study is completed by depicting the effects of the permeation Reynolds number and non-dimensional wall dilation rate on the dimensionless axial velocity near the center and the lower axial velocity near the wall. The behavior of axial velocity for different permeation Reynolds number, over a range of non-dimensional wall dilation rate a ¼ 1 and a ¼ 9, was plotted in Figs. 5 and 6. As can be seen clearly in Fig. 5, for every level of injection or suction, in the case of expanding wall that is, the expansion ratio of the wall, as a increases from 0.1 to 0.9, the higher will be the axial velocity near the center, and the lower near the wall. That is because the flow toward the center becomes greater to make up for the space caused by the expansion of the wall and as a result the axial velocity also becomes greater near the center. Similarly, for the case of contracting wall as shown in Fig. 6, increasing contraction ratio leads to lower axial velocity near the center, and the higher near the wall because the flow toward the wall becomes greater and as a result the axial velocity near the wall becomes greater.
6. Results and discussion
7. Conclusion
The objective of the present study was to apply HAM and HPM to obtain an explicit analytic solution of laminar, isothermal, incompressible viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions. In order to demonstrate the efficiency and accuracy of this method, the obtained results for the temperature distribution for different permeation Reynolds number and non-dimensional wall dilation rate are compared with the numerical procedure as shown in Fig. 3 respectively. Validity of HAM and HPM
In this letter, the analytical approaches called Homotopy Analysis Method and Homotopy Perturbation Methods are successfully applied into the governing differential equation of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. Also, the transformed ODE equation is numerically solved with Runge–Ku tta–Fehlberg. Moreover, the effects of wall dilation rate ðaÞ and permeation Reynolds number (Re) on the dimensionless axial velocity distributions are represented. The following conclusions are drawn from the present research.
The solutions fðyÞ were too long to be mentioned here; therefore, they are shown graphically. 5. Convergence of the HAM solution As pointed out by Liao [18–20], the convergence region and rate of solution series can be adjusted and controlled by means ndash
of the auxiliary parameter h . In order to check the converndash
gence of the present solution, the so-called
h -curve of ndash
f000 ð0Þ is shown in Fig. 2. The solutions converge for h values ndash
which are corresponding to the horizontal line segment in h curve. In order to investigate the range of admissible values of ndash
ndash
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow The results of the present study show that for every level of injection or suction, in the case of expanding wall, increasing a leads to higher axial velocity near the center and the lower axial velocity near the wall. Also, it is found that these methods are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering specially some heat transfer equations. References [1] Berman AS. Laminar, flow in channels with porous walls. J Appl Phys 1953;24:1232–5. [2] Choi JJ, Rusak Z, Tichy JA. Maxwell fluid suction flow in a channel. J Non-Newton Fluid Mech 1999;85:165–87. [3] Rosenhead L. Laminar boundary layers. Oxford: Clerendon Press; 1963. [4] Yahyazadeh A, Yahyazadeh H, Khalili M, Malekzadeh M. Analytical solution of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J Math Comput Sci 2012;5:331–6. [5] Sushila, Singh J, Shishodia YS. A reliable approach for twodimensional viscous flow between slowly expanding or contracting walls with weak permeability using sumudu transform. Ain Shams Eng J 2014;5:237–42. [6] Freidoonimehr N, Rostami B, Rashidi MM. Predictor homotopy analysis method for nano-fluid flow through expanding or contracting gaps with permeable walls. Int J Biomath 2015. http://dx.doi.org/10.1142/S1793524515500503. [7] Rashidi MM, Shahmohamadi H, Dinarvand S. Analytic approximate solutions for unsteady two-dimensional and axisymmetric squeezing flows between parallel plates. Math Probl Eng 2008. http://dx.doi.org/10.1155/2008/93509. Article ID 935095. [8] Rashidi MM, Siddiqui AM, Asadi M. Application of homotopy analysis method to the unsteady squeezing flow of a second grade fluid between circular plates. Math Probl Eng 2010. http://dx.doi. org/10.1155/2010/ 706840. Article ID 706840. [9] Rashidi MM, Siddiqui AM, Rastegari MT. Analytical solution of squeezing flow between two circular plates. Int J Comput Methods Eng Sci Mech 2012;13(5):342–9. [10] Aziz A, Na TY. Perturbation method in heat transfer. Washington, DC: Hemisphere Publishing Corporation; 1984. [11] Bildik N, Konuralp A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 2006;7:65–70. [12] He JH. Non-perturbative methods for strongly nonlinear problems. Dissertation. de-Verlag im Internet GmbH, Berlin; 2006. [13] Ganji DD, Sadighi A. Application of homotopy-perturbation and variational iteration methods to non-linear heat transfer and porous media equations. J Comput Appl Math 2007;207: 24–34. [14] He JH. Variational iteration method – a kind of non-linear analytical technique: some examples. Int J Non-Linear Mech 1999;34:699–708. [15] Adomian G. Nonlinear stochastic operator equations. New York: Academic Press Inc; 1986. [16] He JH. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999;178:257–62.
7 [17] Rostamiyan Y, Ganji DD, Rahimipetroudi I, Khazayinejadabaei M. Analytical investigation of nonlinear model arising in heat transfer through the porous fin. Therm Sci 2014;18:409–17. [18] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003. [19] Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;47(2):499–513. [20] Liao SJ. Homotopy analysis method in nonlinear differential equation. Berlin & Beijing: Springer & Higher Education Press; 2012. [21] Ghasempour M, Rokni E, kimiaeifar A, Rahimpour M. Assessment of HAM and PEM to find analytical solution for calculating displacement functions of geometrically nonlinear prestressed cable structures with concentrated mass. Int J World Appl Sci 2009;9:2264–71. [22] Abbasi M, Hamzehnava GH, Rahimipetroudi I. Analytic solution of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid with convective surface boundary condition. Indian J Sci Res 2014;1:15–9. [23] Abbasi M, Ahmadian CHashmi A, Rahimipetroudi I, Hosseinzadeh KH. Analysis of a fourth grade fluid flow in a channel by application of VIM and HAM. Indian J Sci Res 2014;1:389–95. [24] 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;15:1691–9. [25] Fooladi M, Abaspour SR, Kimiaeifar A, Rahimpour M. on the analytical solution of nonlinear normal mode for continuous systems by means of HAM. World Appl Sci J 2009;6:297–302. [26] Shahbabaei M, Saedodin S, Soleymanibeshei M, Rahimipetroudi I. MHD effect on thermal performance of cylindrical spin porous fin with temperature dependent heat transfer coefficient and emissivity. Int J Energy Technol 2014;6:1–10. [27] Majdalani J, Zhou C, Dawson CA. Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J Biomech 2002;35:1399–403. [28] Ganji ZZ, Ganji DD, Janalizadeh A. Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. Math Comput Appl 2010;15:957–61. [29] Jafaryar M, Iman Pourmousavi S, Hosseini M, Mohammadian E. Application of DTM for 2D viscous flow through expanding or contracting gaps with permeable walls. New Trends Math Sci 2014;2(3):145–58.
Ehsan Rahimi received his M.S degree of mechanical engineering at Faculty of Mechanical Engineering in Science and Research Branch of Azad University, located at Arak, Iran in 2013. His research interests include Thermal Hydraulic Analysis of PWR Nuclear Power Plant, Magneto-hydrodynamic and analytical solutions. The author can be contacted through Tel./fax: +989112543474 and at ehsan.rahimipetroudi@ gmail.com. The author has published the following articles: Transverse magnetic field on Jeffery–Hamel problem with Cu–water Nano fluid between two nonparallel plane walls by using collocation method, Case Studies in Thermal Engineering (Elsevier); Semi-analytical method for solving non-linear equation arising of natural convection porous fin, Thermal science.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
8 Amir Rahimifar is a M.S student of mechanical engineering at Faculty of Mechanical Engineering in Babol University of Technology located at PO Box 484, Babol, Iran. The author can be contacted through Tel./fax: +989119496092 and at amirrahimifar@ yahoo.com. His research interests include renewable energy, direct methanol fuel cell, numerical simulation, and analytical solutions. The author has published the following article: Transverse magnetic field on Jeffery– Hamel problem with Cu–water Nano fluid between two non-parallel plane walls by using collocation method, Case Studies in Thermal Engineering (Elsevier). Reza Mohammadyari received his Ph.D degree of mathematics at Azad University, located at Tehran, Iran in 2012. At present, he is a assistant professor of mathematics at Azad University, located at Buinzahra. The author can be contacted through Tel./fax: +98 9125240975 and at r.mohammadyari@ buiniau.ac.ir. The author has published the following articles in ISI: Topological indices of the double odd graph 2O_K, Creative mathematics and informatics, 20 (2011), No. 2, 163–170, Topological indices of the Kneser graph KG_n,k, Filomat 26:4 (2012), 665–672, The PI index of polyomino chains of 2k-cycles, Creative mathematics and informatics, creative mathematics and informatics, 22(2013), No.1, 89–94, Differential Transformation method to determine magneto hydrodynamics flow of compressible fluid in a channel with porous walls, Bol. Soc. Paran. Mat. V. 32 2 (2014):249–261, On recognition of simple group L_2(r) by the number of sylow subgroups, Maringa, v. 36, n. 3, p. 487–489, july–sept., 2014, Analytical and Numerical investigation of natural convection in a heated cylinder using Homotopy Perturbation Method. Maringa´, v.36, n. 4, p. 669–677, Oct.–Dec., 2014, Invettigation of flow and heat transfer of nanofluid in a diverging sinusoidal channel, int. j. nano dimensions. 2014 6(3): 241–253, Numerical simulation of flow and heat transfer of turbulent nanofluid flow in a triangular rod bundle, Indian journal of fundamental and applied life sciences issn:2231–6345.2014 v.4(s4) 1448–14462, Flow behavior of boundary layer flow of nanoflouid over a flat plate using ham, advances in environmental biology, 8(24) December 2014, 324–331, Topological indices of the bipartite kneser graphs h_n,k, filomat18:10(2014), 1989–1996, On sylow numbers of some finite groups. Siberian electronic mathematical reports, tam 12, cmp. 309–317 (2015), Analytical solution of settling behavior of a particle in incompressible Newtonian fluid by using Parameterized Perturbation Method. Bol. Soc. Paran. Mat. V. 33 2 (2015):143–158, Analytical solution of settling behavior of a particle in incompressible Newtonian fluid by using parameterized method, bol soc. paran. Mat. (3s) v. 33 2 (2015):143–158, Implementation of homotopy analysis
E. Rahimi et al. method on circular permeable slider containing of incompressible Newtonian fluid, bol. Soc. Paran. Mat. (3s) v. 34 1(2016):21–31, Homotopy analysis method to determine magneto hydrodynamics flow of compressible fluid in a channel with porous walls. Bol. Soc. Paran. Mat. (3s) v. 34 1(2016):173–186, Thermo-hydraulic investigation of nanofluid as a coolant in vver-440 fuel rod bundle, transport phenomena in nano and micro scales,3 (2015) 77–88 and Recognizing alternating groups by their order and one conjugacy class length, Journal of algebra and its applications, Vol.15, No. 2 (2015). Iman Rahimipetroudi received his M.S degree of mechanical engineering at Faculty of Mechanical Engineering in Azad University, located at Sari, Iran in 2013. At present, he is a PhD student at Faculty of Mechanical Engineering at Sogang University, Seoul, South Korea. His research interests include Multiphase flow, Magneto-hydrodynamic, numerical, and analytical solutions. The author can be contacted through Tel./fax: +989113526500 and at iman.rahimipetroudi@ gmail.com. The author has published the following articles: Transverse magnetic field on Jeffery–Hamel problem with Cu–water Nano fluid between two non-parallel plane walls by using collocation method, Case Studies in Thermal Engineering (Elsevier); On the MHD squeeze flow between two parallel disks with suction or injection via HAM and HPM, Frontiers of Mechanical Engineering (Springer); Analytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces, Central European Journal of Engineering (Springer); Application of Galerkin and Collocation method to the Electro-hydrodynamic flow analysis in a circular cylindrical conduit, Journal of the Brazilian Society of Mechanical Sciences and Engineering (Springer), Analysis of MHD flow characteristics of an UCM viscoelastic flow in a permeable channel under slip conditions, Journal of the Brazilian Society of Mechanical Sciences and Engineering (Springer). Analytic solution of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid with convective surface boundary condition, Indian J. Sci. Res; Analysis of a fourth grade fluid flow in a channel by application of VIM and HAM, Indian J.Sci.Res; Analytical investigation of nonlinear model arising in heat transfer through the porous fin, Thermal science; Prediction of an Semi-Exact Analytic Solution of a Convective Porous Fin with Variable Cross Section by Different Types of Methods, Walailak Journal for science and Technology; Comparative Analysis of MHD Boundary-Layer Flow of Viscoelastic Fluid in Permeable Channel with Slip Boundaries by using HAM, VIM, HPM, Walailak Journal for science and Technology; MHD effect on thermal performance of cylindrical spin porous fin with temperature dependent heat transfer coefficient and emissivity, International Journal of Energy & Technology; Semi-analytical investigation of a transverse magnetic field on Viscous Flow over a Stretching Sheet, MAGNT Research Report.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013
Analytical approach for solving laminar viscous flow M. Rahimi-Esbo is a PhD student of mechanical engineering at Faculty of Mechanical Engineering in Babol University of Technology located at PO Box 484, Babol, Iran. The author can be contacted through Tel./fax: +98 9116277073 and at rahimi.mazaher@ gmail.com. The author has published the following articles in ISI: Numerical study of turbulent forced convection jet flow in a convergence sinusoidal channel, International journal of Thermal Science (Elsevier); Numerical study of turbulent forced convection jet flow of nanofluids in a converging duct, Numerical Heat Transfer Part A (Taylor & Francis); Numerical simulation of forced convection of nanofluid in a confined jet, Journal of Heat and Mass Transfer; Analytical and Numerical Investigation of fin Efficiency and Temperature Distribution of Conductive, Convective and Radiative Straight fins, International journal of heat transfer Asian research (Wiley), Analytical and Numerical
9 Investigation of Natural Convection in a Heated Cylinder using Homotopy Perturbation Method, journal of Acta Scientiarum Technology; and Determining the Fin Efficiency of Convective Straight Fins with Temperature Dependent Thermal Conductivity by using Homotopy Perturbation Method, International Journal of Numerical Methods for Heat and Fluid Flow (Emerald). The author has also published the following articles: Numerical Investigation of Fin Efficiency and Temperature Distribution of Conductive, Convective and Radiative Straight fins using Gallerkin method, International journal of research and review in applied science; An Analytical Solution For Laminar Flow through a Tube with Porous Walls, International journal of nonlinear dynamics in engineering science; Homotopy Perturbation Method and Variational Iteration Method for solving Burger’s equation, International journal of nonlinear dynamics in engineering science; and Differential Transformation Method to determine Magneto Hydrodynamics flow of compressible fluid in a channel with porous walls, Boletim da Sociedade Paranaense de Matema´tica.
Please cite this article in press as: Rahimi E et al., Analytical approach for solving two-dimensional laminar viscous flow between slowly expanding and contracting walls, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2015.07.013