Homotopy Analysis Method Applied to Electrohydrodynamic Flow

Commun Nonlinear Sci Numer Simulat 16 (2011) 2730–2736

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Commun Nonlinear Sci Numer Simulat

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Homotopy analysis method applied to electrohydrodynamic ﬂow ⇑ Antonio Mastroberardino

School of Science, Penn State Erie, The Behrend College, Erie, Pennsylvania 16563-0203, USA article info abstract

Article history: In this paper, we consider the nonlinear boundary value problem (BVP) for the electrohy- Received 20 July 2010 drodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. We Received in revised form 30 September present analytical solutions based on the homotopy analysis method (HAM) for various 2010 values of the relevant parameters and discuss the convergence of these solutions. We also Accepted 2 October 2010 compare our results with numerical solutions. The results provide another example of a Available online 25 October 2010 highly nonlinear problem in which HAM is the only known analytical method that yields convergent solutions for all values of the relevant parameters. Keywords: Ó 2010 Elsevier B.V. All rights reserved. Homotopy analysis method Electrohydrodynamic flow Nonlinear boundary value problem

1. Introduction

The electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit was first reviewed by McKee [1]. In that article, a full description of the problem was presented in which the governing equations were reduced to the nonlinear boundary value problem (BVP) d2w 1 dw w þ þ H2 1 ¼ 0; 0 < r < 1; ð1:1Þ dr2 r dr 1 aw subject to the boundary conditions

w0ð0Þ¼0; ð1:2Þ wð1Þ¼0; ð1:3Þ where w(r) is the ﬂuid velocity, r is the radial distance from the center of the cylindrical conduit, H is the Hartmann electric number, and the parameter a is a measure of the strength of the nonlinearity. Perturbative and numerical solutions to (1.1)– (1.3) for small/large values of a were provided. Paullet [2] proved the existence and uniqueness of a solution to (1.1)–(1.3), and in addition, discovered an error in the perturbative and numerical solutions given in [1] for large values of a. The purpose of this present work is to present accurate analytical solutions to (1.1)–(1.3) for all values of the relevant parameters using the homotopy analysis method (HAM), introduced by Liao [3–6]. We show that the analytical solutions are in excellent agreement with numerical solutions obtained with MATLAB. We also show that the homotopy perturbation method (HPM) yields divergent solutions for all of the cases considered. This is further illustration of the utility of HAM in comparison with other analytical methods used to solve highly nonlinear differential equations. We refer the reader to [7– 14] for enlightening comparisons between HAM and HPM.

⇑ Tel.: +1 814 898 6328; fax: +1 814 898 6213. E-mail address: [email protected]

HAM is a nonperturbative analytical method for obtaining series solutions to nonlinear equations and has been success- fully applied to numerous problems in science and engineering [15–22]. In comparison with other perturbative and nonperturbative analytical methods, HAM offers the ability to adjust and control the convergence of a solution via the so-called convergence-control parameter. Because of this, HAM has proved to be the most effective method for obtaining analytical solutions to highly nonlinear differential equations. Previous applications of HAM have mainly focused on nonlinear differential equations in which the nonlinearity is a poly- nomial in terms of the unknown function and its derivatives. As seen in (1.1), the nonlinearity present in electrohydrodynamic ﬂow takes the form of a rational function, and thus, poses a greater challenge with respect to ﬁnding approximate solutions analytically. Our results show that even in this case, HAM yields excellent results.

2. Homotopy analysis method

In this section, we apply HAM to solve (1.1)–(1.3) for the ﬂuid velocity w(r). We choose the initial guess to be

w0ðrÞ¼0; ð2:4Þ which satisﬁes the boundary conditions in (1.2) and (1.3). Since the domain of the unknown function is bounded, it is appro- priate to choose the linear operator to be [23] Lðf Þ¼f 00; ð2:5Þ with the property

L½c1r þ c2¼0; ð2:6Þ where c1 and c2 are constants of integration. The zeroth-order deformation equation is

ð1 pÞL½w^ ðr; pÞw0ðrÞ ¼ phN½w^ ðr; pÞ; ð2:7Þ with the boundary conditions @w^ ð0; pÞ¼0 and w^ ð1; pÞ¼0; ð2:8Þ @r where ! @2w^ 1 @w^ N½w^ ðr; pÞ ¼ ð1 aw^ Þ þ þ H2ð1 ð1 þ aÞw^ Þ: ð2:9Þ @r2 r @r

Here p 2 [0,1] is an embedding parameter, and h is the convergence-control parameter. Note that for p = 0 and p = 1 we have

w^ ðr; 0Þ¼w0ðrÞ and w^ ðr; 1Þ¼wðrÞ: ð2:10Þ

Thus as p increases from 0 to 1, w^ ðr; pÞ varies from the initial guess w0(r) to the desired solution w(r). Expanding w^ ðr; pÞ in a Taylor series with respect to p yields

X1 m w^ ðr; pÞ¼w0ðrÞþ wmðrÞp ; ð2:11Þ m¼1 where m 1 @ w^ ðr; pÞ wmðrÞ¼ m : ð2:12Þ m! @p p¼0 If the auxiliary linear operator, the initial guess, and the convergence-control parameter h are properly chosen, the series in (2.11) converges at p = 1, yielding the homotopy-series solution

X1 wðrÞ¼w0ðrÞþ wmðrÞ; ð2:13Þ m¼1 to (1.1)–(1.3). Differentiating (2.7) m times with respect to the embedding parameter p, dividing by m!, and then setting p = 0, we obtain the mth-order deformation equation

L½wmðrÞvmwm1ðrÞ ¼ hR mðw~m1Þ; ð2:14Þ where

1 mX1 a Xm1 R ðw~ Þ¼w00 þ w0 þ H2½1 v ð1 þ aÞw a w w00 w w0 ; ð2:15Þ m m1 m1 r m1 m m1 i m1i r i m1i i¼1 i¼1 2732 A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730–2736 and 0; if m 6 1; v ¼ ð2:16Þ m 1; if m > 1; subject to the boundary conditions

0 wmð0Þ¼0; wmð1Þ¼0: ð2:17Þ The general solution to (2.14) is

] wmðrÞ¼wmðrÞþc1r þ c2; ð2:18Þ ] where wmðrÞ is the particular solution. The constants c1 and c2 are determined by the boundary conditions in (2.17) and are given by

] c1 ¼ 0; c2 ¼wmð1Þ: ð2:19Þ

Starting with the initial guess in (2.4), wm(r) for m P 1 are obtained iteratively by solving (2.14) and (2.17) with symbolic computational software. This procedure is terminated after a ﬁxed number iterations N to yield the approximate analytical solution

XN wðrÞw~ NðrÞ¼ wmðrÞ; ð2:20Þ m¼0 to (1.1)–(1.3). To facilitate the analysis in the next section, we substitute (2.20) into (1.1) to obtain the residual function 2 d w~ N 1 dw~ N 2 w~ N RðrÞ¼ 2 þ þ H 1 : ð2:21Þ dr r dr 1 aw~ N We also deﬁne the square residual error [24] for the Nth order approximation to be Z 1 2 ENðhÞ¼ ½RðrÞ dr: ð2:22Þ 0

3. Convergence of the HAM solution

In this section, we discuss the convergence of the HAM solution in (2.20) for N = 20. The convergence depends on the convergence-control parameter h, and so, we plot h-curves for w(0) in Fig. 1. As discussed in [3], the interval of convergence is determined by the ﬂat portion of the h-curve. It is clear from Fig. 1 that the admissible values of h are contained in [0.7,0] for all of the cases considered and that as H2 increases, this interval shrinks due to the increase in nonlinearity. Since h = 1is not contained in the interval of convergence, solutions obtained with HPM-a special case of HAM in which h = 1 [7] – are divergent. To determine the optimal values of h, we minimize the square residual error given in (2.22). As discussed in [24], com- puting EN(h) directly with symbolic computational software is impractical. Thus, we approximate (2.22) using Gaussian

Fig. 1. h-curves for the 20th order approximation for a = 0.5, 1. A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730–2736 2733

Fig. 2. Square residual error for the 20th order approximation for a = 0.5, 1. quadrature with eight nodes and plot these approximations in Fig. 2. The optimal values of h for all of the cases considered are obtained by minimizing (2.22) using the Mathematica function Minimize and are given in Table 1. In addition, we plot the residual function R(r)inFigs. 3–6 for all of the cases considered. These plots demonstrate the accuracy of the HAM solution given in (2.20). It is worth noting the residual has been plotted as a function of r for a ﬁxed value of h and not as a function of h for a ﬁxed value of r as this is a better illustration of convergence.

4. Comparison with numerical solutions

Here we solve (1.1)–(1.3) numerically and compare with the analytical solutions obtained in the previous section for spe- cific values of h. We first convert (1.1)–(1.3) to an initial value problem for a two-dimensional first order system and use a shooting method in order to satisfy the right boundary condition in (1.3). To handle the singularity at r = 0, the numerical method involves a combination of Euler’s implicit method for the first step of Dr = 0.05 and MATLAB’s differential equation solver ode45 for the remainder of the interval. Fig. 7(a) and (b) demonstrate that the analytical solutions for various values of the relevant parameters compare extremely well with the numerical solutions. For all of the cases considered, the maximum difference between the analytical solution and the numerical solution was determined to be less than 103.

Table 1 The optimal values of h.

2 a H Optimal value of h Minimum value of EN 0.5 0.5 0.375 7.772 1012 0.5 1 0.276 1.230 109 0.5 2 0.275 5.319 108 0.5 4 0.205 4.568 105 1 0.5 0.303 4.634 1011 110.292 4.996 109 120.254 2.363 106 140.198 3.461 104

Fig. 3. The residual of the 20th order approximation for H2 = 0.5. 2734 A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730–2736

Fig. 4. The residual of the 20th order approximation for H2 =1.

Fig. 5. The residual of the 20th order approximation for H2 =2.

Fig. 6. The residual of the 20th order approximation for H2 =4.

5. Conclusions

In this paper, the homotopy analysis method (HAM) has been applied to obtain analytical solutions for a nonlinear boundary value problem governing electrohydrodynamic ﬂow. It has been noted that the nonlinearity confronted in this problem is in the form of a rational function, and thus, poses a signiﬁcant challenge in regard to obtaining analytical solutions. Despite this fact, we have shown that the solutions obtained are convergent and that they compare extremely well with numerical solutions. It has also been shown that the homotopy perturbation method yields divergent solutions for all of the cases considered. These results demonstrate that HAM is a very effective analytical method for solving highly nonlinear problems in science and engineering. A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730–2736 2735

(a)

0.5

0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(b)

0.5

0.4

0.3

0.2

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 7. Comparison of 20th order HAM solution (solid line) and numerical solution for (a) a = 0.5 and (b) a =1.

Acknowledgment

The author thanks the referee for helpful suggestions that improved the content of the paper.

References

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