Commun Nonlinear Sci Numer Simulat

Commun Nonlinear Sci Numer Simulat 16 (2011) 2730 2736 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns 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: Received 20 July 2010 Received in revised form 30 September 2010 Accepted 2 October 2010 Available online 25 October 2010 Keywords: Homotopy analysis method Electrohydrodynamic flow Nonlinear boundary value problem In this paper, we consider the nonlinear boundary value problem (BVP) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. We present analytical solutions based on the homotopy analysis method (HAM) for various values of the relevant parameters and discuss the convergence of these solutions. We also compare our results with numerical solutions. The results provide another example of a highly nonlinear problem in which HAM is the only known analytical method that yields convergent solutions for all values of the relevant parameters. Ó 2010 Elsevier B.V. All rights reserved. 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) d 2 w dr þ 1 2 r dw dr þ H2 1 w ¼ 0; 0 < r < 1; ð1:1þ 1 aw subject to the boundary conditions w 0 ð0þ ¼0; wð1þ ¼0; ð1:2þ 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: axm62@psu.edu 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.10.004

A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730 2736 2731 HAM is a nonperturbative analytical method for obtaining series solutions to nonlinear equations and has been successfully 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 polynomial 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 w 0 ðrþ ¼0; which satisfies the boundary conditions in (1.2) and (1.3). Since the domain of the unknown function is bounded, it is appropriate to choose the linear operator to be [23] Lðf Þ¼f 00 ; with the property L½c 1 r þ c 2 Š¼0; where c 1 and c 2 are constants of integration. The zeroth-order deformation equation is ð1 pþl½ ^wðr; pþ w 0 ðrþš ¼ phn½^wðr; pþš; with the boundary conditions where @ ^w ð0; pþ ¼0 and ^wð1; pþ ¼0; ð2:8þ @r N½^wðr; pþš ¼ ð1 a ^wþ @ 2 ^w @r 2 þ 1 r! @ ^w þ H 2 ð1 ð1þaþ^wþ: @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 ^wðr; 0Þ ¼w 0 ðrþ and ^wðr; 1Þ ¼wðrÞ: ð2:10þ Thus as p increases from 0 to 1, ^wðr; pþ varies from the initial guess w 0 (r) to the desired solution w(r). Expanding ^wðr; pþ in a Taylor series with respect to p yields ð2:4þ ð2:5þ ð2:6þ ð2:7þ ð2:9þ ^wðr; pþ ¼w 0 ðrþþ X1 w m ðrþp m ; m¼1 where w m ðrþ ¼ 1 @ m ^wðr; pþ m! @p m : p¼0 ð2:11þ ð2:12þ 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 wðrþ ¼w 0 ðrþþ X1 m¼1 w m ðrþ; 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 ð2:13þ L½w m ðrþ v m w m 1 ðrþš ¼ hr m ð~w m 1 Þ; ð2:14þ where R m ð~w m 1 Þ¼w 00 m 1 þ 1 r w0 m 1 þ H2 ½1 v m ð1þaþw m 1 Š a Xm 1 w i w 00 m 1 i a r i¼1 X m 1 i¼1 w i w 0 m 1 i ; ð2:15þ

2732 A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730 2736 and 0; if m 6 1; v m ¼ 1; if m > 1; ð2:16þ subject to the boundary conditions w 0 m ð0þ ¼0; w mð1þ ¼0: ð2:17þ The general solution to (2.14) is w m ðrþ ¼w ] m ðrþþc 1r þ c 2 ; ð2:18þ where w ] mðrþ is the particular solution. The constants c 1 and c 2 are determined by the boundary conditions in (2.17) and are given by c 1 ¼ 0; c 2 ¼ w ] mð1þ: ð2:19þ Starting with the initial guess in (2.4), w m (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 wðrþ ~w N ðrþ ¼ XN m¼0 w m ðrþ; 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 RðrÞ ¼ d2 ~w N dr þ 1 d ~w N 2 r dr þ ~w N H2 1 : ð2:21þ 1 a ~w N We also define the square residual error [24] for the Nth order approximation to be ð2:20þ E N ðhþ ¼ Z 1 0 ½RðrÞŠ 2 dr: ð2:22þ 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 H 2 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], computing E N (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) in Figs. 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 specific 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 10 3. Table 1 The optimal values of h. a H 2 Optimal value of h Minimum value of E N 0.5 0.5 0.375 7.772 10 12 0.5 1 0.276 1.230 10 9 0.5 2 0.275 5.319 10 8 0.5 4 0.205 4.568 10 5 1 0.5 0.303 4.634 10 11 1 1 0.292 4.996 10 9 1 2 0.254 2.363 10 6 1 4 0.198 3.461 10 4 Fig. 3. The residual of the 20th order approximation for H 2 = 0.5.

2734 A. Mastroberardino / Commun Nonlinear Sci Numer Simulat 16 (2011) 2730 2736 Fig. 4. The residual of the 20th order approximation for H 2 =1. Fig. 5. The residual of the 20th order approximation for H 2 =2. Fig. 6. The residual of the 20th order approximation for H 2 =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 [1] McKee S. Calculation of electrohydrodynamic flow in a circular cylindrical conduit. Z Angew Math Mech 1997;77:457 65. [2] Paullet JE. On the solutions of electrohydrodynamic flow in a circular cylindrical conduit. Z Angew Math Mech 1999;79:357 60. [3] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton, FL: Chapman & Hall-CRC Press; 2003.

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