View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 http://www.jtaphys.com/content/6/1/45 RESEARCH Open Access Semi-analytic algorithms for the electrohydrodynamic flow equation Ram K Pandey1,2, Vipul K Baranwal1, Chandra S Singh1 and Om P Singh1* Abstract In this paper, we consider the nonlinear boundary value problem for the electrohydrodynamic (EHD) flow of a fluid in an ion-drag configuration in a circular cylindrical conduit. This phenomenon is governed by a nonlinear second-order differential equation. The degree of nonlinearity is determined by a nondimensional parameter α.Wepresenttwo semi-analytic algorithms to solve the EHD flow equation for various values of relevant parameters based on optimal homotopy asymptotic method (OHAM) and optimal homotopy analysis method. In 1999, Paullet has shown that for large α, the solutions are qualitatively different from those calculated by Mckee in 1997. Both of our solutions obtained by OHAM and optimal homotopy analysis method are qualitatively similar with Paullet’s solutions. Keywords: Optimal homotopy asymptotic method (OHAM), Optimal homotopy analysis method, Electrohydrodynamic flow, Square residual error, Gauss quadrature MSC: 34B15, 34B16, 76E30 Background solutions given by Equations 4 and 6 depending on the The electrohydrodynamic flow of a fluid in an ion-drag value of the nonlinearity control parameter α. configuration in a circular cylindrical conduit is gov- For α << 1 and assuming a solution of the form erned by a nonlinear second-order ordinary differential X1 αn ; α : equation. Perturbation solutions of fluid velocities for urðÞ¼ unðÞr ð3Þ different orders of nonlinearities were given by McKee n¼0 et al. [1]. In their study, a description of the problem Mckee et al. [1] obtained the O(α3) perturbation solu- was presented in which the governing equations were tion as I0ðÞHr reduced to the following nonlinear boundary value prob- urðÞ¼; α 1 À þ α½þðÞu1ðÞþHr C1 I0ðÞþHr v1ðÞHr K0ðÞHr I0ðÞH lem (BVP): 2 α ½ðÞu2ðÞþHr C2 I0ðÞþHr v2ðÞHr K0ðÞHr : d2u 1 du u ð4Þ þ þ H2 1 À ¼ 0; 0 < r < 1; ð1Þ dr2 r dr 1 À αu Similarly, for α >> 1, the authors [1] proposed that the solution to the BVP could be expanded in the series of subject to boundary conditions the form X1 0 Àn u ðÞ¼0 0; uðÞ¼1 0; ð2Þ urðÞ¼ α unðÞr; α ð5Þ n¼0 where u(r)isthefluidvelocity,r is the radial distance from with an O(1) leading-order term and obtained the per- the centre of the cylindrical conduit, H is the Hartman turbation solution as electric number and the parameter α is a measure of the 0 1 Zr strength of the nonlinearity. In [1], the authors used a H2 1 ÀÁ1 logðÞ 1 À s2 π2 urðÞ¼; α 1 þ 1 À r2 þ @2 ds þ A: regular perturbation technique to obtain two perturbation 4 α α2 s 6 0 * Correspondence: [email protected] ð6Þ 1Department of Applied Mathematics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India Full list of author information is available at the end of the article © 2012 Pandey et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 2 of 10 http://www.jtaphys.com/content/6/1/45 Paullet [2] proved the existence and uniqueness of the so- lution to the BVP (1) and (2) in the following theorem: 1400 Theorem 1 α 2 ≠ . For any > 0 and any H 0, there exists 1200 a solution to the BVP (1) and (2). Furthermore, this solu- 1000 tion is monotonically decreasing and satisfies 0 < u(r) < E3 1/(α + 1) for all r ∈ (0,1). 800 Remark 1. By a solution of Equations 1 and 2, we mean 600 2 afunctionu(r) ∈ C[0,1] \ C (0,1) that satisfies Equation 1 400 for 0 < r < 1 along with Equation 2. In order for such a 200 function to be a solution, we must necessarily have u(r)< 2 0 1/α on (0,1); if u(r) ever equals 1/α,itisnolongerC , 0.40 0.35 0.30 0.25 0.20 0.15 0.10 owing to the term u(r)/(1− αu(r)) in Equation 1 [2]. C1 Paullet [2] claimed an error in the perturbation and Figure 1 Square residual error E for the third-order OHAM numerical solutions given in [1] for large values of α. 3 (C2 = −0.1809055, C3 = 0.275378). This stems from the fact that for large α,thesolutions are O(1/α), not O(1) as proposed in the perturbation ex- pansionusedin[1].Forα << 1, our solutions obtained method) as compared to the deviations in ℏ (in the case by the two semi-analytic algorithms (proposed in the of HAM). A comparison is made between OHAM, opti- ‘Application of OHAM to EHD flow problem’ and mal homotopy analysis method and HAM via exact ‘Application of optimal homotopy analysis method to square residual errors. It is shown that for higher values EHD flow problem’ subsections) are in complete agree- of α and H2, the respective third- and fourth-order ment with those of [1] and [2], but for α >> 1, the pro- OHAM and optimal homotopy analysis method solu- posed solution profiles are similar to those of [2]. Thus, tions are more accurate than the 20th-order HAM solu- based on our work in this paper, we support Paullet’sso- tions. Further, the central processing unit (CPU) time is lution profiles for α >> 1. also calculated and compared for these methods, estab- Recently, Mastroberardino [3] proposed an analytical lishing the superiority of OHAM and optimal homotopy method based on the homotopy analysis method (HAM) analysis method over the HAM solution. Also, the solu- to find the solutions of Equations 1 and 2 for α ∈ (0,1] tion profiles shown for α = 4, 10, H2 = 1 and α = 4, 10, and H2 up to 4. The author [3] has shown that the H2 = 10 by Figures 3 and 4 respectively match Paullet’s homotopy perturbation method (HPM) yields a diver- solution profiles shown in Figures one and two of [2] for gent solution for all of the cases considered. The HAM the corresponding values of the parameters. solutions are quite accurate for lower values of the para- meters α and H2, but the accuracy decreases rather fast Analysis of the method for higher values of these parameters even though fairly Optimal homotopy asymptotic method higher order (20 to be precise) solutions were consid- Since the last two decades, homotopy perturbation ered, as shown in Table one of [3]. Further, from Figure method [4] and homotopy analysis method [5] based on two of [3], we observe that even a slight deviation from the topological concept of homotopy have become very the optimal value of ℏ causes a huge square residual popular in solving nonlinear ordinary/partial differential error for α = 0.5, 1 and H2 = 4. This, along with the qualitative difference between the solution profiles of 1000 Mckee et al. [1] and Paullet [2] for α >> 1, motivated us to look for algorithms giving accurate solutions for 800 higher values of the parameters as well. 600 The aim of the present work is to propose two algo- E4 rithms for the solutions of the above BVP (1) and (2) for 400 all values of relevant parameters using optimal homo- 200 topy asymptotic method (OHAM) and optimal homo- topy analysis method. We show that even the third- and 0 fourth-order solutions obtained from OHAM and opti- 200 mal homotopy analysis method, respectively, are highly 0.10 0.08 0.06 0.04 0.02 0.00 accurate for α >> 1. From Figures 1 and 2, we see that c1 the square residual errors E3/E4 are stable even for larger Figure 2 Square residual error E4 of the fourth-order optimal deviations from the optimal value of C1 (in the case of HAM (c2 = 0.0242099, c3 = −6.19495). OHAM) or c1 (in the case of optimal homotopy analysis Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 3 of 10 http://www.jtaphys.com/content/6/1/45 where, A = L + N, L is a linear operator, N is a nonlinear operator, r denotes the independent variable, u(r)isan unknown function, f(r) is a known function and B is a boundary operator. A homotopy h(ϕ(r,q),q): R × [0,1] → R is constructed satisfying ðÞ1 À q ½¼LðÞþ’ðÞr; q frðÞ HqðÞ½LðÞþ’ðÞr; q frðÞ þ NðÞ’ðÞr; q ; BðÞ¼’ðÞr; q 0; ð8Þ Figure 3 OHAM solution u(r) for H2 = 1 and α = 4 (red), α =10 ∈ (blue). where, q [0,1] is an embedding parameter, H(q) is a non- zero auxiliary function for q ≠ 0andH(0) = 0. As the em- bedding parameter q increases from 0 to 1, the ϕ(r,q)varies equations [6,7]. Later, in 2008, Marinca et al. [8-11] from the initial approximation u0(r)tothesolutionu(r). introduced a new analytical method known as OHAM The auxiliary function H(q) is chosen as to solve a variety of nonlinear problems. This method is 2 3 straightforward and reliable, and it does not need to look HqðÞ¼qC1 þ q C2 þ q C3 þ ⋯; ð9Þ for ℏ curves like HAM.