An efficient analytical approach for MHD viscous flow over a stretching sheet via homotopy perturbation sumudu transform method

ENGINEERING PHYSICS AND MATHEMATICS

An efficient analytical approach for MHD viscous

flow over a stretching sheet via homotopy perturbation

sumudu transform method

Sushila

*

_{, Jagdev Singh}

_{, Y.S. Shishodia}

a

Department of Physics, Jagan Nath University, Village-Rampura, Tehsil-Chaksu, Jaipur 303 901, Rajasthan, India b_{Department of Mathematics, Jagan Nath University, Village-Rampura, Tehsil-Chaksu, Jaipur 303 901, Rajasthan, India} c

Jagan Nath University, Village-Rampura, Tehsil-Chaksu, Jaipur 303 901, Rajasthan, India

Received 28 September 2012; revised 22 November 2012; accepted 8 December 2012 Available online 29 January 2013

KEYWORDS Sumudu transform; Homotopy perturbation method; He’s polynomials; Stretching sheet; MHD viscous flow

Abstract In this paper, we present an efficient analytical approach based on new homotopy per-turbation sumudu transform method (HPSTM) to investigate the magnetohydrodynamics (MHD) viscous flow due to a stretching sheet. The viscous fluid is electrically conducting in the presence of magnetic field and the induced magnetic field is neglected for small magnetic Reynolds number. Finally, some numerical comparisons among the new HPSTM, the homotopy perturbation method and the exact solution have been made. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.

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1. Introduction

The boundary layer flow of an incompressible viscous fluid over a continuously stretching sheet is often occurs in the sev-eral engineering and industrial processes. It has attracted con-siderable interest during the last few decades. Such flows have important applications in industries, for example in the aero-dynamic extrusion of a polymer sheet from a die, hot rolling,

the cooling of an infinite metallic plate in a cooling bath, the boundary layer along a liquid film in condensation process, glass–fiber production and so on [1–3]. Many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. The mechanical properties of the final product strictly depend on the stretching and cold drawing rates in the process. The pioneering work in this area was conducted by Sakiadis[4,5] and the boundary layer flow on a continuously stretching surface with a constant speed was investigated by several researchers in the field. Spe-cifically Crane[6]found a closed form solution for flow of an incompressible viscous fluid past a stretching plate. Gupta and Gupta [7] added suction or injection on the surface which models condensation or evaporation, a uniform transverse magnetic field, when the fluid is electrically conducting, by Anderson[8]. The joint effect of viscoelasticity and magnetic * Corresponding author. Tel.: +91 9461673365.

E-mail addresses: [email protected] ( Sushila), jagdevsingh [email protected](J. Singh),[email protected](Y.S. Shishodia). Peer review under responsibility of Ain Shams University.

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field on Crane’s problem has been investigated by Ariel [9]. The flow past a stretching sheet need not be necessarily two-dimensional because the stretching of the sheet can take place in a number of ways. It can be three-dimensional when the stretching is radial. The three-dimensional and axisymmetric stretching flat surface was studied by Wang[10]. The flow in-side a stretching channel or tube was conin-sidered by Brady and Acrivos [11] and the flow outside a stretching tube by Wang[12]. The unsteady stretching sheet was investigated by Wang[13]and Usha and Sridharan[14]. A non-iterative solu-tion for the MHD flow has been given by Ariel[15]using the technique of Samuel and Hall[16], Ariel et al. [17]and Ariel

[18]applied HPM and extended HPM to derive the analytical solution to the axisymmetric flow past a stretching sheet.

Magnetohydrodynamics (MHD) is the study of the interac-tion of conducting fluid with electromagnetic phenomena. The flow of an electrically conducting fluid in the presence of mag-netic field is important in several areas of science and engineer-ing such as MHD power generation, MHD flow generators, and MHD pumps. Most boundary-layer models can be re-duced to systems of nonlinear ordinary differential equations which are solved by numerical and analytical methods. Analyt-ical methods have significant advantages over numerAnalyt-ical meth-ods in providing analytic, verifiable, rapidly convergent approximation. There exists a wide class of literature dealing with the problems of approximate solutions to nonlinear dif-ferential equations with various different methodologies, called perturbation methods. The perturbation methods have some limitations, e.g., the approximate solution involves series of small parameters which poses difficulty since majority of non-linear problems have no small parameters at all. Although appropriate choices of small parameters some time leads to ideal solution but in most of the cases unsuitable choices lead to serious effects in the solutions. Therefore, an analytical method is welcome which does not require a small parameter in the equation modeling the phenomenon. The homotopy per-turbation method (HPM) was first introduced by He[19]and was further developed by him[20–26]. The HPM is in fact, a coupling of the traditional perturbation method and homoto-py in topology [27]. This method was applied to helmhotz equation and fifth-order KdV equation [28], nonlinear equa-tion arising in heat transfer[29], determining frequency–ampli-tude relation of a nonlinear oscillator with discontinuity[30], gas dynamics equation [31] and fractional nonlinear Schro¨-dinger equation[32]. Very recently, Singh et al.[33]have pro-posed a new approach named homotopy perturbation sumudu transform method (HPSTM) to solve the nonlinear partial dif-ferential equations. The homotopy perturbation sumudu transform method (HPSTM) is a combination of sumudu transform method, HPM and He’s polynomials and is mainly due to Ghorbani[34,35].

The objective of this paper is to present a simple recursive algorithm based on the new HPSTM which produces the series solution of MHD boundary-layer equations for stretching sheet problem. The advantage of this technique is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear equations. The fact that HPSTM solves linear and nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this technique over this decomposition method. It is worth mentioning that the proposed method is capable of reducing the volume of the computational work

as compared to the classical methods while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach.

2. Sumudu transform

In early 90s, Watugala [36] introduced a new integral trans-form, named the sumudu transform and applied it to the solu-tion of ordinary differential equasolu-tion in control engineering problems. The sumudu transform is defined over the set of functions

A¼ ffðtÞj9M;s1;s2>0;jfðtÞj<Mejtj=sj;ift2 ð1Þ

j

½0;1Þg by the following formula

fðuÞ ¼S½fðtÞ ¼ Z 1

0

fðutÞet dt; u2 ðs1;s2Þ: ð1Þ For further detail and properties of this transform, see

[37–39].

3. Basic idea of HPSTM

In order to elucidate the solution procedure of the new HPSTM, we consider the following general form of 3rd order non-homogenous nonlinear ordinary differential equation with the initial conditions is given by

f000_þb

1ðxÞf00þb2ðxÞf0þb3ðxÞf¼gðfÞ; ð2Þ fð0Þ ¼a; f0ð0Þ ¼b; f00ð0Þ ¼c: ð3Þ Taking the sumudu transform on both sides of Eq.(2), we get

S½f000 _þS½b

1ðxÞf00þb2ðxÞf0þb3ðxÞf ¼S½gðfÞ: ð4Þ Using the differentiation property of the sumudu trans-form, we have

S½f ¼aþbuþcu2_þu3_{S½gðfÞ u}3_S½b

1ðxÞf00þb2ðxÞf0

þb3ðxÞf: ð5Þ

Operating with the sumudu inverse on both sides of Eq.(5)

gives

f¼GðxÞ þS1_½u3_{S½gðfÞ S}1_½u3_S½b

1ðxÞf00þb2ðxÞf0

þb3ðxÞf; ð6Þ

where G(x) represents the term arising from the source term and the prescribed initial conditions. Now we apply the HPM

f¼X 1

m¼0 pm_f

m; ð7Þ

and the nonlinear term can be decomposed as

gðfÞ ¼X 1

m¼0

pmHm; ð8Þ

for some He’s polynomialsHm[35,40]that are given by

Hm¼ 1 m! @m @pm g X1 i¼0 pi_f i ! " # p¼0 ; m¼0;1;2;3;. . . ð9Þ Substituting Eqs. (7) and (8) in Eq.(6), we get

X1 m¼0 pm_f m¼GðxÞ þp S1 u3S X1 m¼0 pm_H m " # " #! p S1 u3 S b1ðxÞ X1 m¼0 pm f00 mþb2ðxÞ X1 m¼0 pm f0 mþb3ðxÞ X1 m¼0 pm fm " # " #! ; ð10Þ which is the coupling of the sumudu transform and the HPM using He’s polynomials. Comparing the coefficients of like powers of p, the following approximations are obtained.

p0: f0¼GðxÞ; p1 : f₁¼S1_½u3_{S ½H} 0 S1½u3S ½b1ðxÞf000þb2ðxÞf00 þb3ðxÞf0; p2 : f₂¼S1_½u3_S½H 1 S1½u3S ½b1ðxÞf001þb2ðxÞf01 þb3ðxÞf1; .. . ð11Þ 4. Mathematical model

The MHD boundary layer flow over a flat plate is governed by the continuity and the Navier–Stokes equations for an incom-pressible viscous fluid. The fluid is electrically conducting un-der the influence of an applied magnetic fieldB(x) normal to the stretching sheet. The induced magnetic field is neglected. The resulting boundary-layer equations are:

@U @xþ @V @y¼0; ð12Þ U@U @xþV @U @y ¼v @2U @y2 r B2ðxÞ q U; ð13Þ

where U andV are the velocity components in thex andy

directions, respectively. Alsov,qandrare the kinematic vis-cosity, density and electrical conductivity of the fluid. TheB(x) is taken as[41]BðxÞ ¼ B0xn1=2.

The boundary conditions applicable to the present flow are

Uðx;0Þ ¼cxn_{; Vðx;0Þ ¼0; and Uðx;1Þ ¼0: ð14Þ}

To solve the problem, momentum and energy equations are firstly nondimensionalized by introducing the following dimen-sionless variables: g¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðnþ1Þ 2v r xn21_y; ð₁₅Þ U¼cxn_f0_{ðgÞ; ð16Þ} V¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cvðnþ1Þ 2 r xn21 fðgÞ þn1 nþ1gf 0_ðgÞ : ð17Þ

Using Eqs. (15)–(17), the governing equations can be re-duced to nonlinear differential equation of the form:

f000þff00bðf0Þ2Mf0¼0; fð0Þ ¼0; f0ð0Þ ¼1 and f0ð1Þ ¼00; ð18Þ where b¼ 2n nþ1 M¼ 2rb2₀ qcð1þnÞ: ð19Þ

5. HPSTM solution and discussion

In this section, we apply the new HPSTM to obtain an approx-imate analytical solution of (18), (19). By applying the sumudu transform on the both sides of Eq.(18), we have

S½fðgÞ ¼uþau2_þu3_S½bðf0_Þ2

ff00þMf0; ð20Þ wheref00_{ð0Þ ¼a.}

The inverse sumudu transform implies that

fðgÞ ¼gþag 2 2 þS 1_½ u3_S½bðf0_Þ2 ff00þMf0: ð21Þ Now applying the HPM, we get

X1 m¼0 pm_f mðgÞ ¼gþ ag2 2 þp S 1 u3_{S b} X 1 m¼0 pm_H mðgÞ ! " " X 1 m¼0 pm_H0 mðgÞ ! þM X 1 m¼0 pm_f0 mðgÞ !##! ; ð22Þ

whereHm(g) andH0mðgÞare He’s polynomials[35,40]that

rep-resents the nonlinear terms. So, He’s polynomials are given by X1 m¼0 pm_H mðgÞ ¼ ðf0Þ 2 ðgÞ: ð23Þ

The first few components of He’s polynomials, are given by

H0ðgÞ ¼ ðf00Þ 2_ð gÞ; H1ðgÞ ¼2f00ðgÞf01ðgÞ; H2ðgÞ ¼ ðf01Þ 2_ð gÞ þ2f0₀ðgÞf0₂ðgÞ; .. . ð24Þ

Table 1 Comparison of the values off00_{ð0Þobtained by HPM}

[42], HPSTM and the exact solution.

b M HPM[42] HPSTM Exact solution 1 0 1 1 1 1 1.41421 1.41421 1.41421 5 2.44948 2.44948 2.44948 10 3.31662 3.31662 3.31662 50 7.14142 7.14142 7.14142 100 10.0499 10.0499 10.04987 500 22.383 22.383 22.38302 1000 31.6386 31.6386 31.63858

Table 2 Variation inf00_{ð0Þat the different values ofbandM}

obtained by HPM[42]and HPSTM. b M HPM[42] HPSTM 1.5 0 1.1486 1.1547 1 1.5252 1.5252 5 2.5161 2.5161 10 3.3663 3.3663 50 7.1647 7.1647 100 10.0664 10.0776 500 22.3901 22.3904 1000 31.6432 31.6438

Figure 1 The results offobtained by HPSTM, HPM[42]and exact solutions forb= 1, when (a)M= 0, (b)M= 1, (c)M= 50, and (d)M= 50.

And forH0 mðgÞwe find that X1 m¼0 pm_H0 mðgÞ ¼fðgÞf00ðgÞ; ð25Þ H0 0ðgÞ ¼f0ðgÞf000ðgÞ; H0 1ðgÞ ¼f0ðgÞf001ðgÞ þf1ðgÞf000ðgÞ; H0₂ðgÞ ¼f0ðgÞf002ðgÞ þf1ðgÞf001ðgÞ þf2ðgÞf000ðgÞ; .. . ð26Þ

Comparing the coefficients of like powers ofp, we have

p0_:f 0¼gþ ag2 2 ; ð27Þ p1:f1¼ ð2b1Þ 120 a 2 g5þðM1þ2bÞ 24 ag 4_þðMþbÞ 6 g 3 ; ð28Þ p2_:f 2¼ ð20b232bþ11Þ 40320 a 3_g8 þð11þ10Mbþ20b 2 8M32bÞ 5040 a 2_g7 þð312b8Mþ10b 2 þ10MbþM2_Þ 720 ag 6 þðM 2_þ2b2 2b2Mþ3MbÞ 120 g 5 ; ð29Þ p3 :f3¼ð 600b3 1596b2 þ1398b375Þ 39916800 a 4 g11 þð375546bMþ300Mb 2_þ 600b3 1596b2_þ 1398bþ243MÞ 3628800 a 3 g10 þð129þ42bM 2 þ300b3 39M2 þ548b710b2 þ243M546bMþ300Mb2 Þ 362880 a 2 g9 þð15þ120Mb 2_þ M3_þ 80b3 166b2_þ 103bþ80M39M2_þ 42bM2 196bMÞ 40320 ag 8 þðM 3 10M2_þ 11bM2_þ 20b2 M26bMþ8Mþ8b16b2_þ 10b3_Þ 5040 g 7 ;... ð30Þ and so on. In this way, the remaining terms of the HPSTM ser-ies can be calculated.

The series solution is given by

f¼f0þf1þf2þf3þ ;

p!1: ð31Þ

Substituting Eqs. (27)–(30) into Eq.(31), we obtain the follow-ing series solution

fðgÞ ¼gþag 2 2 þ ð2b1Þ 120 a 2_g5_þðM1þ2bÞ 24 ag 4 þðMþbÞ 6 g 3_þð20b232bþ11Þ 40320 a 3_g8 þð11þ10Mbþ20b 2_8M32bÞ 5040 a 2_g7 þð312b8Mþ10b 2_þ10MbþM2_Þ 720 ag 6 þðM 2_þ2b2_2b2Mþ3MbÞ 120 g 5 þð600b 3_1596b2_þ1398b375Þ 39916800 a 4_g11 þð375546bMþ300Mb 2_þ600b3_1596b2_{þ1398bþ243MÞ} 3628800 a 3_g10 þð129þ42bM 2_þ300b3_39M2_þ548b710b2_{þ243M546bMþ300Mb}2_Þ 362880 a 2_g9 þð15þ120Mb 2_þM3_þ80b3_166b2_{þ103bþ80M39M}2_þ42bM2_196bMÞ 40320 ag 8 þðM 3_10M2_þ11bM2_þ20b2_{M26bMþ8Mþ8b16b}2_þ10b3_Þ 5040 g 7_{þ ð32Þ} To selectf00_{ð0Þ ¼a and g}

1, we begin with some initial guess ofaand draw f0_{. The solution process is repeated with} another values ofauntilf0_{receive to zero. Also,g}

1is obtained from highest value ofginf0_.

Tables 1 and 2clearly indicate that present solution of new HPSTM shows excellent agreement with the HPM[42]and the exact solution and more convergent than the HPM[42]. This analysis shows that the new HPSTM is valid for MHD viscous flow problems.

6. Results and discussion

Eq.(18)is solved analytically using the new HPSTM. The ana-lytical and the exact solutions offfor different values ofMand bare depicted inFigs. 1 and 2. Furthermore, the comparison between the new HPSTM and the exact solutions off0 _is per-formed inFig. 3. The effects of the magnetic parameter M

are shown inFig. 3. It is observed fromFig. 3that whenM, increases, the velocity profile is more and more far away from the wall, and the boundary layer thickness is more and more thicker. Therefore, it is concluded fromFig. 3thatf0_decreases whenMincreases. A very good agreement is achieved between the results obtained by the new HPSTM, the HPM[42]and the exact solutions for all values ofg. The results of new HPSTM are better than HPM[42]in large amount ofM.

7. Conclusion

In this paper, a simple algorithm based on new HPSTM has been successfully applied for solving the magnetohydrodynam-ics (MHD) viscous flow due to stretching sheet. An excellent agreement is achieved by comparing the present solution very well with the HPM[42]and exact solution. The method is ap-plied here in direct manner without the use of linearization, transformation, discretization, perturbation, or restrictive assumptions. The approach gives more realistic series solutions that converge very rapidly in physical problems. The proposed method’s ability to solve nonlinear problems without the use of

Figure 3 Comparison of the solution (f0_{) obtained by HPSTM}

Adomian polynomials is evidence of its clear advantage over the decomposition method. In conclusion, the HPSTM could be a promising tool for solving more complex boundary equa-tions that the one studied in this paper.

Acknowledgements

The authors are extending their heartfelt thanks to the review-ers for their valuable suggestions for the improvement of the article. The authors are also thankful to Shri Manish Gupta, Chancellor and Shri Deepak Gupta, Vice-Chairman, Jagan Nath University, Jaipur for providing support and facilities for the research work

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Sushila is an Assistant Professor in the Department of Physics, Jagan Nath Univer-sity, Jaipur-303901, Rajasthan, India. Her fields of research include nuclear physics, solid state physics, plasma physics, fluid mechanics, fractional calculus, homotopy methods, Laplace transform method and sumudu transform method.

Jagdev Singhis an Assistant Professor in the Department of Mathematics, Jagan Nath University, Jaipur-303901, Rajasthan, India. His fields of research include special functions, fractional calculus, fluid mechanics, homoto-py methods, Adomian decomposition method, Laplace transform method and sumudu transform method.

Y.S. Shishodia,Pro-Vice-Chancellor in Jagan Nath University, Jaipur-303901, Rajasthan, India. His area of interest is nuclear physics, solid state physics, positron spectroscopy, laser and plasma technology, solar cells, fluid mechanics, homotopy methods, Laplace transform method and sumudu transform method.