Available online at http://ijim.srbiau.ac.ir/

Int. J. Industrial Mathematics (ISSN 2008-5621)

Vol. 7, No. 4, 2015 Article ID IJIM-00716, 8 pages

Research Article

Dirichlet series and approximate analytical solutions of MHD flow

over a linearly stretching sheet

Vishwanath B. Awati ∗†, Mahesh Kumar N ‡, Krishna B. Chavaraddi §

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Abstract

The paper presents the semi-numerical solution for the magnetohydrodynamic (MHD) viscous flowdue to a stretching sheet caused by boundary layer of an incompressible viscous flow. The governingpartial differential equations of momentum equations are reduced into a nonlinear ordinary differentialequation (NODE) by using a classical similarity transformation along with appropriate boundaryconditions. Both nonlinearity and infinite interval demand novel the mathematical tools for theiranalysis. The solution of the resulting third order nonlinear boundary value problem with an infiniteinterval is obtained using fast converging Dirichlet series method and approximate analytical methodviz. method of stretching of variables. These methods have the advantages over pure numericalmethods for obtaining the derived quantities accurately for various values of the parameters involvedat a stretch and they are valid in much larger parameter domain as compared with HAM, HPM,ADM and the classical numerical schemes. Also, these methods require less computer memory spaceas compared with pure numerical methods.

Keywords : Magnetohydrodynamics (MHD); Boundary layer flow; Shrinking sheet; Dirichlet series;Powell’s method; Method of stretching variables.

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1 Introduction

The boundary layer flow of conducting fluidfast stretching surface has numerous impor-

tant engineering applications. The flow situationsencountered in many industrial applications areextrusion of polymer sheet from a die or in thedrawing of plastic films and heat-treated materi-als travelling between a feed roll and a wind-uproll or materials manufactured by extrusion pro-cess, glass-fiber and paper production, cooling of

∗Corresponding author. await vb@yahoo.com†Department of Mathematics, Rani Channamma Uni-

versity, Belagavi -591 156, India.‡Department of Mathematics, Rani Channamma Uni-

versity, Belagavi -591 156, India.§Department of Mathematics, Govt. First Grade Col-

lege, Naragund 582 207, India.

metallic sheets, crystal growing etc. In all thesecases, the mechanical properties of the final prod-uct strictly depend on the stretching and coolingrates during the process. In the manufacture ofthese sheets the melt issues from a slit and is sub-sequently stretched to achieve the desired thick-ness. The pioneering work of Sakiadis [1, 2] givesvarious aspects of boundary layer flow on a con-tinuously stretching surface with constant speed.Crane [3, 4] analysed this configuration for thestretching surfaces. Specifically Crane’s problemfor flow of an incompressible viscous fluid faststretching surface has become classic in the liter-ature. It admits an exact analytical solution andmany researchers have analysed various aspectsof this problem. Among those Mcleod and Raj-gopal [5] discussed the uniqueness of exact analyt-ical solution. Gupta and Gupta [6] examined the

343

344 Vishwanath B. Awati et al, /IJIM Vol. 7, No. 4 (2015) 343-350

heat and mass transfer on a stretching flow sub-ject to suction or injection. Brady and Acrivos [7]analysed the flow inside a stretching channel ortube with accelerating velocity. Wang [8] exam-ined the fluid flow due to a stretching cylinder.In another paper, Wang [9], analysed the threedimensional axi-symmetric flow due to stretch-ing surfaces. Wang [10] and Usha and Sridharan[11] have discussed the unsteady flows inducedby stretching film. The extended work of Crane’sproblem for second grade, micropolar and power-law fluid models have been studied by Rajagopalet al[12], Sankara and Watson[13] and Anders-son et al. [14] respectively. The contributionsdue to the boundary layer flow over a stretchingsurface have been made by Pavlov[15], Sachdevet al. [16], Mahapatra et al. [17], Ahmed andAsghar[18], Makinde and Charles [19], Makindeet al. [20], Recently, Vajravelu and Prasad [21]have analysed various aspects of MHD stretch-ing/shrinking sheet problems involving bound-ary layer theory. Vajravelu et al.[22] have anal-ysed the fluid flow and heat transfer over per-meable stretching cylinder. Hayat et al. [23]have analysed MHD flow of nanofluids over an ex-ponentially stretching sheet in a porous mediumwith convective boundary conditions. Nadeem etal. [24] have examined MHD flow of a Cassonfluid over an exponentially shrinking sheet. Za-imi et al. [25] have discussed boundary layer flowand heat transfer over a nonlinearly permeablestretching/shrinking sheet in a nanofluid.

The present investigation is to analyse theMHD viscous flow caused by a stretching sheet.The solution of the resulting third order nonlin-ear boundary value problem with an infinite in-terval is obtained using Dirichlet series methodand method of stretching of variables. The thirdorder nonlinear differential equation with infiniteboundary admits a Dirichlet series solution; nec-essary conditions for the existence and unique-ness of these solutions may also be found in[26, 27]. The specific type of the boundary con-dition i.e. f ′(∞)=0 , the Dirichlet series solu-tion is particularly useful for obtaining solutionand the derived quantities exactly. A generaldiscussion of the convergence of the Dirichlet se-ries may also be found in Riesz [28]. The ac-curacy as well as uniqueness of the solution canbe confirmed using other powerful semi-numericalschemes. Sachdev et al. [16] have analysed vari-

ous problems from fluid dynamics of stretchingsheet using this approach and found more ac-curate solution compared with earlier numericalfindings. Vishwanath et al. [29, 30, 31] andKudenatti et al. [32] have analysed the prob-lems from MHD boundary layer flow with non-linear stretching sheet using these methods andfound more accurate results compared with theclassical numerical methods. In this paper, wepresent Dirichlet series solution and an approxi-mate analytical method-method of stretching ofvariables. This method is quite easy to use es-pecially for nonlinear ordinary differential equa-tions and requires less computer time comparedwith pure numerical methods and easy to solve,compared with other approximate methods (forexample, Homotopy perturbation method (HPM)Pade’ technique, Adomain decomposition meth-ods (ADM)) etc.

The present paper is structured as follows. InSection 2 the mathematical formulation of theproposed problem with relevant boundary con-ditions is given. Section 3 is devoted to the so-lution of the problem using Dirichlet series. Sec-tion 4 gives the solution by means of method ofstretching of variables. In Section 5 detailed re-sults obtained are compared with the correspond-ing numerical schemes and Section 6 is about theconclusion.

2 Mathematical Formulation

Consider a steady two-dimensional incompress-ible boundary layer flow of an electrically con-ducting isothermal Newtonian liquid over a lin-early stretching sheet as shown in Fig. 1. The

Figure 1: Schematic of the stretching sheet prob-lem.

uniform transverse magnetic field H0 acts paral-lel to the y-axis and the conducting liquid in the

Vishwanath B. Awati et al, /IJIM Vol. 7, No. 4 (2015) 343-350 345

space y>0 is considered. Two equal and oppo-site forces are applied along the x-axis, so thatthe wall is stretched and keep the origin fixed.A Hartmann formulation is done for the MHDproblem. The conservation of mass and momen-tum boundary layer equation for the quadraticstretching sheet problem are as (Makinde etal.[20])

∂u

∂x+∂v

∂y= 0, (2.1)

u∂u

∂x+ v

∂u

∂y= v

∂2u

∂y2− σH2

0

ρu. (2.2)

The relevant boundary conditions for the presentflow are

u(x, y) = cx, v = vw at y = 0 (2.3)

u(x, y) = 0 as y → ∞ (2.4)

where u and v are the liquid velocity compo-nents in the x and y directions respectively, c > 0is the streching rate, ν is the kinematic viscosity,H0 is the applied magnetic field and σ is the elec-trical conductivity of the fluid. The constant vwis the suction/ blowing parameter, where vw < 0corresponds to the suction and vw > 0 to theblowing of the fluid and vw=0 , characterize theimpermeable. Introducing the dimensionless vari-ables f and η as

ψ =√cνxf(η) and η = y

√c

ν(2.5)

The velocity components u and ν are related tothe physical stream function ψ defined by

ψ =∂ψ

∂y, v = −∂ψ

∂x. (2.6)

is introduced such that the continuity equation isautomatically satisfied. Introducing the streamfunction ψ(x, y) and the non-dimensional form ofEqs. (2.2-2.4) becomes

v∂3ψ

∂y3+∂(ψ, ∂ψ∂y )

∂(x, y)− σH2

0

ρ

∂ψ

∂y= 0, (2.7)

∂ψ

∂y= cx,

∂ψ

∂x= vw at y = 0, (2.8)

∂ψ

∂y= 0, as y → ∞, (2.9)

Substituting the similarity transformation forstream function in equation 2.5 into equations 2.7- 2.9, we obtain

f ′′′ + ff ′′ − f ′2 −Mf ′ = 0,′=d

dη. (2.10)

f(0) = fw, f′(0) = 1, f ′(∞) = 0. (2.11)

where M =H2

0σcρ is the Magnetic parameter and

fw = vw√cρ suction /blowing parameter. In this

case, we assume f(0)=fw and fw > 0 correspondto suction, fw < 0 correspond to blowing andfw = 0 is the case when the surface is imperme-able. The functions f(η) allow us to determinethe skin friction coefficient given as Cf = −f ′′(0).

The exact solution of MHD flow of Newto-nian fluid is obtained by Chakrabarti and Gupta[33] and non-Newtonian fluid by Bhattacharyyaet al [34]. We assume the solution in moregeneral form as

f(η) = a+ b exp(−λη). (2.12)

where a, b and λ are constants with λ > 0. Sub-stituting Eq. 2.12 into Eq. 2.10 and Eq. 2.11, weget

b = − 1

λ, a = fw +

1

λand

λ =1

2(fw +

√4 + f2w + 4M)

(2.13)

So, the analytical solution reduces to

f =fw − 2

fw ±√

4 + f2w + 4M

2 exp((12)(−fw ∓√

4 + f2w + 4M)η)

fw ±√4 + f2w + 4M

(2.14)

3 Dirichlet Series Solution

We seek an elegant semi-numerical method i.e.Dirichlet series solution of Eq. 2.10 satisfying lastboundary condition f ′(∞) = 0 automatically inthe form (Kravchenko and Yablonskii [26, 27])

f = γ1 + 6γ

∞∑i=1

biai exp(−iγη) (3.15)

where γ and a are parameters which are to bedetermined. Substituting Eq. 3.15 into Eq. 2.10,

346 Vishwanath B. Awati et al, /IJIM Vol. 7, No. 4 (2015) 343-350

we get

∞∑i=1

(−γ2i3 + γγ1i2 +Mi)bia

i exp(−iγη)

+ 6γ2∞∑i=2

i−1∑k=1

[k2 − k(i− k)]bkbi−kai exp(−iγη)

= 0

(3.16)

For i = 1 we have

γ1 = γ2 −M. (3.17)

Substituting Eq. 3.17 into Eq. 3.16 the recur-rence relation for obtaining coefficients is givenby

bi =6γ2

i(i− 1)[γ2i+M ]

i−1∑k=1

[k2 − k(i− k)]bkbi−k

(3.18)for i = 2, 3, 4, ... . If the Eq. 3.15 converges ab-solutely when γ = 0 for some η0 , this series con-verges absolutely and uniformly in the half planeReη ≥Reη0 and represents an analytic 2π

γ peri-

odic function f = f(η0) such that f ′(∞) = 0([kra26]). The Eq. 3.15 contains two free un-known parameters namely a and γ which are tobe determined from the remaining boundary con-ditions of Eq. 2.11 at η = 0

f(0) =γ2 −M

γ+ 6γ

∞∑i=1

biai = α1 (3.19)

and

f ′(0) = 6γ2∞∑i=1

(−i)biai = β1 (3.20)

The solution of the above transcendental Eq.3.19 and Eq. 3.20 yield constants a and γ . Thesolution of the above transcendental equations isequivalent to the unconstrained minimization ofthe functional

[γ2 −M

γ+6γ

∞∑i=1

biai−α1]

2+[6γ2∞∑i=1

(−i)biai−β1]2

(3.21)We use Powell’s method of conjugate directions(Press et. al. [35]) which is one of the most ef-ficient techniques for solving unconstrained opti-mization problems. This helps in finding the un-known parameters a and γ uniquely for different

values of the parameters α1 ,β1 and M . Alterna-tively, Newton’s method is also used to determinethe unknown parameters a and γ accurately. Thephysical quantity of the interest for the problemis shear stress at the surface and velocity profiles.The shear stress at the surface of the problem isgiven by

f ′′(0) = 6γ∞∑i=1

biai(iγ)2 (3.22)

The velocity profiles of the problem is given by

f ′(η) = 6γ2∞∑i=1

(−i)biai exp(−iγη) (3.23)

4 Approximate Analytical solu-tion

Most of the nonlinear ODE arising in MHDproblems are not amenable for obtaining analyt-ical solutions. In such situations, attempts havebeen made to develop an approximate analyti-cal method for the solution of these problems.The numerical approach is always based on theidea of stretching of variables of the flow prob-lems. Method of stretching of variables is usedhere for the solution of such problems. In thismethod, we have to choose suitable derivativefunction H ′ such that the derivative boundaryconditions are satisfied automatically and inte-gration of H ′ will satisfy the remaining boundarycondition. Substitution of this resulting functioninto the given equation gives the residual of theform R(ξ, α) which is called defect function. Us-ing Least squares method, the residual of the de-fect function can be minimized. For details seeAriel [36]. Using the transformation f = fw + Finto Eq. 2.10, we get

F ′′′+(fw+F )F′′−F ′2−MF ′ = 0,′=

d

dη(4.24)

and the relevant boundary conditions 2.11 be-come

F (0) = 0, F ′(0) = 1, F ′(∞) = 0 (4.25)

We introduce two variables ξ and G in the form

G(ξ) = αF (η) and ξ = αη (4.26)

Vishwanath B. Awati et al, /IJIM Vol. 7, No. 4 (2015) 343-350 347

where α > 0, is an amplification factor. In viewof Eq. 4.26, the system 4.24-4.25 are transformedto the form

α2G′′′ + (fwα+G)G′′ −G′2 −MG′ = 0, (4.27)

where ′ = ddξ and the boundary conditions in Eq.

4.25 become

G(0) = 0, G′(0) = 1, G′(∞) = 0 (4.28)

We choose a trail velocity profile

G′ = exp(−ξ) (4.29)

which satisfies the derivative conditions in Eq.4.28. Integrating Eq. 4.29 with respect to ξ from0 to ξ using conditions 4.28, we get

G = 1− exp(−ξ). (4.30)

Substituting Eq. 4.30 into Eq. 4.27, we get theresidual of defect function

R(ξ, α) = (−α2+ fwα− 1+M) exp(−ξ). (4.31)

Using the Least squares approximation methodas discussed in Ariel [?], the equation 4.31 can beminimized for which

∂

∂α(

∫ ∞

0R2(ξ, α)dξ) = 0. (4.32)

Substituting 4.31 into equation 4.32 and solvingcubic equation in α for a positive root, we get

α =1

2(fw ±

√−4 + 4M + f2w) (4.33)

Once the amplification factor is calculated, thenusing Eq. 4.24, original function f can be writtenas

f = fw +1

α(1− exp(−αη)). (4.34)

With α defined in Eq. 4.33. Thus Eq. 4.34 givesthe solution of Eq. 2.10 for all values of fw andM .

5 Result and discussion

The megnetohydrodynamic (MHD) viscousflow due to a stretching sheet caused by boundarylayer of an incompressible viscous flow is analysedby the more suggestive ways by using the Fortranprogramming and Mathematica. The third or-der nonlinear ordinary differential Eq. 2.11 sub-ject to the infinite boundary conditions 2.11 ( has

been solved semi-numerically using Dirichlet se-ries method and method of stretching of variablesfor different values of the parameters fw and M .Table 1. shows that the results of skin friction fordifferent parameters fw andM which agree accu-rately with available pure numerical method. Fig.2 presents the effect of increasing magnetic fieldstrength on the momentum boundary layer thick-ness. The effect of increasing magnetic strengthon the boundary layer creating the drag forcethat opposes the fluid motion causing the velocityto decrease. Increase the Magnetic parameter itslow down the motion of the fluid and decreasesthe boundary layer thickness. Fig. 3. showsa similar trend with increasing suction fw > 0while an injection fw < 0 causes the boundarylayer to thicken by increasing the fluid velocity.Fig. 4. shows that the skin friction increases gen-erally with increase in the intensity of magneticfield. As the intensity of fluid suction increases,a skin friction increases and an injection causes adecrease in the local skin friction at the surface.

Figure 2: Effect of increasing magnetic field in-tensity on velocity profile for fw = 0.1.

Figure 3: Effect of success/injection on velocityprofile for M = 0.5.

348 Vishwanath B. Awati et al, /IJIM Vol. 7, No. 4 (2015) 343-350

Table 1: Comparison of result obtained by Dirichlet series, Method of stretching of variables [MSV] andnumerical method

M fw Dirichlet series Numerical [20] MSV

a γ f”(0) f”(0) f”(0)

0 0 -1.000000 0.999999 -0.999999 -1.000000 -1.000000

0 0.5 -0.609612 1.280776 -1.280776 -1.280777 -1.280776

.5 0.5 -0.074074 1.500000 -1.500000 -1.500000 -1.500000

1.0 0.5 -0.058622 1.686141 -1.686141 -1.686140 -1.686140

0.5 -0.5 -0.166666 1.000000 -0.999999 -1.000000 -1.000000

0.5 -1.0 -0.246139 0.822876 -0.822875 -0.822875 -0.822875

Figure 4: Effect of parameter variation on skinfriction coefficients.

Nomenclature:

c constant rate of stretching [s−1]

f similarity function

ν kinematic viscosity [m2s−1]

σ electrical conductivity [mhom−1]

ρ density [kgm−3]

H0 strength of the magnetic field [wm−2]

ψ(x, y) stream function [m2s−1]

kl − k1 =k0cµ viscoelastic parameter

l characteristic length [m]

H −H0

√σcρ Hartmann number

M −H2 Magnetic Parameter

u velocity component along the sheet[ms−1]

v velocity component normal to the sheet [ms−1]

vw suction / blowing parameter

x coordinate along the sheet [m]

y coordinate normal to the sheet [m]

η similarity variable

α amplification factor

6 Conclusion

In this article, we describe the analysis of bound-ary value problem for third order nonlinear ordi-nary differential equation over an infinite inter-val arising in MHD boundary layer. The semi-numerical schemes described here offer advan-tages over solutions obtained by HAM, HPM,ADM and other numerical methods etc. The con-vergence of the Dirichlet series method is given.The results are presented in Table and graphi-cally, the effects of the emerging parameters arediscussed.

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Dr. Vishwanath B. Awati: is Asso-ciate Professor in the Departmentof Mathematics at the Rani Chan-namma University, Belagavi. Hereceived his doctorate in Fluid Dy-namics from Karnatak University,Dharwad in 2003. He has varied

interest in research activities, his research papershave been published in various journals of na-tional and international repute.

Dr. Krishna B. Chavaraddi : isAssistant Professor in the Depart-ment of Mathematics at Govt FirstGrade College,Naragund. He re-ceived his doctorate in Fluid Dy-namics from Bangalore University,Bangalore in 2008. He has varied

interest in research activities, his research papershave been published in various journals of na-tional and international repute.

Mr Mahesh Kumar N : is Assis-tant Professor at the Departmentof Mathematics, Rani ChannammaUniversity, Belagavi. He is pursingdoctorate degree in Fluid Dynam-ics from Rani Channamma Univer-sity, Belagavi. He has varied in-

terest in research activities,his research papershave been published in various journals of inter-national repute.