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formerly Central European Journal of Engineering

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Optimal Auxiliary Functions Method for viscous flow due to a stretching surface with partial slip

Vasile Marinca
  • University Politehnica Timişoara, Department of Mechanics and Strength of Materials, Timişoara, 300222, Romania
  • Department of Electromechanics and Vibration, Center for Advanced and Fundamental Technical Research, Romania Academy, Timişoara, 300223, Romania
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  • De Gruyter OnlineGoogle Scholar
/ Remus-Daniel Ene / Valentin Bogdan Marinca
Published Online: 2018-08-31 | DOI: https://doi.org/10.1515/eng-2018-0028

Abstract

The viscous flow induced by a stretching surface with partial slip is investigated. The flow is governed by a third-order nonlinear differential equation, corresponding to the planar and axisymmetric stretching. The derived equation of motion is solved analytically by Optimal Auxiliary Functions Method (OAFM). This procedure is highly efficient and it controls the convergence of the approximate solution. A few examples are given, showing the exceptionally good agreement between the analytical and numerical solutions of the nonlinear problem. OAFM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

Keywords: Optimal auxiliary functions method; viscous flow; partial slip; stretching surface

PACS: 02.60.-x; 47.11.-j; 47.50.-d

1 Introduction

The study of boundary layer viscous flow due to a stretching surface is very important because of its several engineering applications such as: crystal growing, drawing of elastic films, the heat treated materials traveling between a feed roll and the wind-up roll or a conveyor belt poses the features of a moving continuous surface. There exist situations where there may be a partial slip between the fluid and the boundary. In the last few decades, nonlinear problems are widely used as models to describe complex physical phenomena in various scientific field of science. The boundary layer flow over a continuously stretching surface was first studied by Sakiadis [1, 2]. Crane [3] found a closed form exponential solution for the planar viscous flow of a linear stretching case. Sparrow et al. [4, 5] studied a number of flow problems taking the velocity slip into account. In [6] is given an algorithm for the moving boundaries and for the flow of a fluid past a rotating disk. Rajagopal et al. [7] obtained numerical solution for the flow of viscoelastic second order fluid past a stretching sheet. Ariel [8] and Anderson [9] have reported the analytical closed form solutions of the second grade fluid and of the fourth order nonlinear differential equations arising due to the MHD flow. Wang [10] studied axisymmetric cases for the viscous flow. Mirogolbabei et al. [11] considered a number of boundary layer flows induced by the axisymmetric stretching of a sheet. The unsteady axisymmetric flow and heat transfer of a viscous fluid is treated by Sajid et al. [12]. Also, the analytical solution for these problems has been considered by Sajid et al. in [13], Miklavčič and Wang [14] first analyzed properties of the flow to a shrinking sheet with suction. Many other researchers denoted themselves on investigating these problems [15, 16, 17, 18, 19].

In general in fluid mechanics of viscous fluids, by means of similarity transformations, the set of partial differential equations are reduced to that of ordinary differential equations. Much attention of the researchers was focused in obtaining the analytical solutions for the nonlinear problems. But, in general, the nonlinear problems cannot be solved analytically by means of traditional methods. For the weakly nonlinear problems, many methods exist for approximating the solutions. The most widely applied method by researches is perturbation technique [20] which is based on the existence of small parameters. Others methods for approximating the solutions of nonlinear problems are the modified Lindstedt-Poincare method [21], the Adomian decomposition method [22], the parameter-expansion method [23], the optimal homotopy perturbation method [24, 25, 26], the optimal homotopy asymptotic method [27, 28, 29].

Recently, several methods has been used for solving different nonlinear differential equations such as: traveling-wave transformation method [30], Cole-Hopf transformation method [31].

The objective of the present paper is to propose a new and accurate approach to nonlinear differential equation of viscous fluid due to a stretching surface with partial slip, using an analytical technique, namely the optimal auxiliary functions method. The motivation of our approach is to present a new procedure that is effective, explicit and very accurate for nonlinear approximations rapidly convergent to the exact solution using only one iteration. This procedure must provide a rigorous way to control and adjust the convergence of the approximate solutions using some convergence-control parameters. The proposed method is easy, concise and can be applied to other nonlinear problems. The validity of our procedure, which does not imply the presence of a small or large parameter in the equation or into the boundary / initial conditions, is based on the construction and determination of the auxiliary functions, combined with a convenient way to optimally control the convergence of the solutions. The efficiency of the proposed procedure is proven when an accurate solution is explicitly analytically obtained in an iterative way after only one iteration. Our procedure can be applied to a variety of engineering domains as [32, 33, 34, 35, 36, 37, 38, 39, 40]. The validity of this method is demonstrated by comparing the results obtained with the numerical solutions.

2 The governing equations

Consider the three dimensional flow of a viscous fluid bounded by a stretching surface. If (u, v, w) is the velocity components in the Cartesian directions (x, y, z) respectively, then the continuity and steady constant property Navier-Stokes equations for viscous fluid flow are [7], [10, 13]:

ux+vy+wz=0(1)

uux+vuy+wuz=pxρ+ν2u(2)

uvx+vvy+wvz=pyρ+ν2v(3)

uwx+vwy+wwz=pzρ+ν2w(4)

where p is the pressure, ρ is the density and ν is the kinematic viscosity. If a is the stretching constant, a > 0, W is the suction velocity and m is a parameter describing the type of stretching, then the velocity components on the stretching surface are:

u=ax,v=a(m1)y,w=W.(5)

It is known that for m = 1, we have planar stretching case, while for m = 2 we have axisymmetric stretching case. In order to simplify the governing equations we use the similarity transform [10]:

u=axΦ(η),v=a(m1)yΦ(η),w=maνΦ(η)andη=zaν,(6)

where prime denotes the differentiation with respect to η. Eq. (1) is identically satisfied and Eq. (4) becomes

p=νρwz12ρW2+C,(7)

with C a constant. From the Eqs. (2) and (3) we deduce that:

Φ(η)+mΦ(η)Φ(η)Φ(η)2=0,(8)

as there is no lateral pressure gradient at infinity. If N denotes a slip constant, then on the surface of the stretching sheet, the velocity slip is assumed to be proportional to the local sheer stress [41]:

uax=ρνNuz<0,va(m1)y=ρνNvz<0.(9)

From the similarity transform (6) we obtain

Φ(0)=1+λΦ(0),(10)

with λ=ρNaν > 0 a non-dimensional parameter indicating the relative importance of partial slip. If λ = 0 there is no slip. Given a suction velocity of − W on the stretching surface, we have the boundary condition

Φ(0)=α,(11)

where α=Wmaν is a non-dimensional constant which determines the transpiration rate at the surface and α < 0 if injection from the surface occurs, α > 0 for suction and α = 0 for an impermeable sheet. Also, since there is no lateral velocity at infinity, we have yet the condition

Φ()=0.(12)

3 Basic ideas of optimal auxiliary functions method

Eq. (8) with the initial / boundary conditions (10), (11) and (12) can be written in a more general form as [42, 43]:

L[Φ(η)]+g(η)+N[Φ(η)]=0,(13)

where L is a linear operator, g is a known function and N is a given nonlinear operator, η denotes independent variable and Φ(η) is an unknown function. The initial / boundary conditions are

B(Φ(η),dΦ(η)dη)=0.(14)

An exact solution for strongly nonlinear equation (13) and with initial / boundary conditions (14) are frequently scarce. To find an approximate analytic solution of Eqs. (13) and (14), we assume that the approximate solution can be written in the form with only two components:

Φ¯(η)=Φ0(η)+Φ1(η,Ci),i=1,2,...,s,(15)

where the initial approximation Φ0(η) and the first approximation Φ1(η, Ci) will be determined as follows. Substituting Eq. (15) into Eq. (13), it results in:

L[Φ0(η)]+L[Φ1(η,Ci)]+g(η)+N[Φ0(η)+Φ1(η,Ci)]=0.(16)

The initial approximation Φ0(η) can be determined from the linear equation

L[Φ0(η)]+g(η)=0,B(Φ0(η),dΦ0(η)dη)=0(17)

and the first approximation Φ1(η, Ci), from the following equation

L[Φ1(η,Ci)]+N[Φ0(η)+Φ1(η,Ci)]=0,B(Φ1(η,Ci),dΦ1(η,Ci)dη)=0.(18)

Now, the nonlinear term from Eq. (18) is expanded in the form

N[Φ0(η)+Φ1(η,Ci)]=N[Φ0(η)]+k=1Φ1k(η,Ci)k!N(k)[Φ0(η)].(19)

In order to avoid the difficulties that appears in solving of the nonlinear differential equation (18) and to accelerate the rapid convergence of the first approximation Φ1(η, Ci) and implicit of the approximate solution Φ(η), instead of the last term arising into Eq. (18), we propose another expression, such that the nonlinear differential equation (18) can be written as a linear differential equation taking into account the Eq. (19), in the form:

L[Φ1(η,Ci)]+A1[Φ0(η),Ci]N[Φ0(η)]+A2[Φ0(η),Cj]=0,(20)

B(Φ1(η,Ci),dΦ1(η,Ci)dη)=0,(21)

where A1 and A2 are two arbitrary auxiliary functions depending on the initial approximation Φ0(η) and several unknown parameters Ci and Cj, i = 1, 2, …, p, i = p + 1, p + 2, …, s.

The auxiliary functions A1 and A2 (called optimal auxiliary functions) are not unique, and are of the same form like Φ0(η) or of the form of the N[Φ0(η)] or combinations of the forms of Φ0(η) and N[Φ0(η)].

For example, if Φ0(η) or N[Φ0(η)] contain polynomial functions, then A1[Φ0(η), Ci] and A2[Φ0(η), Cj] are sums of polynomial functions. If Φ0(η) or N[Φ0(η)] contain exponential functions, then A1 and A2 would be sums of the exponential functions. If Φ0(η) or N[Φ0(η)] contain trigonometric functions, then A1 and A2 would be sums of the trigonometric functions, and so on. In all these sums, the coefficients of the polynomial, exponential, trigonometric and so on functions, are the parameters C1, C2, …, Cs.

If in a special case N[Φ0(η)] = 0, then it is clear that Φ0(η) is an exact solution of Eqs. (13) and (14). The unknown parameters Ci and Cj can be optimally identified via different method such as: the Galerkin method, the least square method, the collocation method, the Ritz method, the Kantorovich method or by minimizing the square residual error, using:

J(Ci,Cj)=abR2(m,λ,α,η)dη,(22)

where

R(m,λ,α,η)=L[Φ¯(η,Ci,Cj)]+g(η)++N[Φ¯(η,Ci,Cj)],i=1,2,,p,j=p+1,p+2,,s,(23)

a and b are two values depending on the given problem.

The unknown parameters Ci, Cj can be optimally identified from the equations

JC1=JC2=...=JCp=JCp+1=...=JCs=0.(24)

With these parameters Ci, Cj known, by this novel approach, the approximate solution (15) is well determined.

Our procedure proves to be a powerful tool for solving nonlinear problems not depending on small or large parameters. It should be emphasize that our method contains the optimal auxiliary functions A1 and A2 which provides us with a simple way to adjust and control the convergence of the approximate solutions after only one iteration. Also, it is remarkable that a nonlinear differential problem is transformed into two linear differential problems.

4 Application of OAFM to viscous fluid given by Eqs. (8), (10), (11) and (12)

We introduce the basic ideas of the OAFM by considering Eq. (8) with the initial / boundary conditions given by Eqs. (10), (11) and (12). We choose the linear operator L by the form [43]:

L(Φ(η))=Φ+KΦ.(25)

We can choose the linear operator in the form L(Φ(η)) = Φ″′ + K2 Φ′, and so on, where K > 0 is an unknown parameter.

Eq. (17) becomes (g(η) = 0):

Φ0(η)+KΦ0(η)=0,Φ0(0)=α,Φ0(0)=1+λΦ0(0),Φ0()=0,(26)

with the solution

Φ0(η)=α+1eKηK(1+λK).(27)

The nonlinear operator N(Φ(η)) is obtained from Eqs. (8) and (25):

N(Φ(η))=KΦ(η)+mΦ(η)Φ(η)Φ(η)2.(28)

By means of the Eqs. (27) and (28) it is obtain

N(Φ0(η))=βeKη+(m1)e2Kη(1+λK)2,(29)

where

β=K(Kmα)(1+λK)m.(30)

Having in view Eqs. (20), (25) and (29), the first approximation is obtained from the equation:

Φ1+KΦ1+A2(eKη,e2Kη,Cj)+A1(eKη,e2Kη,Ci)×βeKη+(m1)e2Kη(1+2λK)2=0,(31)

with the initial / boundary conditions

Φ1(0)=0,Φ1(0)=λΦ1(0),Φ1()=0.(32)

We have the freedom to choose the optimal auxiliary functions A1 and A2 in the following forms:

A1[Φ0(η),Ci]=(1+λK)2(C1+C2η),(33)

A2[Φ0(η),Cj]=(C3+C4η)(e2Kη+K1e3Kη)(C5+C6η)(e3Kη+K2e4Kη),(34)

or

A1[Φ0(η),Ci]=0,(35)

A2[Φ0(η),Cj]=(C1+C2η)eKη(C3+C4η+C5η2)e2Kη,(36)

or yet

A1[Φ0(η),Ci]=(1+λK)2C1,(37)

A2[Φ0(η),Cj]=(C2+C3η+C4η2)e2Kη(C5+C6η+C7η2)e3Kη,(38)

and so on.

If the auxiliary functions A1 and A2 are given by Eqs. (33) and (34) then Eq. (20) may be written as:

Φ1+KΦ1=(βC1+βC2η)eKη+[(mC2C2+C4)η+(m1)C1+C3]e2Kη+[(K1C4+C6)η+K1C3+C5]e3Kη+[K2C6η+K2C5]e4Kη,(39)

with the initial / boundary conditions given in Eq. (32), whose solution is:

Φ1(η,Ci)=M1+[βC22K2η2+(βC1K2+2βC2K3)η+M2]eKη+[(1m)C2C44K3η+1m4K3C1+1m2K4C2C34K3C42K4]e2Kη[K1C4+C618K3η+K1C3+C518K3+7(K1C4+C6)108K4]e3Kη[K248K3C6η+K248K3C5+5K2288K4C6]e4Kη,(40)

where

M1=1m4β+λK(33m8β)4K3(1+λK)C19+4K1+λK(16K1+27)36K3(1+λK)C3+1m8β+λK(22m12β)4K4(1+λK)C28K1+27+λK(2K118)108K4(1+λK)C49K2+16+λK(45K2+64)144K3(1+λK)C527K2+112+λK(81K2+208)864K4(1+λK)C6,M2=(2β+m1)(1+2λK)2K3(1+λK)C1+8β+3m3+4λK(m1+3β)4K4(1+λK)C2+K1+3+3λK(K1+2)6K3(1+λK)C3+27+5K1+9λK(K1+4)36K4(1+λK)C4+2+K2+2λK(3+2K2)12K3(1+λK)C5+20+7K2+4λK(9+4K2)144K4(1+λK)C6.

If we consider m = 1 and β = 0 (the planar stretching case for impermeable sheet) into Eq. (29) then we have N[Φ0(η)] = 0 and therefore we can obtain the exact solution of the equation:

Φ+ΦΦΦ2=0,Φ(0)=0,Φ(0)=1+λΦ(0),Φ()=0,(41)

which is:

Φ(η)=α+1eKηK(1+λK),(42)

where K is obtained from the equation:

λK3+(1αλ)K2αK1=0.(43)

Beyond this remarkable case, the approximate analytic solution of the Eqs. (8), (10), (11) and (12) can be obtained from Eqs. (15), (27) and (40).

5 Numerical examples

We illustrate the accuracy of OAFM by comparing obtained approximate solutions with the numerical integration results obtained by means of a fourth-order Runge-Kutta method in combination with the shooting method. In all cases, the unknown parameters are optimally identified via Galerkin method. For this, we use the following eleven weighted functions fi given by [41]:

f1(η)=γe3Kη+η2e2Kη+δηeKη,f2(η)=ηeKη,f3(η)=η2eKη,f4(η)=1+γηeKη,f5(η)=e2Kη,f6(η)=eKη,f7(η)=ηe4Kη,f8(η)=e4Kη,f9(η)=δηe2Kη+η3e2Kη,f10(η)=eKη+K1e2Kη+K3ηe3Kη,f11(η)=ηeKη+K2e3Kη+K4e4Kη,(44)

where γ, δ, K1, K2, K3 and K4 are unknown parameters.

The parameters K, K1, K2, K3, K4, γ, δ, C1, …, C6 are determined from equations (Galerkin method):

Jj=0R(m,λ,α,η)fj(η)dη=0,i=1,...,6,j=1,...,11,(45)

where the residual R(m, α, λ, η) is given by Eq. (23):

R(m,λ,α,η)=Φ¯(η)+mΦ¯(η)Φ¯(η)Φ¯(η)2(46)

and Φ(η) is given by Eq. (15) with the initial / boundary conditions (10)-(12).

Example 5.1

Consider planar stretching case with impermeable sheet, m = 1, α = 0 and λ = 1. In this case, from Eqs. (45) and (46) we obtain

K=0.4631238249,γ=0.6858214854,δ=0.1052314539,K1=0,K2=0,K3=2.13106,K4=0.21106,C1=0.0119658357,C2=0.0007396024,C3=0.0642359221,C4=0.0326512044,C5=0.0530417312,C6=0.0102396495.

The first-order approximate solution given by Eq. (15) can be written in the form

Φ¯(η)=0.7549045180+(0.0011830813η2+0.0280632559η0.1942637382)e0.4631238249η(0.0821765397η+0.5165462911)e0.9262476499η0.0440894886e1.3893714748η.(47)

Example 5.2

In the second case for the same planar stretching case with impermeable sheet m = 1, α = 0 but λ = 5, from Eqs. (45) the values of the parameters are

K=0.5252370049,γ=4.8796805607,δ=1.2890550305,K1=0,K2=0,K3=1.24106,K4=1.02106,C1=0.0279185716,C2=1.9918945423106,C3=6.24935814611012,C4=3.902111975511012,C5=9.6210413121012,C6=0.

Approximate solution (15) becomes:

Φ¯(η)=0.5251657155+(1.3352013705109η20.0000374387η0.5251657155)e0.5252370049η+(6.73247789151012η+1.48537029521011)×e1.0504740099η+(9.12846706011013η+5.71652283551012)e1.5757110148η.(48)

Example 5.3

In the third case for the planar stretching case with impermeable sheet m = 1, α = 0 and with sleep parameter λ = 10, from Eqs. (45) we obtain

K=0.4331821853,γ=3.0639340595,δ=0.5722383481,K1=0,K2=0,K3=1.03106,K4=2.11106,C1=0.0125089730,C2=9.6275057188107,C3=6.60108971911012,C4=3.3603467961012,C5=1.00959311611011,C6=2.05610548811012.

The first-order approximate solution (15) is

Φ¯(η)=0.4331053011+(1.2809679965109η20.0000332989η0.4331053011)e0.433182185η+(1.03368925011011η+2.74231241171011)×e0.8663643706η+(+0.9322450391012η+1.0657161621011)e1.2995465559η.(49)

The values of Φ″(0) and Φ(∞) are given in Table 1 for the same planar stretching case with impermeable sheet, calculated by means OAFM, and by numerical integration.

Table 1

Values of Φ″(0) and Φ(∞) in the case of planar stretching case with impermeable sheet m = 1, α = 0

Example 5.4

In this case, we consider planar stretching case but suction sheet, m = 1, α = 3 and with slip parameter λ = 1. The parameters obtained by means of Eqs. (45) are:

K=1.0265261151,γ=3.011278740,δ=0.9721507438,K1=0,K2=0,K3=1.002106,K4=0.52106,C1=2.67819207611011,C2=4.25251099561012,C3=4.655928821109,C4=1.6964107211109,C5=6.96843414911010,C6=0.0001204043

and the first-order approximate solution (15) becomes

Φ¯(η)=3.0795956234+(2.35251693611012η2+2.01348063021011η4.92110796551011)×e1.0265261151η+(8.71809025561011η6.94320011421011)e2.0530522302η+(1.3753249143106η0.0795955933)e3.0795783454η.(50)

Example 5.5

For planar stretching case and suction sheet, m = 1, α = 3 and slip parameter λ = 5, we have

K=1.0016112565,γ=2.9079023283,δ=0.9038557714,K1=0,K2=0,K3=0.55106,K4=0.62106,C1=1.3658235079106,C2=2.1653508962107,C3=0.0005801616156,C4=0.0002087511,C5=0.9806268698,C6=0.0158334825

and therefore, the first-order approximate solution (15) has the form

Φ¯(η)=3.0205594824+(5.4255801019107η2+4.6777709040106η0.0000118225)e1.0016112565η+(0.0000200454η0.0000156837)e2.0032225131η+(0.0003400695η0.0205318961)e3.0048337696η.(51)

Example 5.6

The planar stretching case and suction sheet, m = 1, α = 3 but slip parameter λ = 10 we obtain from Eqs. (45):

K=1.1395297367,γ=2.9650196892,δ=0.9588570845,K1=0,K2=0,K3=1.002106,K4=1.32106,C1=0.0000646091,C2=0.0000113762,C3=0.0863263871,C4=0.02944192315,C5=0.2391591505,C6=0.06102791273

and the first-order approximate solution (15) can be written as

Φ¯(η)=3.0106792212+(0.0000661253η2+0.0005189796η0.0011722736)e1.1395297367η+(0.0027526798η0.0032398505)e2.2790594735η+(0.0012678992η0.0062671170)e3.4185892103η.(52)

In Table 2, we present the values of Φ″(0) and Φ(∞) in the case of planar stretching case with suction sheet.

Table 2

Values of Φ″(0) and Φ(∞) in the case of planar stretching case with suction sheet m = 1, α = 3

Example 5.7

Now, we consider the axisymmetric flow with impermeable sheet, m = 2, α = 0 and with slip parameter λ = 1. From Eqs. (45) we obtain the following set of parameters

K=1.2284861855,γ=1.2107679860,δ=5.9462560050,K1=K2=0.7335772195,K3=1.11106,K4=1.06106,C1=0.0272717347,C2=0.0098884303,C3=0.0130978383,C4=0.1950919615,C5=0.06294893217,C6=0.0018821549.

In this case, the first-order approximate solution (15) is

Φ¯(η)=0.5509446955+(0.0077420177η20.0679124177η0.6421304231)e1.2284861855η+(0.0469289624η+0.0831404603)e2.4569723711η+(0.0041898146η+0.0075368933)e3.6854585567η+(0.0000154947η+0.0005083699)e4.9139447423η.(53)

Example 5.8

For the case of the axisymmetric flow with impermeable sheet, m = 2, α = 0 and with slip parameter λ = 5, from Eqs. (45) we have

K=0.8408529662,γ=2.5936764722,δ=12.6879512489,K1=K2=0.5953831575,K3=2.03106,K4=1.01106,C1=0.0053548861,C2=0.0012448162,C3=0.00821412956,C4=0.03031382709,C5=0.01091079627,C6=0.00176755942

and therefore, the first-order approximate solution (15) becomes

Φ¯(η)=0.3780173693+(0.0023588678η20.0315158198η0.4396463038)e0.8408529662η+(0.0208604015η+0.0573797361)e1.6817059324η+(0.001423055078η+0.0040581045)e2.5225588986η+(0.0000368811η+0.0001911237)e3.3634118648η.(54)

Example 5.9

In this case we consider the axisymmetric flow with impermeable sheet, m = 2, α = 0 and with slip parameter λ = 10. From Eqs. (45) yields

K=0.5239724222,γ=1.6648535124,δ=1.8794890289,K1=K2=3.4855129921,K3=0.43106,K4=0.31106,C1=0.0112870692,C2=0.0004699006,C3=0.0366309932,C4=0.0059152635,C5=0.0310033957,C6=0.00815982977.

The first-order approximate solution can be written in the form

Φ¯(η)=0.3106373548+(0.0006102510η2+0.0246579539η0.3170175340)e0.5239724222η+(0.0247341994η+0.0442009306)e1.0479448445η+(0.0157948648η0.0156201635)e1.5719172668η+(0.0041191012η0.0222006138)e2.095889689η.(55)

In Table 3 we present the values of Φ″(0) and Φ(∞) for the stretching flow with impermeable sheet.

Table 3

Values of Φ″(0) and Φ(∞) in the case of stretching flow with impermeable sheet m = 2, α = 0

Example 5.10

For the axisymmetric flow with suction sheet m = 2, α = 3 and slip parameter λ = 1 we obtain the following values of the parameters:

K=1.5574274146,γ=0.7860056759,δ=3.6690514687,K1=K2=0.0507748232,K3=1.003106,K4=0.32106,C1=0.0107133300,C2=0.0024557002,C3=9.2340545611,C4=4.93780472001,C5=2.72171778619,C6=23.18498897742928,

with the first-order approximate solution obtained from (15):

Φ¯(η)=3.0235364880+(0.0026339287η2+0.0162169332η0.0279371463)e1.5574274146η+(0.0864947892η0.0510827083)e3.1148548292η+(0.0937682076η+0.0527720186)e4.6722822438η+(0.0064922112η+0.0027118480)e6.2297096585η.(56)

Example 5.11

For the case of the axisymmetric flow with suction sheet m = 2, α = 3 and slip parameter λ = 5, the parameters are

K=1.7780169491,γ=0.5957717503,δ=2.8222930255,K1=K2=0.0131160498,K3=1.013106,K4=1.22106,C1=0.0000862400,C2=0.0000222658,C3=0.3512786202,C4=0.1994379874,C5=1.6374819712,C6=1.6614059805

and therefore, the first-order approximate solution (15) has the form

Φ¯(η)=3.0053612258+(0.0001065840η2+0.0005858601η0.0008994531)e1.7780169491η+(0.0035222493η0.0022439786)e3.55603389829η+(0.0065444365η0.0021759449)e5.3340508474η+(0.0000797662η0.0000397491)e7.1120677965η.(57)

Example 5.12

In the last case we consider the axisymmetric flow with suction sheet m = 2, α = 3 and slip parameter λ = 10, such that the parameters are

K=1.8412120648,γ=0.554362126,δ=2.6330245292,K1=K2=0.0066382243,K3=1.004106,K4=0.52106,C1=9.5831454966106,C2=2.5525898638106,C3=0.0821658733,C4=0.04718459612,C5=0.8534171847,C6=0.5288479009.

The first-order approximate solution (15) can be written as

Φ¯(η)=3.0027282856+(0.0000243991η2+0.0001301957η0.0001939768)e1.8412120648η+(0.0008131418η0.0005329181)e3.6824241297η+(0.0020278091η0.0019878153)e5.5236361946η+(0.0000116973η0.0000135853)e7.3648482595η.(58)

In Table 4 are presented the values of Φ″(0) and Φ(∞) for stretching flow with suction sheet.

Table 4

Values of Φ′(0) and Φ(∞) in the case of stretching flow with suction sheet m = 2, α = 3

From Tables 1-4 we deduced that there exist an excellent agreement between the numerical results and the results obtained by means of OAFM.

On the other hand, considering the effect of slip parameter on the velocity Φ(η) in both flows, Figures 1-4 have been displayed. Figure 1 shows the variation of Φ for planar flow and impermeable sheet. In Figure 2 the variation of Φ for planar flow and suction sheet is plotted.

Variation of Φ by increasing the slip parameter λ for planar flow and impermeable sheet (m = 1, α = 0): — numerical solution; …… OAFM solution
Figure 1

Variation of Φ by increasing the slip parameter λ for planar flow and impermeable sheet (m = 1, α = 0): — numerical solution; …… OAFM solution

Variation of Φ by increasing the slip parameter λ for planar flow and suction sheet (m = 1, α = 3): — numerical solution; …… OAFM solution
Figure 2

Variation of Φ by increasing the slip parameter λ for planar flow and suction sheet (m = 1, α = 3): — numerical solution; …… OAFM solution

Figure 3 shows the variation of Φ for axisymmetric flow and impermeable sheet. Figure 4 shows the variation of Φ for axisymmetric flow and suction sheet. It is clear that the velocity components decrease with an increase in the slip parameter for all cases.

Variation of Φ by increasing the slip parameter λ for axisymmetric flow and impermeable sheet (m = 2, α = 0): — numerical solution; …… OAFM solution
Figure 3

Variation of Φ by increasing the slip parameter λ for axisymmetric flow and impermeable sheet (m = 2, α = 0): — numerical solution; …… OAFM solution

Variation of Φ by increasing the slip parameter λ for axisymmetric flow and suction sheet (m = 2, α = 3): — numerical solution; …… OAFM solution
Figure 4

Variation of Φ by increasing the slip parameter λ for axisymmetric flow and suction sheet (m = 2, α = 3): — numerical solution; …… OAFM solution

In Figures 5-7 the planar cases for every value of slip parameter λ have been plotted and in Figures 8-10 the stretching cases for different values of λ have been plotted. It is evident that the velocity is less for the axisymmetric flow when compared with the planar case.

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 1): — numerical solution; …… OAFM solution
Figure 5

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 1): — numerical solution; …… OAFM solution

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 5): 
— numerical solution; …… OAFM solution
Figure 6

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 5): — numerical solution; …… OAFM solution

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 10): — numerical solution; …… OAFM solution
Figure 7

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 10): — numerical solution; …… OAFM solution

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 1): — numerical solution; …… OAFM solution
Figure 8

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 1): — numerical solution; …… OAFM solution

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 5): — numerical solution; …… OAFM solution
Figure 9

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 5): — numerical solution; …… OAFM solution

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 10): 
— numerical solution; …… OAFM solution
Figure 10

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 10): — numerical solution; …… OAFM solution

Finally, the residual functions obtained for the approximate analytic solutions given by Eqs. (47)-(58) are plotted in Figures 11-22.

The residual R(1, 1, 0, η) for Eq. (47) obtained by OAFM
Figure 11

The residual R(1, 1, 0, η) for Eq. (47) obtained by OAFM

The residual R(1, 5, 0, η) for Eq. (48) obtained by OAFM
Figure 12

The residual R(1, 5, 0, η) for Eq. (48) obtained by OAFM


The residual R(1, 10, 0, η) for Eq. (49) obtained by OAFM
Figure 13

The residual R(1, 10, 0, η) for Eq. (49) obtained by OAFM

The residual R(1, 1, 3, η) for Eq. (50) obtained by OAFM
Figure 14

The residual R(1, 1, 3, η) for Eq. (50) obtained by OAFM

The residual R(1, 5, 3, η) for Eq. (51) obtained by OAFM
Figure 15

The residual R(1, 5, 3, η) for Eq. (51) obtained by OAFM

The residual R(1, 10, 3, η) for Eq. (52) obtained by OAFM
Figure 16

The residual R(1, 10, 3, η) for Eq. (52) obtained by OAFM

The residual R(2, 1, 0, η) for Eq. (53) obtained by OAFM
Figure 17

The residual R(2, 1, 0, η) for Eq. (53) obtained by OAFM

The residual R(2, 5, 0, η) for Eq. (54) obtained by OAFM
Figure 18

The residual R(2, 5, 0, η) for Eq. (54) obtained by OAFM

The residual R(2, 10, 0, η) for Eq. (55) obtained by OAFM
Figure 19

The residual R(2, 10, 0, η) for Eq. (55) obtained by OAFM

The residual R(2, 1, 3, η) for Eq. (56) obtained by OAFM
Figure 20

The residual R(2, 1, 3, η) for Eq. (56) obtained by OAFM

The residual R(2, 5, 3, η) for Eq. (57) obtained by OAFM
Figure 21

The residual R(2, 5, 3, η) for Eq. (57) obtained by OAFM

The residual R(2, 10, 3, η) for Eq. (58) obtained by OAFM
Figure 22

The residual R(2, 10, 3, η) for Eq. (58) obtained by OAFM

6 Conclusions

In this work, the Optimal Auxiliary Functions Method (OAFM) is employed to propose an analytic approximate solution to the boundary nonlinear problem of the viscous flow induced by a stretching surface with partial slip. Our procedure is valid even if the nonlinear equations of motion do not contain any small or large parameters. The proposed approach is mainly based on a new construction of the solutions and especially on the auxiliary function. These auxiliary functions lead to an excellent agreement of the solutions with numerical results. This technique is very effective, explicit and accurate for non-linear approximations rapidly converging to the exact solution after only one iteration. Also OAFM provides a simple but rigorous way to control and adjust the convergence of the solution by means of some convergence-control parameters. Our construction is different from other approaches especially referring to the linear operator L depending of an optimal parameter K and to the auxiliary convergence-control function A1 and A2 which ensure a fast convergence of the solutions. In a special case OAFM lead to choice of the exact solution. Numerical results are obtained using the shooting method in combination with the fourth-order Runge-Kutta method, using Wolfram Mathematica 9.0 software.

Further important contributions in the area of this paper can be: OAFM allows us to obtain a very good results for both cases: planar and axisymmetric flow for any value of the slip parameter. We point out, also the influence of the coefficient α in the case of impermeable or suction sheet. The values Φ″(0) and Φ(∞) are given for both flow situations, and found in excellent agreement with numerical results.

The proposed method is straightforward, effective, concise and can be applied to other different nonlinear problems.

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About the article

Received: 2017-06-15

Accepted: 2018-06-12

Published Online: 2018-08-31


Conflict of InterestConflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.


Citation Information: Open Engineering, Volume 8, Issue 1, Pages 261–274, ISSN (Online) 2391-5439, DOI: https://doi.org/10.1515/eng-2018-0028.

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© 2018 V. Marinca et al., published by De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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