Viscous flow and heat transfer over an unsteady stretching surface

Remus-Daniel Ene; Vasile Marinca; Valentin Bogdan Marinca

doi:10.1515/phys-2016-0042

Artikel Open Access

Viscous flow and heat transfer over an unsteady stretching surface

Remus-Daniel Ene , Vasile Marinca und Valentin Bogdan Marinca

Veröffentlicht/Copyright: 20. Oktober 2016

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Abstract

In this paper we have studied the flow and heat transfer of a horizontal sheet in a viscous fluid. The stretching rate and temperature of the sheet vary with time. The governing equations for momentum and thermal energy are reduced to ordinary differential equations by means of similarity transformation. These equations are solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM) which provides us with a convenient way to control the convergence of approximation solutions and adjust convergence rigorously when necessary. Some examples are given and the results obtained reveal that the proposed method is effective and easy to use.

Keywords: optimal homotopy asymptotic method (OHAM); film flow; heat transfer; unsteady stretching surface

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

1 Introduction

The flow and heat transfer in a viscous fluid over a stretching surface is important for engineers and applied mathematicians. Studies have been conducted which take into account the numerous industrial applications. Examples of such applications are crystal growing, continuous casting, polymer extrusion, manufacture and drawing of plastics and rubber sheets, wire drawing and so on. Sakiadis [1, 2], Crane [3], Tsou et al. [4], Gupta and Gupta [5], Maneschy et al. [6], Grubka and Bobba [7], Wang [8], Usha and Rukamani [9], Anderson et al. [10], Ali [11], Magyari et al. [12], Vajravelu [13], Magyari and Keller [14], Elbashbeshy and Bazid [15], Dandapat et al. [16], Ali and Magyari [17], Liu and Anderson [18], Chen [19], Dandapat et al. [20], Cortell [21] have studied different problems related to such applications.

Analytical solutions to nonlinear differential equations play an important role in the study of flow and heat transfer of different types of fluids, but it is difficult to find these solutions in the presence of strong nonlinearity. A few approaches have been proposed to find and develop approximate solutions of nonlinear differential equations. Perturbation methods have been applied to determine approximate solutions to weakly nonlinear problems [22]. But the use of perturbation theory in many problems is invalid for parameters beyond a certain specified range. Other procedures have been proposed such as the Adomian decomposition method [23], some linearization methods [24, 25], various modified Lindstedt-Poincare methods [26], variational iteration method [27], optimal homotopy perturbation method [28] and optimal homotopy asymptotic method [29–34]

In this study we propose an accurate approach to nonlinear differential equations of the flow and heat transfer in a viscous fluid, using an analytical technique, namely the optimal homotopy asymptotic method. Our procedure, which does not imply the presence of a small or large parameter in the equation or in the boundary/initial conditions, is based on the construction and determination of the linear operators and of the auxiliary functions, combined with a convenient way to optimally control the convergence of the solution. The efficiency of the proposed procedure is proven, while an accurate solution is explicitly analytically obtained in an iterative way after only one iteration. The validity of this method is demonstrated by comparing the results obtained with the numerical solution.

2 Equations of motion

Consider an unsteady, two dimensional flow on a continuous stretching surface. If u and v are velocity components, T is the temperature and k is the thermal conductivity, then the governing time-dependent equations for the continuity, momentum and thermal energy are [8, 10, 15, 17, 18]:

∂u∂x+∂v∂y=0(1)

∂u∂t+u∂u∂x+v∂v∂y=v∂2u∂y2(2)

∂T∂t+u∂T∂x+v∂T∂y=k∂2T∂y2(3)

The appropriate boundary conditions are:

u=u0xl1+γt,v=0,T=T∞+T0(1+γt)c(xl)naty=0(4)

u→0,T→T∞aty→∞(5)

where u₀, T₀, T_∞, γ are positive constants, c and n are arbitrary and l is a reference length.

If Re=u0lv and Pr=vk are the Reynolds number and the Prandtl number respectively and if we choose a stream function Ψ(x, y) such that:

u=∂Ψ∂y,v=−∂Ψ∂x(6)

then equation (1) of continuity is satisfied. The mathematical analysis of equations (2) and (3) is simplified by introducing the following similarity transformation:

Ψ=xlf(η)Re(1+γt)1/2(7)

η=Reyl(1+γt)1/2(8)

T=T∞+T0(xl)nθ(η)(1+γt)c(9)

where T₀ is a reference temperature. In this way equations (6) can be written in the form:

u(x,y,t)=u0lx(1+γt)f′(η)(10)

v(x,y,t)=−u0Re(1+γt)1/2f(η)(11)

where prime denotes differentiation with respect to η.

Substituting equations (7), (8), (9), (10) and (11) into equations (2) and (3), we obtain

f‴+ff″−f′2+Λ(f′+12ηf″)=0(12)

1Prθ″+fθ′−nf′θ+Λ(cθ+12ηθ′)=0.(13)

Here Λ=γlu0 is a dimensionless measure of the unsteadiness.

The dimensional boundary conditions (4) and (5) become:

u=u0lx(1+γt)f′(0)aty=0(14)

v=−u0Re(1+γt)1/2f(0)aty=0(15)

T=T∞+T0(xl)nθ(0)(1+γt)caty=0(16)

such that for the dimensionless functions f and θ, the boundary/initial conditions become:

f(0)=fw,f′(0)=1,f′(∞)=0(17)

θ(0)=1,θ(∞)=0.(18)

In addition to the boundary conditions (17) and (18), the requirements

f′(η)≥0,θ(η)≥0,∀η≥0(19)

must also satisfied [17].

3 Basic ideas of optimal homotopy asymptotic method

Equations (12) (or (13)) with boundary conditions (17) (or (18)) can be written in a more general form:

N(Φ(η))=0(20)

where N is a given nonlinear differential operator depending on the unknown function Φ(η), subject to the initial/boundary conditions:

B(Φ(η),dΦ(η)dη)=0.(21)

It is clear that Φ(η) = f(η) or Φ(η) = θ(η).

Let Φ₀(η) be an initial approximation of Φ(η) and L an arbitrary linear operator such as:

L(Φ0(η))=0,B(Φ0(η),dΦ0(η)dη)=0.(22)

We remark that this operator L is not unique.

If p ∈ [0, 1] denotes an embedding parameter and F is a function, then we propose to construct a homotopy [29] - [33]:

H[L(F(η,p)),H(η,Ci),N(F(η,p))](23)

with the following two properties:

H[L(F(η,0)),H(η,Ci),N(F(η,0))]==L(F(η,0))=L(Φ0(η))(24)

H[L(F(η,1)),H(η,Ci),N(F(η,1))]=H(η,Ci)N(Φ(η))(25)

where H(η, C_i) ≠ 0, is an arbitrary auxiliary convergence-control function depending on variable η and on a number of arbitrary parameters C₁, C₂, …, C_m which ensure the convergence of the approximate solution.

Let us consider the function F in the form:

F(x,p)=Φ0(η)+pΦ1(η,Ci)+p2Φ2(η,Ci)+…(26)

By substituting equation (26) into the equation obtained by means of the homotopy (23), then:

H[L(F(η,p)),H(η,Ci),N(F(η,p))]=0(27)

and equating the coefficients of like powers of p, we obtain the governing equation of Φ₀(x) given by equation (22) and the governing equation of Φ₁(η, C_i), Φ₂(η, C_i) and so on. If the series (26) is convergent at p = 1, one has:

F(η,1)=Φ0(η)+Φ1(η,Ci)+Φ2(η,Ci)+…(28)

But in particular we consider only the first-order approximate solution:

Φ¯(η,Ci)=Φ0(η)+Φ1(η,Ci),i=1,2,…,m(29)

and the homotopy (23) in the form:

H[L(F(η,p)),H(η,Ci),N(F(η,p))]=L(itΦ0(η))++p[L(Φ1(η,Ci))−L(Φ0(η))−H(η,Ci)N(Φ0(η))].(30)

Equating only the coefficients of p⁰ and p¹ into equation (30), we obtain the governing equation of Φ₀(η) given by equation (22) and the governing equation of Φ₁(η, C_i) i.e.:

L(Φ1(η,Ci))=H(η,Ci)N(Φ0(η)),B(Φ1(η,Ci),dΦ1(η,Ci)dη)=0,i=1,2,…,m.(31)

It should be emphasized that Φ₀(η) and Φ₁(η, C_i) are governed by the linear equations (22) and (23) respectively, with boundary conditions that come from the original problem, which can be easily solved. The convergence of the approximate solution (29) depends upon the auxiliary convergence-control function H(η, C_i). There are many possibilities to choose the auxiliary function H(η, C_i). Basically, the shape of H(η, C_i) must follow the terms appearing in equation (31). Therefore, we try to choose H(η, C_i) so that in equation (31) the product H(η, C_i) N(Φ₀(η)) will be the same shape with N(Φ₀(η)). Now, by substituting equation (29) into equation (20), the following residual is given:

R(η,Ci)=N(Φ¯(η,Ci)).(32)

At this moment, the first-order approximate solution given by equation (29) depends on the parameters C₂, C₂, …, C_m and these parameters can be optimally identified via various methods, such as the least square method, the Galerkin method, the Kantorowich method, the collocation method or by minimizing the square residual error:

J(C1,C2,…,Cm)=∫abR2(η,C1,C2,…,Cm)dη(33)

where a and b are two values depending on the given problem. The unknown parameters C₂, C₂, …, C_m can be identified from the conditions:

∂J∂C1=∂J∂C2=…=∂J∂Cm=0.(34)

With these parameters known (namely convergence-control parameters), the first-order approximate solution (29) is well-determined.

4 Application of OHAM to flow and heat transfer

We use the basic ideas of the OHAM by considering equation (12) with the boundary conditions given by equation (17). We can choose the linear operator in the form:

Lf(Φ(η))=Φ‴−K2Φ′,(35)

where K > 0 is an unknown parameter at this moment.

Here, we state that the linear operator is not unique.

Equation (22) becomes:

Φ0‴−K2Φ0′=0,Φ0(0)=fw,Φ0′(0)=1,Φ0′(∞)=0.

which has the following solution:

Φ0(η)=fw+1−e−KηK.(36)

The nonlinear operator N_f(Φ(η)) is obtained from equation (12):

Nf(Φ(η))=Φ‴(η)+Φ(η)Φ″(η)−Φ′(η)2+Λ(Φ′(η)+12ηΦ″(η))=0(37)

such that substituting equation (36) into equation (37), we obtain

Nf(Φ0(η))=(αη+β)e−Kη(38)

where

α=12KΛ;β=K2−1−Kfw−Λ.(39)

Having in view that in equation (38) there is an exponential function and that the auxiliary function H_f(η, C_i) must follow the terms appearing in equation (38), then we can choose the function H_f(η, C_i) in the following forms:

Hf(η,Ci)=C1+C2η+(C3+C4η)e−Kη+(C5+C6η)e−2Kη(40)

Hf∗(η,Ci)=C1+(C2+C3η+C4η2)e−Kη(41)

or yet

Hf∗∗(η,Ci)=C1+C2η+C3η2+(C4+C5η)e−Kη+(C6+C7η+C8η2)e−2Kη(42)

and so on, where C₂, C₂, … are unknown parameters at this moment.

If we choose only the expression (40) for H_f(η, C_i), then by using equations (38), (40) and (31), we can obtain the equation in Φ₁(η, C_i):

Φ1‴−K2Φ1′=[βC1+(αC1+βC2)η+αC2η2]e−Kη+[βC3+(αC3+βC4)η+αC4η2]e−2Kη+[βC5+(αC5+βC6)η+αC6η2]e−3Kη,Φ1(0)=Φ1′(0)=Φ1′(∞)=0.(43)

The solution of equation (43) can be found as:

Φ1(η)=M1+[N1+(7αC24K4+3αC14K3+3βC24K3+βC12K2)η+(3αC24K3+αC14K2+βC24K2)η2+αC26K2η3]e−Kη+[−85αC4108K5−11αC336K4−11βC436K4−βC36K3−(11αC418K4+αC36K3+βC46K3)η−αC46K3η2]e−2Kη+(−115αC61728K5−13αC5288K4−13βC6288K4−βC524K3)e−Kη(44)

where

M1=−3α+2Kβ4K4C1−7α+3Kβ4K5C2−5α+6Kβ36K4C3−−19α+15Kβ108K5C4−7α+12Kβ144K4C5−37α+42Kβ864K5C6N1=3α+2Kβ4K4C1+7α+3Kβ4K5C2+4α+3Kβ9K4C3++26α+12Kβ27K5C4+3α+4Kβ32K4C5+7α+6Kβ64K5C6.(45)

The first-order approximate solution (29) for equations (12) and (17) is obtained from equations (36) and (45):

f¯(η)=Φ¯(η)=Φ0(η)+Φ1(η).(46)

In what follows, we consider equations (13) and (18). In this case, we choose the linear operator in the form:

Lθ(φ(η))=φ″+Kφ′(47)

where the parameter K is defined in equation (35).

Equation (22) becomes:

φ0″+Kφ0′=0,φ0(0)=1,φ0(∞)=0.(48)

Equation (48) has the solution

φ0(η)=e−Kη.(49)

The nonlinear operator N_θ(φ(η)) is obtained from equation (12):

Nθ(φ(η))=1Prφ″+Φφ′−nΦ′φ+Λ(cφ+12ηφ′).(50)

Substituting equation (49) into equation (50), we obtain:

Nθ(φ0(η))=(m1η+m2)e−Kη+m3e−2Kη(51)

where

m1=−12KΛ;m2=K2Pr−Kfw−1+cΛ;m3=1−n.(52)

The auxiliary function H_θ(η, C_i) can be chosen in the forms:

Hθ(η,Ci)=C7+C8η+(C9+C10η)e−Kη+(C11+C12η)e−2Kη(53)

Hθ∗(η,Ci)=C7+C8η+C9η2+(C10+C11η)e−Kη+C13e−2Kη(54)

or yet

Hθ∗∗(η,Ci)=C7+(C8+C9η)e−Kη+(C10+C11η)e−2Kη(55)

and so on, where C₂, C₂, … are unknown parameters.

If we choose equation (53) for H_θ, then from equations (51), (53) and (31) we obtain the equation in φ₁(η, C_i) as

φ1″+Kφ1′=[m2C7+(m1C7+m2C8)η+m1C8η2]⋅e−Kη+[m2C9+m3C7+(m1C9+m2C10+m3C8)η+m1C10η2]e−2Kη+[m3C9+m2C11+(m3C10+m1C11+m2C12)η+m1C12η2]e−3Kη+(m3C11+m3C12η)e−4Kη,ϕ1(0)=ϕ1(∞)=0.(56)

Solving equation (56), we obtain:

φ1(η)=[P1−(2m1C8K3+m1C7K2+m2C8K2+m2C7K)η−(m1C8K2+m1C72K+m2C82K)η2−m1C83Kη3]e−Kη+[7m1C104K4+3m1C94K3+3m2C104K3+3m3C84K3+m2C92K2+m3C72K2+(3m1C102K3+m1C92K2+m2C102K2+m3C82K2)η+m1C102K2η2]e−2Kη+[5(m3C10+m1C11+m2C12)36K3+m3C9+m2C116K2+(m3C10+m1C11+m2C126K2+5m1C1218K3)η+m1C126K2η2]e−3Kη+(7m3C1112K2+7m3C12144K3+m3C1212K2η)e−4Kη,ϕ1(0)=ϕ1(∞)=0,(57)

where

P1=−m3C72K2−3m3C84K3−9m1+6Km2+2Km312K3C9−63m1+27Km2+5Km336K4C10−5m1+K(m2+3m3)36K3C11−20m2+7m3144K3C12.(58)

In this way, the first-order approximate solution (29) for equations (13) and (18) becomes

θ¯(η)=φ¯(η)=φ0(η)+φ1(η,Ci).(59)

5 Numerical examples

In order to prove the accuracy of the obtained results, we will determine the convergence-control parameters K and C_i which appear in equations (46) and (59) by means of the least square method. In this way, the convergence-control parameters are optimally determined and the first-order approximate solutions known for different values of the known parameters f_w, Λ, Pr, n and c. In what follows, we illustrate the accuracy of the OHAM by comparing previously obtained approximate solutions with the numerical integration results computed by means of the shooting method combined with fourth-order Runge-Kutta method using Wolfram Mathematica 6.0 software. For some values of the parameters f_w, Λ, Pr, n and c we will determine the approximate solutions.

Example 5.1.a

In the first case we consider f_w = −1, Λ = 1, c=12, n = 1, Pr = 0.7. For equation (46), following the procedure described above, the following convergence-control parameters are obtained:

C1=−0.0881661632,C2=0.0159074525,C3=101.5499315816,C4=−16.3157319695,C5=−99.5951678657,C6=−64.5910051875K=0.7591636981

and consequently the first-order approximate solution (46) can be written in the form:

f¯(η)=0.4921333156+(−0.5668554881+0.0071536463η−0.0087518103η2+0.0017461596η3)e−0.7591636981η+(−0.3980837570−7.4190413787η+2.3591440003η2)e−1.5183273963η+(−0.5271940704+6.1764503488η+2.3348551362η2)e−2.2774910945η(60)

Now, for equation (59), the convergence-control parameters are:

C7=0.0363993085,C8=0.0363993085,C9=−7.1448075448,C10=4.3237724702,C11=42.1871800319,C12=18.0805214975

and therefore the first-order approximate solution (59) becomes:

θ¯(η)=(0.3541683003+0.2415876957η−0.0823906873η2++0.0060665514η3)e−0.7591636981η+(−2.6848440528+6.3660521866η−1.4238603876η2)e−1.5183273963η++(3.3306757524−3.3281240073η−1.9846972772η2)e−2.2774910945η(61)

In Tables 1 and 2 we present a comparison between the first-order approximate solutions given by equations (60) and (61) respectively, with numerical results for some values of variable η and the corresponding relative errors.

Table 1

Comparison between OHAM results given by equation (60) and numerical results for f_w = −1, Λ = 1.

η	f_numeric	f¯OHAM, Eq. (60)	relative error = \|fnumeric−f¯OHAM\|
0	−1	−0.9999999999	1.88 ·10⁻¹⁵
1	−0.1497942276	−0.1501421749	3.47 ·10⁻⁴
2	0.3108643384	0.3106655367	1.98 ·10⁻⁴
3	0.4604991620	0.4600705386	4.28 ·10⁻⁴
4	0.4887865463	0.4894195278	6.32 ·10⁻⁴
5	0.4919308455	0.4920394273	1.08 ·10⁻⁴
6	0.4921388939	0.4918417250	2.97 ·10⁻⁴
7	0.4921471111	0.4919782168	1.68 ·10⁻⁴
8	0.4921472622	0.4922151064	6.78 ·10⁻⁵
9	0.4921472290	0.4923440748	1.96 ·10⁻⁴
10	0.4921472001	0.4923640175	2.16 ·10⁻⁴

Table 2

Comparison between OHAM results given by equation (61) and numerical results for f_w = −1, Λ = 1, c=12, n = 1, Pr = 0.7.

η	θ_numeric	θ¯OHAM, Eq. (61)	relative error = \|θnumeric−θ¯OHAM\|
0	1	0.9999999999	8.88 ·10⁻¹⁶
1	0.5325816311	0.5344078544	1.82 ·10⁻³
2	0.2137609331	0.2123010871	1.45 ·10⁻³
3	0.0624485224	0.0627998626	3.51 ·10⁻⁴
4	0.0129724736	0.0141235139	1.15 ·10⁻³
5	0.0019027817	0.0018865990	1.61 ·10⁻⁵
6	0.0001968219	−0.0002871987	4.84 ·10⁻⁴
7	0.0000144329	−0.0002516924	2.66 ·10⁻⁴
8	8.27 ·10⁻⁷	0.0000468974	4.60 ·10⁻⁵
9	1.08 ·10⁻⁷	0.0002281868	2.28 ·10⁻⁴
10	7.47 ·10⁻⁸	0.0002807824	2.80 ·10⁻⁴

Example 5.1.b

In this case, we consider f_w = −1, Λ = 1, c=12, n = 1, Pr = 2. The solution f¯(η) is given by equation (60). The convergence-control parameters for equation (59) are:

C7=0.3992391297,C8=−0.0398233823,C9=13.9195482545,C10=−9.8140323466,C11=−37.1866029355,C12=−48.0634410813

such that the first-order approximate solution (59) becomes:

θ¯(η)=(−2.1207041376−0.0561685488η+0.0879369070η2−0.0066372303η3)⋅e−0.7591636981η+e−1.5183273963η(7.9716346529−−9.1921128520η+3.2318564396η2)e−0.7591636981η+(−4.8509305153+8.0572360536η+5.2759197606η2)e−2.2774910945η(62)

In Table 3 we present a comparison between the first-order approximate solutions given by equation (62) with numerical results and corresponding relative errors.

Table 3

Comparison between OHAM results given by equation (62) and numerical results for f_w = −1, Λ = 1, c=12, n = 1, Pr = 2.

η	θ_numeric	θ¯OHAM, Eq. (62)	relative error = \|θnumeric−θ¯OHAM\|
0	1	0.9999999999	1.77 ·10⁻¹⁵
1	0.3341908857	0.3295767993	4.61 ·10⁻³
2	0.0347540513	0.0372485103	2.49 ·10⁻³
3	0.0011305745	−0.0002320988	1.36 ·10⁻³
4	0.0000127262	−0.0002852788	2.98 ·10⁻⁴
5	−5.44 ·10⁻⁸	0.0002991504	2.99 ·10⁻⁴
6	−9.15 ·10⁻⁸	0.0002884740	2.88 ·10⁻⁴
7	−7.96 ·10⁻⁸	0.0001372878	1.37 ·10⁻⁴
8	−7.05 ·10⁻⁸	−0.0000295212	2.94 ·10⁻⁵
9	−6.53 ·10⁻⁸	−0.0001505702	1.50 ·10⁻⁴
10	−5.99 ·10⁻⁸	−0.0002044072	2.04 ·10⁻⁴

Example 5.2.a

For f_w = 0, Λ = 1, c=12, n = 1, Pr = 0.7, the convergence-control parameters for equation (46) are:

C1=−1.2640611927,C2=0.1680009020,C3=−34.0575215187,C4=30.7898356526,C5=37.1281425060,C6=13.8590545976,K=1.1203766872

and therefore, the first-order approximate solution (46) can be written in the form:

f¯(η)=0.9662722752+(1.3563995648+0.0351604322η−0.1157605059η2+0.0124958644η3)e−1.1203766872η+(−2.5490820161−1.7111113870η−2.0440805123η2)e−2.2407533744η+(0.2264101761−−0.7552406743η−0.2300192808η2)e−3.3611300616η(63)

For equation (59), the convergence-control parameters are:

C7=−1.6947892627,C8=0.2632100295,C9=−1.2815579214,C10=2.6268699384,C11=15.8897673738,C12=9.5966071873

and the first-order approximate solution (59) is:

θ¯(η)=(0.2987405005+1.1383970916η−0.4581387031η2+0.0438683382η3)⋅e−1.1203766872η+(−0.1000278783++1.4818560845η−0.5861577557η2)⋅e−2.2407533744η+(0.8012873778−−0.5959066165η−0.7137932043η2)⋅e−3.3611300616η(64)

In Tables 4 and 5 we present a comparison between the first-order approximate solutions given by equations (63) and (64) respectively, with numerical results and corresponding relative errors.

Table 4

Comparison between OHAM results given by equation (63) and numerical results for f_w = 0, Λ = 1.

η	f_numeric	f¯OHAM, Eq. (63)	relative error = \|fnumeric−f¯OHAM\|
0	−5.50 ·10⁻²¹	4.44 ·10⁻¹⁶	4.44 ·10⁻¹⁶
1	0.6894348341	0.6894914970	5.66 ·10⁻⁵
2	0.9167696529	0.9166682157	1.01 ·10⁻⁴
3	0.9608821303	0.9609858144	1.03 ·10⁻⁴
4	0.9659196704	0.9659030730	1.65 ·10⁻⁵
5	0.9662619960	0.9661631962	9.87 ·10⁻⁵
6	0.9662759513	0.9662663279	9.62 ·10⁻⁶
7	0.9662762950	0.9663395358	6.32 ·10⁻⁵
8	0.9662763018	0.9663501433	7.38 ·10⁻⁵
9	0.9662763032	0.9663306688	5.43 ·10⁻⁵
10	0.9662763043	0.9663080318	3.17 ·10⁻⁵

Table 5

Comparison between OHAM results given by equation (64) and numerical results for f_w = 0, Λ = 1, c=12, n = 1, Pr = 0.7.

η	θ_numeric	θ¯OHAM, Eq. (64)	relative error = \|θnumeric−θ¯OHAM\|
0	1	1.00	2.22 ·10⁻¹⁶
1	0.4003127445	0.4006174190	3.04 ·10⁻⁴
2	0.1184290061	0.1183366534	9.23 ·10⁻⁵
3	0.0250358122	0.0254649591	4.29 ·10⁻⁴
4	0.0037386526	0.0032572108	4.81 ·10⁻⁴
5	0.0003935287	−0.0000242886	4.17 ·10⁻⁴
6	0.0000291963	0.0001165652	8.73 ·10⁻⁵
7	1.53 ·10⁻⁶	0.0003370019	3.35 ·10⁻⁴
8	6.39 ·10⁻⁸	0.0003255701	3.25 ·10⁻⁴
9	8.43 ·10⁻⁹	0.0002261157	2.26 ·10⁻⁴
10	6.39 ·10⁻⁹	0.0001326393	1.32 ·10⁻⁴

Example 5.2.b

For f_w = 0, Λ = 1, c=12, n = 1, Pr = 2 the firs-order approximate solution (46) is given by equation (63). The convergence-control parameters for equation (59) are determined as:

C7=0.4652101281,C8=−0.0724620728,C9=14.4924736065,C10=−12.1643720147,C11=0.7277740132,C12=−56.0450834796

such that the first-order approximate solution (59) may be written as:

θ¯(η)=(−1.4030799687+0.1042610535η+0.0880913412η2−0.0120770121η3)⋅e−1.1203766872η+(3.1476673319−−3.8089261057η+2.7143486992η2)⋅e−2.2407533744η+(−0.7445873632++5.1973912018η+4.1686190694η2)e−3.3611300616η(65)

In Table 6 we present a comparison between the first-order approximate solutions given by equation (65) with numerical results. The corresponding relative errors are also presented.

Table 6

Comparison between OHAM results given by equation (65) and numerical results for f_w = 0, Λ = 1, c=12, n = 1, Pr = 2.

η	θ_numeric	θ¯OHAM, Eq. (65)	relative error = \|θnumeric−θ¯OHAM\|
0	1	1	0
1	0.1197281855	0.1187072441	1.02 ·10⁻³
2	0.0042249398	0.0041010249	1.23 ·10⁻⁴
3	5.09 ·10⁻⁵	−6.04 ·10⁻⁶	5.69 ·10⁻⁵
4	2.41 ·10⁻⁷	0.0001841967	1.83 ·10⁻⁴
5	1.67 ·10⁻⁸	0.0000163574	1.63 ·10⁻⁵
6	1.42 ·10⁻⁸	−0.0001452859	1.45 ·10⁻⁴
7	1.26 ·10⁻⁸	−0.0001791057	1.79 ·10⁻⁴
8	1.15 ·10⁻⁸	−0.0001403309	1.40 ·10⁻⁴
9	1.09 ·10⁻⁸	−0.0000887807	8.87 ·10⁻⁵
10	1.05 ·10⁻⁸	−0.0000493843	4.93 ·10⁻⁵

Example 5.3.a

We consider f_w = 1, Λ = 1, c=12, n = 1, Pr = 0.7. The convergence-control parameters for equation (46) are given by:

C1=0.6287723857,C2=−0.1379103919,C3=−64.6127509553,C4=52.1862259014,C5=65.7049797786,C6=66.9471031457,K=1.6976766716.

such that the first-order approximate solution (46) may be written as:

f¯(η)=1.6119245343+(−0.1095284867−0.0145745495η+0.0381089526η2−0.0067695650η3)e−1.6976766716η+(−0.6841717456+0.0590171180η−1.5089146740η2)e−3.3953533432η+(0.1817756979−0.6276022634η−0.4839278208η2)e−5.0930300148η(66)

The convergence-control parameters for equation (59), are:

C7=−2.4317156290,C8=0.4199773974,C9=3.2538989273,C10=2.8819189890,C11=2.6307320394,C12=0.0503403055

and the first-order approximate solution (59) becomes:

θ¯(η)=(0.0191058146+1.8994242629η−0.7216780879η2+0.0699962329η3)⋅e−1.6976766716η+(0.9928667494++0.3712879543η−0.4243916166η2)⋅e−3.3953533432η+(−0.0119725641−−0.1259716711η−0.0024710391η2)⋅e−5.0930300148η(67)

In Tables 7 and 8 we present a comparison between the first-order approximate solutions given by equations (66) and (67) respectively, with numerical results and corresponding relative errors.

Table 7

Comparison between OHAM results given by equation (66) and numerical results for f_w = 1, Λ = 1.

η	f_numeric	f¯OHAM, Eq. (66)	relative error = \|fnumeric−f¯OHAM\|
0	1	1.00	2.22 ·10⁻¹⁶
1	1.5177074192	1.5176780223	2.93 ·10⁻⁵
2	1.6030516967	1.6030350430	1.66 ·10⁻⁵
3	1.6114161917	1.6114348208	1.86 ·10⁻⁵
4	1.6119056438	1.6119031833	2.46 ·10⁻⁶
5	1.6119228465	1.6119073012	1.55 ·10⁻⁵
6	1.6119232066	1.6119136285	9.57 ·10⁻⁶
7	1.6119232084	1.6119199331	3.27 ·10⁻⁶
8	1.6119232063	1.6119229505	2.55 ·10⁻⁷
9	1.6119232045	1.6119240509	8.46 ·10⁻⁷
10	1.6119232031	1.6119243982	1.19 ·10⁻⁶

Table 8

Comparison between OHAM results given by equation (67) and numerical results for f_w = 1, Λ = 1, c=12, n = 1, Pr =0.7.

η	θ_numeric	θ¯OHAM, Eq. (67)	relative error = \|θnumeric−θ¯OHAM\|
0	1	1	0
1	0.2625870984	0.2626175913	3.04 ·10⁻⁵
2	0.0500304293	0.0500306615	2.32 ·10⁻⁷
3	0.0067456425	0.0067634155	1.77 ·10⁻⁵
4	0.0006411532	0.0006125219	2.86 ·10⁻⁵
5	0.0000429270	0.0000457402	2.81 ·10⁻⁶
6	2.01 ·10⁻⁶	2.08 ·10⁻⁵	1.88 ·10⁻⁵
7	5.14 ·10⁻⁸	1.35 ·10⁻⁵	1.34 ·10⁻⁵
8	−1.29 ·10⁻⁸	6.14 ·10⁻⁶	6.16 ·10⁻⁶
9	−1.33 ·10⁻⁸	2.24 ·10⁻⁶	2.25 ·10⁻⁶
10	−1.22 ·10⁻⁸	7.13 ·10⁻⁷	7.25 ·10⁻⁷

Example 5.3.b

For f_w = 1, Λ = 1, c=12, n = 1, Pr = 2 the first-order approximate solution for f¯(η) is given by equation (66).

For equation (59) the convergence-control parameters are given by:

C7=−0.3023817542,C8=0.0519503347,C9=−27.1653468182,C10=23.1898068679,C11=−15.3543205956,C12=31.0226487950.

such that the first-order approximate solution (59) may be written as:

θ¯(η)=(0.9205703595−0.1921602046η−0.0487183578η2+0.0086583891η3)⋅e−1.6976766716η+(0.2637846962++0.9429053366η−3.4149327807η2)⋅e−3.3953533432η+(−0.1843550558−−2.0986586159η−1.5227992326η2)⋅e−5.0930300148η(68)

In Table 9 we present a comparison between the first-order approximate solutions given by equations (68) with numerical results. The corresponding relative errors are presented.

Table 9

Comparison between OHAM results given by equation (68) and numerical results for f_w = 1, Λ = 1, c=12, n = 1, Pr = 2.

η	θ_numeric	θ¯OHAM, Eq. (68)	relative error = \|θnumeric−θ¯OHAM\|
0	1	1.00	2.22 ·10⁻¹⁶
1	0.0288461240	0.0286378502	2.08 ·10⁻⁴
2	0.0002627867	0.0004342087	1.71 ·10⁻⁴
3	1.00 ·10⁻⁶	−1.91 ·10⁻⁴	1.91 ·10⁻⁴
4	1.21 ·10⁻⁷	−1.46 ·10⁻⁴	1.46 ·10⁻⁴
5	1.05 ·10⁻⁷	−3.96 ·10⁻⁵	3.97 ·10⁻⁵
6	9.39 ·10⁻⁸	−4.54 ·10⁻⁶	4.63 ·10⁻⁶
7	8.68 ·10⁻⁸	1.08 ·10⁻⁶	9.96 ·10⁻⁷
8	7.95 ·10⁻⁸	8.82 ·10⁻⁷	8.02 ·10⁻⁷
9	7.44 ·10⁻⁸	3.60 ·10⁻⁷	2.85 ·10⁻⁷
10	6.91 ·10⁻⁸	1.18 ·10⁻⁷	4.89 ·10⁻⁸

In Figs 1 and 2 are plotted the profiles of f¯(η) and velocity profile f¯′(η) respectively for different values of f_w. It is clear that the solution f¯(η) increases with an increase of f_w and the velocity decreases with an increase of f_w. The condition f¯′(η)>0 for η > 0 is satisfied.

$Figure 1 Solutions f¯OHAM(η)$\overline{f}_{OHAM}(\eta)$ given by (60), (63) and (66) for different values of fw—numerical solution; ⋯ OHAM solution.$