Heat transfer analysis of Williamson fluid over exponentially stretching surface

Appl. Math. Mech. -Engl. Ed., 35(4), 489–502 (2014)DOI 10.1007/s10483-014-1807-6c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2014

Applied Mathematicsand Mechanics(English Edition)

Heat transfer analysis of Williamson fluid over exponentiallystretching surface∗

S. NADEEM, S. T. HUSSAIN

(Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan)

Abstract This study explores the effects of heat transfer on the Williamson fluid over aporous exponentially stretching surface. The boundary layer equations of the Williamsonfluid model for two dimensional flow with heat transfer are presented. Two cases of heattransfer are considered, i.e., the prescribed exponential order surface temperature (PEST)case and the prescribed exponential order heat flux (PEHF) case. The highly nonlinearpartial differential equations are simplified with suitable similar and non-similar variables,and finally are solved analytically with the help of the optimal homotopy analysis method(OHAM). The optimal convergence control parameters are obtained, and the physical fea-tures of the flow parameters are analyzed through graphs and tables. The skin frictionand wall temperature gradient are calculated.

Key words optimal homotopy analysis method (OHAM), Williamson fluid, expo-nential stretching, permeable wall, heat transfer, prescribed exponential order surfacetemperature (PEST), prescribed exponential order heat flux (PEHF)

Chinese Library Classification TD9532010 Mathematics Subject Classification 35Q35

1 Introduction

In many practical situations, the stretching surface does not need to be linear, e.g., in plasticsheet stretching. The heat transfer analysis of boundary layer flow over a continuous stretchingsurface with prescribed temperature or heat flux has gained considerable attention due to itsapplications in manufacturing processes of polymer sheets, glass fibre, paper production, metalwires, plastic films, etc. In polymer, the glass and plastic industry quality of the final productgreatly depends on the rate of cooling. Sakiadis[1] was the first to study the boundary layerflow over a continuous stretching surface. He developed the two dimensional boundary layerequations. Tsou et al.[2] examined the heat transfer effects on the boundary layer flow over astretching surface. Erickson et al.[3] extended this work for mass transfer by considering suctionand injection. Later on, many researchers have given their insight into boundary layer flowsover linear stretching surfaces[4–6].

Kumaran and Ramanaiah[7] considered the quadratic stretching on the viscous fluid flowover a stretching surface, and obtained the closed form solutions. Ali[8] considered power lawstretching and temperature to investigate the thermal boundary layer. Elbashbeshy[9] dis-cussed the flow and heat transfer of viscous fluid by considering the exponential stretching.

∗ Received Nov. 17, 2012 / Revised Nov. 18, 2013Project supported by the Ph.D. Indigenous Scheme of the Higher Education Commission of Pakistan(No. 112-21674-2PS1-576)Corresponding author S. NADEEM, Associate Professor, Ph.D., E-mail: [email protected]

490 S. NADEEM and S. T. HUSSAIN

Sanjayanand and Khan[10] extended this work for heat and mass transfer of viscoelastic fluidby considering the viscous dissipation and elastic deformation. Magyari and Keller[11] devel-oped the numerical solution for heat and mass transfer of viscous fluid over an exponentiallystretching surface. Nadeem et al.[12] discussed the thermal radiation effects of Jeffery fluidover an exponentially stretching surface, and compared the homotopy analysis method (HAM)results with the numerical results obtained by Magyari and Keller[11]. More recently, Nadeemand Lee[13] studied the nano particle effects on the boundary layer flow of viscous fluid due toexponential stretching.

HAM is a popular technique among researchers[14–17]. The optimal HAM (OHAM) is arefined and better version of HAM[14] . There are various types of OHAM, i.e., basic OHAM,three parameter OHAM, finite parameter OHAM, and infinite parameter OHAM. All thesetypes are discussed in detail by Liao[18]. We employ the basic OHAM to obtain the solution.In the basic OHAM, the squared residual error is minimized to obtain the optimized valuesof the convergence control parameters. Liao[19] presented the comparison of various OHAMapproaches. He suggested that the basic OHAM and the three parameter OHAM are better thanother OHAM approaches when compared in respect to the computational efficiency. Wang[20]

used the three parameter OHAM to obtain the solution of the Kawahara equation. Nadjafi andJafari[21] compared Liao’s OHAM with Niu’s one-step OHAM[22]. They used both techniques toobtain the solutions of linear Volterra integro-differential equations, and presented that Liao’sOHAM is more accurate to determine the convergence-control parameter. Fan and You[23]

discussed in detail the global and step by step approaches to obtain the optimized convergencecontrol parameter.

However, the flow of Williamson fluid over an exponentially stretching sheet has not beenconsidered. In the present paper, we discuss the heat transfer analysis of pseudoplastic fluidover a porous exponentially stretching sheet with the help of the Williamson fluid model[24–29].The governing boundary layer equations are first simplified by using suitable similarity trans-formations. The resulting equations are solved analytically by OHAM for two cases, i.e., pre-scribed exponential order surface temperature (PEST) and prescribed exponential order heatflux (PEHF). The convergence of the obtained solutions is examined with the help of averagesquared residual error tables. Graphs are plotted to observe the effects of the Williamson param-eter, the suction/injection parameter, and the Prandtl number on the velocity and temperatureprofiles. Finally, tables are drawn for the skin friction and wall temperature gradient.

2 Fluid model

For the Williamson fluid model, the Cauchy stress tensor S is defined as[29]

S = −pI + τ, (1)

τ =(μ∞ +

μ0 − μ∞1 − Γγ̇

)A1, (2)

where τ is the extra stress tensor, μ0 is the limiting viscosity at zero shear rate, μ∞ is thelimiting viscosity at the infinite shear rate, Γ > 0 is a time constant, A1 is the first Rivlin-Erickson tensor, and γ̇ is defined as

γ̇ =

√12π, π = trace(A2

1). (3)

Here, we consider the case for which

μ∞ = 0, Γγ̇ < 1.

Heat transfer analysis of Williamson fluid over exponentially stretching surface 491

Then, we obtain

τ =μ0

1 − Γγ̇A1 (4)

or

τ = μ0(1 + Γγ̇)A1. (5)

2.1 Mathematical formulationLet us consider a steady, two dimensional flow of an incompressible Williamson fluid over an

exponentially stretching porous surface. The plate is exponentially stretched along the x-axiswith a velocity Uw = U◦ exp(x

l ) at y = 0. Here, we assume the no slip boundary condition.The fluid near the wall has the velocity Uw and the temperature Tw. The stretched plate isalso assumed to be porous. v = −β(x) is the velocity on the wall. The schematic flow diagramis shown in Fig. 1. The governing boundary layer equations for flow and heat transfer in theabsence of body force and viscous dissipation can be written as[30]

∂u

∂x+

∂v

∂y= 0, (6)

u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2+√

2νΓ∂u

∂y

∂2u

∂y2, (7)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2, (8)

where u(x, y) and v(x, y) are the velocity components along the flow direction and normal tothe flow direction, respectively, T is the fluid temperature, and α is the thermal diffusivity. Thecorresponding boundary conditions are⎧⎪⎨

⎪⎩u = Uw = U◦exp

(x

l

), v = −β(x), T = Tw at y = 0,

u → 0, T = T∞ as y → ∞,

(9)

where ν is the kinematic viscosity, Uw is the velocity at wall, β(x) > 0 is the suction velocity,and β(x) < 0 is the injection velocity. Tw and T∞ are the wall and ambient fluid temperature,respectively. Our interest is to consider two special types of heat transfer problems, i.e., thePEST case and the PEHF case.

Fig. 1 Schematic diagram of boundary layer flow over stretching sheet

The boundary conditions for the above mentioned cases are

T = Tw = T∞ + T0exp( x

2l

)at y = 0 for the PEST case, (10)

− k(∂T

∂y

)w

= T1exp( x

2l

)at y = 0 for the PEHF case. (11)


For both cases,

T → T∞ as y → ∞. (12)

Introduce the following transformations in Eqs. (7)–(9):⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

u = U◦exp(x

l

)f ′(η),

v = −√

νU◦2l

exp( x

2l

)(f(η) + ηf ′(η)),

η =

√U◦2νl

y exp( x

2l

).

(13)

For the PEST case,

T = T∞ + T0

( x

2l

)θ(η). (14)

For the PEHF case,

T = T∞ +T1

kexp

( x

2l

)√2νl

U◦φ(η). (15)

After using the above transformations, our governing equations with the correspondingboundary equations take the following forms:⎧⎪⎪⎪⎨

⎪⎪⎪⎩f ′′′ − 2f ′2 + ff ′′ + λf ′′f ′′′ = 0,

f = −s, f ′ = 1 at η = 0,

f ′ → 0 as η → ∞,

(16)

{θ′′ + Pr(θ′f − f ′θ) = 0,

θ(0) = 1, θ(∞) = 0,(17)

{φ′′ + Pr(φ′f − f ′φ) = 0,

φ′(0) = −1, φ(∞) = 0,(18)

where s is the suction/injection parameter (s < 0 for suction and s > 0 for injection), and λ isthe dimensionless Williamson parameter defined as

λ = Γ

√U3◦ exp(3x/l)

νl. (19)

For λ = 0, Eq. (16) reduces to the classical boundary layer equation for viscous flow. Physicalquantities of interest are the coefficient of the skin friction cf and the wall temperature gradientθ′(0). After using the boundary layer approximations, τw is given by

τw = μ0

(∂u

∂y+

Γ√2

(∂u

∂y

)2)∣∣∣y=0

. (20)

The coefficient of skin friction is defined as

cf =τw

ρU2w

. (21)


In the dimensionless form, the skin friction becomes√

2Recf =(f ′′ +

λ

2f ′′2

)η=0

, (22)

where Re = Uwlν is the Reynolds number.

3 Homotopy analysis solutions

HAM is a strong analytic technique to solve linear and non-linear, ordinary and partialdifferential equations. HAM was developed by Liao in 1992. This technique unified severalprevious techniques like the Adomian decomposition method, the δ-expansion method, and theLyapunov artificial small parameter method. HAM can be equally applied to weak and strongnon-linear problems because it is independent of small/large physical parameter restriction. Italso provides a way to check and adjust the convergence of the obtained solution with the helpof auxiliary parameters and base functions. We choose the set of base functions

{ηkexp(−nη), k � 0, n � 0}. (23)

Then, we can write ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f(η) = a00, 0 +

∞∑n=0

∞∑k=0

akm, nηk exp(−nη),

θ(η) = b00, 0 +

∞∑n=0

∞∑k=0

bkm, nηk exp(−nη),

φ(η) = c00, 0 +

∞∑n=0

∞∑k=0

ckm, nηk exp(−nη),

(24)

where akm, n, bk

m, n, and ckm, n are the coefficients. With the help of the boundary conditions

defined in Eqs. (16)–(18), the initial approximation and the auxiliary linear operator are chosenas ⎧⎪⎪⎪⎨

⎪⎪⎪⎩f0(η) = 1 − s − exp(−η),

θ0(η) = exp(−η),

φ0(η) = exp(−η),

(25)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Lf (f) =d3f

dη3− df

dη,

Lθ(θ) =d2θ

dη2+

dθ

dη,

Lφ(φ) =d2φ

dη2+

dφ

dη,

(26)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Lf (C1 + C2 exp(η) + C3 exp(−η)) = 0,

Lθ(C4 + C5 exp(−η)) = 0,

Lφ(C6 + C7 exp(−η)) = 0,

(27)

where Ci (i = 1, 2, · · · , 7) are the arbitrary constants.


3.1 Deformation equationsIf q ∈ [0, 1] is the embedding parameter and the non-zero auxiliary parameter is h, then the

problem at the zeroth-order deformation is given by⎧⎪⎪⎪⎨⎪⎪⎪⎩

(1 − q)Lf (f(η; q) − f0(η)) = qhfNf (f(η; q)),

(1 − q)Lθ(θ(η; q) − θ0(η)) = qhθNθ(f(η; q), θ(η; q)),

(1 − q)Lφ(φ(η; q) − φ0(η)) = qhφNφ(f(η; q), φ(η; q)),

(28)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

(f(η; q))η=0 = −s,(∂f(η; q)

∂η

)η=0

= 1,(∂f(η; q)

∂η

)η→∞

= 0,

(θ(η; q))η=0 = 1, (θ(η; q))η→∞ = 0,

(∂φ(η; q)∂η

)η=0

= −1, (φ(η; q))η→∞ = 0,

(29)

where the nonlinear operators are defined as

Nf(f(η; q)) =∂3f(η; q)

∂η3− 2

(∂f(η; q)∂η

)2

+ f(η; q)∂2f(η; q)

∂η2+ λ

(∂3f(η; q)∂η3

) ∂2f(η; q)∂η2

,

Nθ(θ(η; q)) =∂2θ(η; q)

∂η2+ Pr

(∂θ(η; q)∂η

f(η; q) − ∂f(η; q)∂η

θ(η; q)),

Nφ(φ(η; q)) =∂2φ(η; q)

∂η2+ Pr

(∂φ(η; q)∂η

f(η; q) − ∂f(η; q)∂η

φ(η; q)).

When q = 0 and q = 1, we have⎧⎪⎪⎪⎨⎪⎪⎪⎩

f(η; 0) = f0(η), f(η; 1) = f(η),

θ(η; 0) = θ0(η), θ(η; 1) = θ(η),

φ(η; 0) = φ0(η), φ(η; 1) = φ(η).

(30)

With the help of Taylor series expansion, we can write⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f(η; q) = f0(η) +∞∑

m=1

fm(η)qm,

θ(η; q) = θ0(η) +∞∑

m=1

θm(η)qm,

φ(η; q) = φ0(η) +∞∑

m=1

φm(η)qm,

(31)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

fm(η) =1m!

∂mf(η; q)∂qm

∣∣∣q=0

,

θm(η) =1m!

∂mθ(η; q)∂qm

∣∣∣q=0

,

φm(η) =1m!

∂mφ(η; q)∂qm

∣∣∣q=0

.

(32)


The auxiliary parameter h is chosen in such a way that Eq. (31) converges at q = 1. Thus,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f(η) = f0(η) +∞∑

m=1

fm(η),

θ(η) = θ0(η) +∞∑

m=1

θm(η),

φ(η) = φ0(η) +∞∑

m=1

φm(η).

(33)

The mth-order approximation is defined as

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Lf (fm(η) − χmfm−1(η)) = hfRfm(η),

Lθ(θm(η) − χmθm−1(η)) = hθRθm(η),

Lφ(φm(η) − χmφm−1(η)) = hφRφm(η),

(34)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

fm(0) = f ′m(0) = f ′

m(∞) = 0,

θm(0) = θm(∞) = 0,

φ′m(0) = φm(∞) = 0,

(35)

where

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Rfm(η) = f ′′′

m−1 +m−1∑k=0

(fm−1−kf ′′k − f ′

m−1−kf ′k) + λ

m−1∑k=0

f ′′m−1−kf ′′′

k−l,

Rθm(η) = θ′′m−1 + Pr

m−1∑k=0

(θ′m−1−kfk − f ′m−1−kθk),

Rφm(η) = φ′′

m−1 + Pr

m−1∑k=0

(φ′m−1−kfk − f ′

m−1−kφk),

(36)

χm =

{0, m � 1,

1, m > 1.(37)

The general solution to Eq. (34) is defined as

⎧⎪⎪⎪⎨⎪⎪⎪⎩

fm(η) = f∗m(η) + C1 + C2 exp(η) + C3 exp(−η),

θm(η) = θ∗m(η) + C4 + C5 exp(−η),

φm(η) = φ∗m(η) + C6 + C7 exp(−η),

(38)

where f∗m(η), θ∗m(η), and φ∗

m(η) are special functions. We can determine the constants by using


Eq. (35) as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩

C2 = C4 = C6 = 0, C3 =∂f∗

m(η)∂η

∣∣∣η=0

, C1 = −C3 − f∗m(0),

C5 = −θ∗m(0), C7 =∂φ∗

m(η)∂η

∣∣∣η=0

.

(39)

The solution is obtained with the help of OHAM[18]. We can calculate any order approxi-mation for m = 1, 2, 3, · · · with the help of Mathematica.3.2 Optimal values of convergence parameters

It is noteworthy that our solutions f(η), θ(η), and φ(η) contain unknown convergence control(auxiliary) parameters (hf , hθ, hφ ). To find out the optimal values of the convergence controlparameters, we define the exact squared residuals at the mth-order approximation as follows:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εfm =

∫ ∞

0

(Nf

( m∑i=0

fi(η)))2

dη,

εθm =

∫ ∞

0

(Nθ

( m∑i=0

fi(η),m∑

i=0

θi(η))2

dη,

εφm =

∫ ∞

0

(Nφ

( m∑i=0

fi(η),m∑

i=0

φi(η)))2

dη,

(40)

and ⎧⎨⎩

εtm = εf

m + εθm,

εsm = εf

m + εφm,

(41)

where εtm and εs

m are the total squared residual errors at the mth-iteration for the PEST caseand the PEHF case, respectively. It is obvious that more quickly εt

m and εsm approach to zero,

faster the corresponding solutions converge. Thus, at the mth-iteration, the optimal values ofconvergence control parameters are given by the minimums of εt

m and εsm, corresponding to the

set of these equations

∂εtm

∂hf=

∂εtm

∂hθ=

∂εsm

∂hf=

∂εsm

∂hφ= 0. (42)

The total squared residual error defined by Eq. (41) takes too much CPU time to calculate theerror even if the order of approximation is not very high. Thus, to increase the computationalefficiency, we define the discrete squared residual error (as defined by Liao[19]) at the mth-iteration by ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Efm ≈ 1

N + 1

N∑j=0

(Nf

( m∑i=0

fi(ηj)))2

,

Eθm ≈ 1

N + 1

N∑j=0

(Nθ

( m∑i=0

fi(ηj),m∑

i=0

θi(ηj)))2

,

Eφm ≈ 1

N + 1

N∑j=0

(Nφ

( m∑i=0

fi(ηj),m∑

i=0

φi(ηj)))2

,

(43)


where

ηj = jΔη, Δη = 0.5, N = 20.

The total discrete squared residual errors are defined as⎧⎨⎩

Etm = Ef

m + Eθm.

Esm = Ef

m + Eφm.

(44)

In the present paper, the total discrete squared residual error is used to obtain the optimal con-vergence control parameters. In order to obtain the local optimal convergence control parame-ters, we directly employ the minimize command in the computational software Mathematica.

4 Results and discussion

We use the basic OHAM to obtain the results and the discrete squared residual error ap-proach to obtain the optimal values of the convergence control parameters when

Pr = 8/10, s = −2/10, λ = 2/10.

Tables 1 and 2 are drawn to get the local optimal convergence control parameters at differentorders of approximation for PEST and PEHF cases, respectively. For both cases, it is foundthat the total discrete squared residual error decreases with the number of iterations. Tables3 and 4 show the average squared residual errors for velocity and temperature equations cor-responding to the set of optimal values obtained in Tables 1 and 2, respectively. It is veryclear that the average squared residual errors and total squared residual errors decrease as theorder of approximation increases, which assures that the solution is convergent at higher orderapproximations. The results will be similar if we choose the values of the optimal convergenceparameters from any higher order approximation. We choose the 8th iteration set of optimalvalues to plot figures and draw tables in the coming sections.

Table 1 Local optimal convergence control parameters for PEST case at different orders of approxi-mation

m hf hθ Etm

2 –1.04 -0.42 4.92 × 10−4

4 –0.96 –0.81 3.02 × 10−5

6 –0.95 –0.74 1.88 × 10−6

8 –0.94 –0.69 1.37 × 10−7

Table 2 Local optimal convergence control parameters for PEHF case at different orders of approx-imation

m hf hφ Esm

2 –1.04 –0.52 5.18 × 10−4

4 –0.96 –0.99 3.08 × 10−5

6 –0.94 –0.83 2.08 × 10−6

8 –0.93 –0.76 1.50 × 10−7


Table 3 Average squared residual errors for PEST case at different orders of approximation

m hf hθ Efm Eθ

m Etm

2 –1.04 –0.42 3.88 × 10−4 1.04 × 10−4 4.92 × 10−4

4 –0.96 –0.81 2.11 × 10−5 9.17 × 10−6 3.02 × 10−5

6 –0.95 –0.74 1.41 × 10−6 4.71 × 10−7 1.88 × 10−6

8 –0.94 –0.69 1.06 × 10−7 3.01 × 10−8 1.37 × 10−7

Table 4 Average squared residual errors for PEHF case at different orders of approximation

m hf hφ Efm Eφ

m Esm

1 –1.04 –0.52 3.88 × 10−4 1.29 × 10−4 5.18 × 10−4

3 –0.96 –0.99 2.15 × 10−5 9.27 × 10−6 3.08 × 10−5

5 –0.94 –0.83 1.47 × 10−6 6.07 × 10−7 2.08 × 10−6

8 –0.93 –0.76 1.10 × 10−7 3.98 × 10−8 1.50 × 10−7

Since the momentum equation is decoupled from heat equation, the velocity profile is notaffected by the choice whether the PEST or PEHF case. Figures 2 and 3 are drawn to observethe impact of the Williamson and suction/injection parameters on the velocity. It is observedthat the velocity profile and momentum boundary layer thickness decrease with the increases inthe Williamson parameter and the suction parameter (i.e., negative increase in s) while increasefor the case of injection (i.e., positive increase in s), which is justified by the fact that as thefluid is injected to the stretching surface, it covers the region near the wall which causes anincrease in the momentum boundary layer thickness. For the case of suction, the fluid movesout from the region closed to the wall, making it difficult for the boundary layer to establish.Thus, the net effect of suction is to slow down the flow and decrease the boundary layer.

Fig. 2 Velocity profile for different λ Fig. 3 Velocity profile for different s

In both PEST and PEHF cases, the temperature profile and thermal boundary layer thick-ness increase with the increases in the Williamson parameter and the injection parameter whiledecrease with the increase in the suction parameter (see Figs. 4–7). It is also observed that thetemperature profile and the thermal boundary layer thickness decrease with the increase in thePrandtl number for both cases, which is consistent with the fact that in heat flow problems,the Prandtl number controls the relative thickness of the momentum and thermal boundarylayers.


Fig. 4 Temperature profile in PEST case fordifferent λ

Fig. 5 Temperature profile in PEST case fordifferent s

Fig. 6 Temperature profile in PEHF case fordifferent λ

Fig. 7 Temperature profile in PEHF case fordifferent s

Figures 8 and 9 show the temperature profiles in PEST and PEHF cases for different Pr. Itcan be seen from Figs. 8 and 9 that the heat diffuses very quickly for small values of the Prandtlnumber.

Tables 5–7 are drawn to observe the effects of the effective parameters on the skin frictionand wall temperature gradient. Since the skin friction is given by −√

2Recf , where viscosityappears in the denominator of the Reynolds number, it decreases with the increase in the non-Newtonian Williamson parameter. Mathematically, from Eq. (22), we can see that the skinfriction is the sum of f ′′(0) and its square. Since f ′′(0) is negative while its square is positivewhich is multiplied by a fraction (λ) less than 1, the difference of them reduces as we increaseλ. Thus, the skin friction decreases with the increase in λ, as appeared in Table 5. It is alsonoticed that the skin friction decreases with the increase in the suction/injection parameter s.From Table 6, it is observed that the wall temperature gradient decreases with the increasesin the Williamson and suction/injection parameters. The wall temperature gradient increaseswith the increase in the suction parameter because suction causes heat drain which resultantlyincreases the temperature gradient. Therefore, suction can be used to control the temperatureof a process. Since the fluid is injected through wall, the wall temperature gradient decreasesfor the case of injection. Table 7 shows that the wall temperature gradient increases with theincrease in the Prandtl number. Tables 6 and 7 are drawn only for the PEST case because forthe PEHF case, we have φ′(0) = −1.


Fig. 8 Temperature profile in PEST case fordifferent Pr

Fig. 9 Temperature profile in PEHF case fordifferent Pr

Table 5 Values of –√

2Recf for different values of λ and s

λ s = −0.2 s = −0.1 s = 0.0 s = 0.1 s = 0.2

0.0 1.378 89 1.329 30 1.281 80 1.236 38 1.192 98

0.1 1.346 59 1.298 01 1.251 53 1.207 10 1.164 68

0.2 1.310 31 1.263 10 1.217 94 1.17481 1.133 65

0.3 1.267 88 1.222 76 1.179 56 1.138 25 1.098 81

Table 6 Values of wall temperature gradient −θ′(0) in PEST case for different values of λ and swhen Pr = 0.5

λ s = −0.2 s = −0.1 s = 0.0 s = 0.1 s = 0.2

0.0 0.643 54 0.618 35 0.594 33 0.571 51 0.549 88

0.1 0.637 37 0.612 56 0.588 92 0.566 48 0.545 21

0.2 0.630 34 0.606 01 0.582 85 0.560 86 0.540 03

0.3 0.622 02 0.598 35 0.575 82 0.554 42 0.534 14

Table 7 Values of wall temperature gradient −θ′(0) in PEST case for different values of Pr and swhen λ = 0.2

Pr s = −0.2 s = −0.1 s = 0.0 s = 0.1 s = 0.2

0.5 0.630 34 0.606 01 0.582 85 0.560 86 0.540 03

0.8 0.893 17 0.850 50 0.809 86 0.771 23 0.734 62

1.0 1.046 58 0.992 12 0.939 50 0.889 63 0.842 46

1.2 1.164 81 1.119 82 1.057 50 0.996 18 0.938 44

5 Concluding remarks

In the present paper, we analyze the heat transfer effects on the boundary layer flow ofWilliamson fluid over an exponentially stretching surface for two special cases, i.e., PEST andPEHF. It is found that OHAM can be successfully employed to solve the coupled system ofnon-linear differential equations. The important findings are concluded as follows:

(i) The velocity and momentum boundary layer decrease with the increases in the Williamsonparameter and the suction parameter while increase with the increase in the injection parameter.


(ii) Both the PEST case and the PEHF case have qualitatively the same results. For bothcases, the temperature and thermal boundary layer decrease with the increase in the Prandtlnumber.

(iii) The skin friction decreases with the increase in the Williamson parameter λ.(iv) The wall temperature gradient decreases with the increases in the Williamson and

suction/injection parameters and increases with the increase in the Prandtl number.(v) Suction can be used as a mean to the regulate temperature of a process.

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Heat transfer analysis of Williamson fluid over exponentially stretching surface

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