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Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet
International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet M. Mustafa a,n, Junaid Ahmad Khan b, T. Hayat c,d, A. Alsaedi d a
School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan Research Centre for Modeling and Simulation (RCMS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan c Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan d Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah 21589, Saudi Arabia b
art ic l e i nf o
a b s t r a c t
Article history: Received 25 June 2014 Received in revised form 23 November 2014 Accepted 9 January 2015
This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nanofluid Axisymmetric flow Non-linear stretching sheet Optimal homotopy analysis method Keller–Box method
1. Introduction Increasing energy requirements has resulted in a massive hunt for renewable energy sources at global level. Solar energy is perhaps the most appropriate source for current energy requirements of the world. On the other hand heat transfer applications always desired fluids with high thermal conductivity. The idea of using small particles to collect solar energy was first investigated by Hunt [1] in the 1970s. Researchers concluded that heat transfer and the solar collection processes can be improved with the addition of nanoparticles in the base fluids. Masuda et al. [2] found that both viscosity and thermal conductivity can be enhanced by dispersing nanometersized metallic particles in the fluid. Choi and Eastman [3] suggested the use of nanofluids for improving heat transfer efficiency. Due to the significant thermal conductivity enhancement, nanofluids have already been used in several industrial and engineering applications including power generation, transportation, electronic and space cooling, treatment of cancer-infected tissues etc. [4–6]. Buongiorno [7] provided the mathematical model for understanding convective transport in nanofluids. Nield and Kuznetsov [8] discussed the effects of thermal convection on the horizontal layer of porous medium
n
Corresponding author Tel.: þ 92 51 90855596. E-mail addresses: [email protected], [email protected] (M. Mustafa).
saturated by a nanofluid. They also derived the constitutive equations using Buongiorno's model. In another paper, Nield and Kuznetsov [9] examined the Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Kuznetsov and Nield [10] considered the classical problem of natural convective boundary layer flow of nanofluid past a vertical plate maintained at constant temperature. The generalized stretching wall problem for nanofluids was first discussed by Khan and Pop [11]. The study of convective heat transfer on the steady boundary layer flow of nanofluid was presented by Makinde and Aziz [12]. Mustafa et al. [13] derived analytic solutions for stagnation-point flow of a nanofluid past a stretching sheet by HAM. Rana and Bhargava [14] performed a finite element analysis for boundary layer flow of nanofluid caused by a non-linearly stretching sheet. After these fundamental studies, researchers have extensively investigated the nanofluid dynamics in diverse situations [15–25]. Boundary layer development due to a moving plate in a quiescent ambient fluid was initially discussed by Sakiadis [26]. Sakiadis problem for a stretching sheet was considered by Crane [27]. He showed that the problem exhibits a closed form solution for the two-dimensional Navier–Stokes equations. Crane's problem has vital importance in numerous industrial and technological applications such as polymer extrusion in melt spinning process, annealing and tinning of copper wires, fabrication of glass and plastic, manufacturing of metallic sheets, paper production etc. In these processes the quality of desired product largely depends on the rate of heat transfer that is governed by the
http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005 0020-7462/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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structure of boundary layer near the sheet. Due to such applications, Crane's problem has been widely investigated by the researchers for different physical effects. In particular flow past a stretching surface with exponential [28], non-linear [29], quadratic [30] and even oscillatory [31] velocity distributions has also been addressed previously. The wall skin friction in case of exponentially stretching sheet reported by Magyari and Keller [28] is bigger than that computed by Vajravelu [29] for the non-linearly stretching sheet even with uw ¼ cx5 . On the other hand the wall skin friction has oscillatory behavior in case of oscillatory stretching sheet as observed by Abbas et al. [31]. The classical problem of axisymmetric flow due to radially stretching sheet was reported by Ariel [32]. He computed the numerical solution by explicit finite difference method. In another paper, Ariel [33] considered velocity slip effects on the axisymmetric flow over a stretching sheet and provided an analytic solution of the arising differential system. Series solutions for axisymmetric flow past a non-linearly stretching sheet were derived by Sajid et al. [34]. Khan and Shahzad [35] showed that an exact solution for steady axisymmetric flow due to non-linearly stretching sheet exists when the velocity of the stretching sheet is proportional to x3 : Recently Mustafa et al. [36] studied the flow of nanofluid due to plane radially stretching sheet and computed both numerical and homotopy solutions. The purpose of current study is to analyze the axisymmetric boundary layer flow of nanofluid due to non-linear stretching sheet along the radial direction. The study is important in the sense that cooling process of the extruded polymer/plastic sheet can benefit from the nanoparticle working fluid. The revised model proposed by Kuznetsov and Nield [24] is adopted. In the past, the boundary conditions for nanoparticle volume fraction analogous to temperature were employed. However in industrial applications, it is very difficult to maintain a constant value of nanoparticle volume fraction at the wall. On the other hand the model proposed in [24] suggests that mass flux of nanoparticles at the wall is zero and the particle fraction value adjusts there accordingly, which seems physically plausible. The boundary layer equations are first reduced to self-similar forms and then solved analytically by optimal homotopy analysis method (OHAM) and numerically through Keller–Box method. Error analysis of OHAM shows that 15th-order solutions are accurate correct up to nine decimal places. Analytic solutions are in a very good agreement with the numerical ones. Plots of the dimensionless quantities showing the underlying physics of the problem are discussed in detail.
2. Problem formulation Let us choose the cylindrical coordinate system ðr; θ; zÞ: Consider the laminar incompressible flow of nanofluid over a circular sheet aligned with the rθ-plane as shown in Fig. 1. The fluid resides in the half space z Z 0 of the vertical axis. The sheet is stretched in its own plane with the power-law variation of velocity along the radial direction. The sheet is maintained at constant temperature T w whereas nanoparticle mass flux at the wall is taken equal to zero. T 1 and C 1 denote the ambient values of temperature and nanoparticle volume fraction respectively. Under the usual boundary layer assumptions, the equations governing the conservation of mass, momentum, energy and nanoparticle volume fraction can be expressed as (see [33,34], etc.) ∂u u ∂w þ þ ¼ 0; ∂r r ∂z u
u
ð1Þ
∂u ∂u ∂2 u þ w ¼ νf 2 ; ∂r ∂z ∂z
ð2Þ
" # ∂T ∂T ∂2 T ∂C ∂T DT ∂T 2 þ w ¼ α 2 þ τ DB :þ ; ∂r ∂z ∂z ∂z T 1 ∂z ∂z
ð3Þ
Fig. 1. Physical configuration and coordinate system.
∂C ∂C ∂ 2 C DT ∂ 2 T u þw ¼ DB 2 þ ; ∂r ∂z T 1 ∂z2 ∂z
ð4Þ
where u and w are the velocity components along r- and z-directions respectively, p^ is the pressure, νf is the kinematic viscosity of the fluid, α is the thermal diffusivity, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, k is the thermal conductivity and τ is the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid. The boundary conditions for the considered problem are as under: u ¼ uw ðrÞ ¼ ar n ;
T ¼ Tw;
T-T 1 ;
C-C 1
u-0;
∂C DT ∂T þ ¼ 0 at y ¼ 0; ∂z T 1 ∂z as y-1; DB
ð5Þ
in which a 40 is the stretching constant and n 4 0 is the power-law index. Now we introduce the following similarity transformations rffiffiffiffiffi νf n þ 3 n1 0 0 f ðηÞ þ ηf ðηÞ ; u ¼ ar n f ðηÞ; w ¼ ar ðn 1Þ=2 2 2 a sffiffiffiffiffi T T1 C C1 a ðn 1Þ=2 r ; ϕ¼ ; η¼ z: ð6Þ θ¼ νf Tw T1 C1 It is worth mentioning to point out through the above equation that similarity transformations in Ref. [34] have been modified so that these satisfy the continuity equation (1). Obviously when n ¼ 3, Eq. (6) yields the transformations used in Ref. [35]. Through Eqs.(2)–(5) we have f ″0 þ
nþ3 02 f f ″ nf ¼ 0; 2
ð7Þ
1 nþ3 0 θ″ þ f θ þ Nbθ0 ϕ0 þ Ntθ02 ¼ 0; Pr 2 ϕ″ þ
nþ3 Nt Scf ϕ0 þ θ″ ¼ 0; 2 Nb
f ð0Þ ¼ 0; 0
f ð1Þ-0; Pr ¼
ð8Þ
νf ; α
0
f ð0Þ ¼ 1;
θð0Þ ¼ 1;
θð1Þ-0; Sc ¼
νf ; DB
ð9Þ Nbϕ0 ð0Þ þNtθ0 ð0Þ ¼ 0;
φð1Þ-0;
Nb ¼
ðρcÞp DB C 1 ; ðρcÞf νf
ð10Þ Nt ¼
ðρcÞp DT ðT w T 1 Þ : ðρcÞf T 1 νf ð11Þ
In the above equations Pr is the Prandtl number, Sc is the Schmidt number, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter. Using variables (6), the local skin
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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friction coefficient C f ¼ μð∂u=∂zÞz ¼ 0 =ρf u2w and the local Nusselt number Nu ¼ rð∂T=∂zÞz ¼ 0 =ðT w T 1 Þ can be put into the following forms: 1=2
Rer C f ¼ f ″ð0Þ;
1=2
Rer
Nu ¼ θ0 ð0Þ;
ð12Þ
where Rer ¼ uw r=νf is the local Reynolds number. The reduced Sherwood number, which is the dimensionless mass flux, is now identically zero. When n ¼ 3, the closed form exact solutions of Eqs. (7) and (8) can be sought in the case of regular fluid as (see [35] for details) pffiffiffiffi 1 f ðηÞ ¼ pffiffiffið1 e 3η Þ; ð13Þ 3 pffiffiffiffi ΓðPr; 0Þ ΓðPr; e 3η PrÞ ; θðηÞ ¼ ΓðPr; 0Þ ΓðPr; PrÞ
ð14Þ
in which Γða; xÞ is the incomplete Gamma function. 3. Optimal homotopy analysis solutions
θ0 ðηÞ ¼ expð ηÞ;
ϕ0 ðηÞ ¼
Nt expð ηÞ; Nb ð15Þ
and the auxiliary linear operators are selected as under: 3
d f df Lf ðf Þ ¼ 3 ; dη dη
2
d θ Lθ ðθÞ ¼ 2 θ; dη
2
d ϕ Lϕ ðϕÞ ¼ 2 ϕ: dη
h
i
¼ 0;
¼ 1;
Θðη; qÞ
η-1
η¼0
η¼0
Nt ∂Θðη; qÞ þ ¼ 0; Nb ∂η η ¼ 0 ∂f^ ðη; qÞ ¼ 0; Θðη; qÞη-1 ¼ 0; ∂η
f m ðηÞ ¼
;
1 X
1 ∂m Θðη; qÞ ; m m! ∂q q¼0
θm ðηÞqm ;
θm ðηÞ ¼
ϕm ðηÞqm ;
ϕm ðηÞ ¼
m¼1
Φðη; qÞ ¼ ϕ0 ðηÞ þ
1 ∂m f^ ðη; qÞ m! ∂qm
¼ 1;
m¼1
1 ∂m Φðη; qÞ : m! ∂ηm q ¼ 0
Lϕ ϕm ðηÞ χ m ϕm 1 ðηÞ ¼ ℏR ϕm ðηÞ;
h i 1 ∂2 Θðη; qÞ n þ 3 ∂Θðη; qÞ f^ ðη; pÞ þ Nθ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ¼ Pr ∂η2 2 ∂η ∂Θðη; qÞ ∂Φðη; qÞ ∂Θðη; qÞ 2 þNt ; þ Nb ∂η ∂η ∂η
Rθm ðηÞ ¼
(
1 ∂3 f m 1 mX nþ3 ∂2 f ∂f ∂f f m 1 k 2k n m 1 k k ; þ 3 2 ∂η ∂η ∂η ∂η k¼0 1 1 ∂2 θm 1 mX nþ3 ∂θk ∂θ ∂ϕ f þ Nb m 1 k k þ Pr ∂η2 2 m 1 k ∂η ∂η ∂η k¼0 ∂θ ∂θ þNt m 1 k k ; ∂η ∂η 1 ∂2 ϕm 1 mX nþ3 ∂ϕ Nt ∂2 θm 1 Scf m 1 k k þ þ ; 2 2 ∂η Nb ∂η2 ∂η k¼0
0;
m r 1;
1;
m 4 1:
h i ∂2 Φðη; qÞ nþ3 ∂Φðη; qÞ Scf^ ðη; pÞ Nϕ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ¼ þ 2 ∂η ∂η2
ð30Þ
ð31Þ
ð32Þ
ð33Þ
ð34Þ
The optimal values of the convergence control parameter ℏ can be determined by minimizing the squared residuals of the governing Eqs. (7)–(9), ζ f M ; ζ θ M and ζ ϕ M , in the domain η A ½0; 30. 2 0 132 Z 1 M X f 4 @ Nf f j ðηÞA5 dη; ð35Þ ζ M¼
ζ θM
ζ ð22Þ
ð29Þ
df m ðηÞ ¼ 0; θm ð0Þ ¼ 0; f m ð0Þ ¼ 0; dη η ¼ 0 dϕm ðηÞ Nt dθm ðηÞ þ ¼ 0; dη Nb dη η¼0 ∂f m ðηÞ ¼ 0; θm ð1Þ ¼ 0; ϕm ð1Þ ¼ 0; ∂η η- þ 1
0
in which the non-linear operators Nf ,Nθ and Nϕ can be obtained through Eqs. (7)–(9) as below: !2 h i ∂3 f^ ðη; qÞ n þ 3 ∂2 f^ ðη; qÞ ∂f^ ðη; qÞ ^ ^ f ðη; qÞ þ n ; ð21Þ Nf f ðξ; η; qÞ ¼ 2 ∂η ∂η3 ∂η2
ð26Þ
ð28Þ
χm ¼
ð20Þ
ð25Þ
The final solutions can be obtained by substituting q ¼ 1 in the above equations. The functions f m and θm can be determined from the deformation of Eqs. (7)–(10). Explicitly mth-order deformation equations corresponding to Eqs. (7)–(10) are as under Lf f m ðηÞ χ m f m 1 ðηÞ ¼ ℏRfm ðηÞ; ð27Þ
ð19Þ
η¼0
ð24Þ
p¼0
1 X
Θðη; qÞ ¼ θ0 ðηÞ þ
Rϕm ðηÞ ¼
∂Φðη; qÞ ∂η
Φðη; qÞη-1 ¼ 0;
f m ðηÞqm ;
m¼1
ð18Þ
h i ð1 qÞLϕ ½Φðη; qÞ ϕ0 ðηÞ ¼ qℏNϕ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ;
η¼0
1 X
f^ ðη; qÞ ¼ f 0 ðηÞ þ
Rfm ðηÞ ¼
ð1 qÞLθ ½Θðη; qÞ θ0 ðηÞ ¼ qℏNθ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ;
∂f^ ðη; qÞ ∂η
ð23Þ
By Taylor's series expansion one obtains
ð16Þ
Let q A ½0; 1 be an embedding parameter and ℏ be the non-zero auxiliary parameter, then one can express the generalized homotopic equations corresponding to (7)–(10) as h i ð17Þ ð1 qÞLf ½f^ ðη; qÞ f 0 ðηÞ ¼ qℏNf f^ ðη; qÞ ;
f^ ðη; qÞ
Nt ∂2 Θðη; qÞ : Nb ∂η2
Lθ θm ðηÞ χ m θm 1 ðηÞ ¼ ℏR θm ðηÞ;
We solve the boundary value problems given in Eqs. (7)–(10) by optimal homotopy analysis method (OHAM) [37,38]. Through the rule of solution expression and the boundary conditions (10) we select the following initial guesses f 0 ;θ0 and ϕ0 of f ðηÞ;θðηÞ and ϕðηÞ f 0 ðηÞ ¼ 1 expð ηÞ;
þ
3
ϕ
Z ¼
1 0
¼
2
0
4Nθ @
M X
f j ðηÞ;
j¼0
Z M
j¼0
1 0
2
0
4Nϕ @
M X j¼0
M X
θj ðηÞ;
j¼0
f j ðηÞ;
M X j¼0
M X
13 2 ϕj ðηÞA5 dη;
ð36Þ
j¼0
θj ðηÞ;
M X
13 2 ϕj ðηÞA5 dη:
ð37Þ
j¼0
Such kind of error has been considered in other works [39–42]. The smaller ζ M 0 s; the more accurate the Mth order approximation of the solution. A sample of the optimal values of ℏ for the functions θ and ϕ are given in Table 1.
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Table 1 Optimal values of ℏ for the functions θ and ϕ when Pr ¼ Sc ¼ 1 and Nb ¼ Nt ¼ 0:5: n
Optimal ℏ for θ
Minimum ζ θ M
Optimal ℏ for ϕ
Minimum ζ ϕ M
1 3 4 5
0.873 0.606 0.473 0.403
1.25 10 10 3.13 10 9 3.37 10 8 7.29 10 7
0.827 0.608 0.493 0.407
1.41 10 9 1.35 10 8 5.89 10 7 5.56 10 6
hj ðZ þ Z j 1 Þ; 2 j hj n þ 3 hj ðf þ f j 1 ÞðV j þ V j 1 Þ n ðSj þ Sj 1 Þ2 ¼ 0; Vj Vj1 þ 2 4 j 4 ϕj ϕj 1 ¼
ð47Þ hj n þ3 Pr ðf j þf j 1 ÞðY j þY j 1 Þ Yj Yj1 þ 2 4 hj þ Pr Nb ðY j þ Y j 1 ÞðZ j þ Z j 1 Þ 4 hj þ Pr Nt ðY j þ Y j 1 Þ2 ¼ 0; 4 Zj Zj 1 þ
Fig. 2. Discretization for difference approximations.
Eqs. (7)–(10) has been solved numerically by using implicit finite difference scheme known as Keller–Box method [43]. It is a four step method outlined as under: 1. The differential equations are reduced into first order system by using appropriate substitutions. 2. First-order differential equations are converted into difference equations using central differences. 3. The difference equations are linearized using Newton's method. 4. The resulting equations are written in matrix–vector form and then solved by block-tridiagonal-elimination procedure. Introduction the new dependent variables SðηÞ; VðηÞ; YðηÞ and ZðηÞ such that 0
S0 ¼ V ;
θ0 ¼ Y and ϕ0 ¼ Z;
ð38Þ
Eqs. (7)–(10) reduce to the following first-order system as below: V0 þ
nþ3 f V nS2 ¼ 0; 2
ð39Þ
Y0 þ
nþ3 Pr f Y þ Pr Nb YZ þ Pr Nt Y 2 ¼ 0; 2
ð40Þ
Z0 þ
nþ3 Nt Scf Z þ Y 0 ¼ 0; 2 Nb
ð41Þ
f ð0Þ ¼ 0;
Sð0Þ ¼ 1;
Sð1Þ-0;
θð0Þ ¼ 1;
θð1Þ-0;
Nb Zð0Þ þ Nt Yð0Þ ¼ 0;
ϕð1Þ-0:
ð42Þ
Let us consider a small interval along the η axis as shown in figure where the points within the interval having arbitrary step size hj are defined as below η0 ¼ 0;
ηj ¼ ηj 1 þ hj ;
j ¼ 1; 2; :::; J;
ð48Þ
hj nþ3 Nt Sc ðf j þ f j 1 ÞðZ j þ Z j 1 Þ þ ðY j Y j 1 Þ ¼ 0; 2 Nb 4 ð49Þ
4. Numerical solution by Keller–Box method
f ¼ S;
ð46Þ
ηJ ¼ η1
f 0 ¼ 0;
S0 ¼ 1;
SJ ¼ 0;
θJ ¼ 0;
θ0 ¼ 1;
Nt Z 0 ¼ Nb Y0;
ϕJ ¼ 0:
ð50Þ
Now we use Newton's method to linearize the system of Eqs. (43)– (50) by introducing following iterate: kþ1
fj
k
k
¼ f j þ δf j ;
Skj þ 1 ¼ Skj þ δSkj ;
θkj þ 1 ¼ θkj þ δθkj ;
Y kj þ 1 ¼ Y kj þ δY kj ;
ϕkj þ 1 ¼ ϕkj þδϕkj ;
Z kj þ 1 ¼ Z kj þ δZ kj ;
V kj þ 1 ¼ V kj þ δV kj ; ð51Þ
where k is iteration index. After substituting Eq. (51) into Eqs. (43)–(50) and dropping the quadratic and higher order terms of k δf j ; δSkj ; δV kj ; δθkj ; δY kj ; δϕkj and δZ kj we obtain the following linear tridiagonal system: Aδ ¼ r;
ð52Þ
in which the matrices A; δ and r can be expressed as below: 2 3 2 3 ½δ1 ½A1 ½C 1 6 ½B ½A 7 6 7 ½C 2 2 6 2 7 6 ½δ2 7 6 7 6 7 7; ½δ ¼ 6 ⋮ 7; ⋱ ⋱ ⋱ ½A ¼ 6 6 7 6 7 6 7 6 7 ½BJ 1 ½AJ 1 ½C J 1 5 4 4 ½δJ 1 5 2
3
½BJ
½AJ
½r 1 6 ½r 7 6 2 7 6 7 ⋮ 7; ½r ¼ 6 6 7 6 ½r 7 4 J 1 5 ½r J
½δJ
ð53Þ
The solution of the system (52) can be determined by employing block-tridiagonal elimination technique. The whole procedure can be found in Ibrahim and Shanker [44].
5. Numerical results and discussion
Using central differences the functions F and F 0 at point ηj 1=2 are expressed as F j 1=2 ¼ ðF j þF j 1 Þ=2 and F 0j 1=2 ¼ ðF j F j 1 Þ=hj (Fig. 2). In view of the above definitions, Eqs. (38)–(42) can be approximated within the segment P 1 P 2 as f j f j1 ¼
hj ðS þ Sj 1 Þ; 2 j
ð43Þ
Sj Sj 1 ¼
hj ðV þ V j 1 Þ; 2 j
ð44Þ
θj θj 1 ¼
hj ðY þ Y j 1 Þ; 2 j
ð45Þ
A comparison of numerical solution with the exact solution given in Eq. (14) is shown in Fig. 3. We notice that data obtained by two solutions is identical showing the validation of numerical computations. In Figs. 4 and 5, we have compared the 15th-order OHAM solutions for temperature and nanoparticle volume fraction distribution with the numerical solutions for various values of n. The two solutions yield virtually similar results in all the cases as demonstrated in these figures. Increasing values of n reduce the thickness of thermal and nanoparticle volume fraction boundary layers. This in turn augments the rate of heat transfer from the sheet. This rise in θ'ð0Þ is due to the fact that increasing n enhances the convective properties of the fluid since it increases the deformation by the shear
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Fig. 3. Temperature profiles for different values of Pr. Lines: numerical solution and circles: exact solution (15).
5
Fig. 6. Effect of Pr on θ.
Fig. 7. Effect of Sc on θ. Fig. 4. Temperature profiles for different values of n. Lines: numerical solution; circles: 15th-order analytic solution.
Fig. 8. Effect of Nt on θ Fig. 5. Nanoparticle volume fraction profiles for different values of n. Lines: numerical solution; circles: 15th-order analytic solution.
stress from the wall to the fluid. In Fig. 6, temperature θ is plotted versus the similarity variable η with the variation in Pr. Temperature decreases and becomes equal to the ambient temperature at smaller values of η as Pr is increased indicating a diminution in the penetration depth for thermal boundary layer. This results in the
enhancement of heat transfer rate from the sheet. This fact can also be understood from the fact that profiles become steeper with an augmentation of Pr signifying an increase in wall slope of temperature distribution. Fig. 7 depicts the influence of Schmidt number on the temperature distribution. Schmidt number is analogous of Pr for mass transfer and relates the viscous or hydrodynamic boundary
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Fig. 9. Effect of Pr on ϕ.
Fig. 10. Effect of Sc on ϕ.
Fig. 12. Effect of Nb on ϕ.
Fig. 13. Effect of Prand n on –θ0 ð0Þ.
Fig. 11. Effect of Nt on ϕ
layer to nanoparticle volume fraction boundary layer. Increasing Sc corresponds to stronger viscous diffusion compared to mass diffusion which enhances molecular motions and their interactions and hence temperature rises. The variation in temperature θ with increase in Sc is only accounted in the vicinity of the stretching sheet. Fig. 8 shows the impact of thermophoretic diffusion on the thermal boundary layer. A stronger thermophoretic force drives the particles from the hot sheet towards the ambient fluid and thus affects larger extent of
Fig. 14. Effect of Sc and n on –θ0 ð0Þ.
the fluid. Due to this reason, thermal boundary layer grows with an increment in Nt. In accordance with Kuznetsov and Nield [24], the outcome of increase in Brownian motion parameter on temperature is negligible. It can also be seen that profiles become less steep with an increase in Nt revealing a decrease in the magnitude of reduced Nusselt number.
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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6. Concluding remarks Steady axisymmetric flow of nanofluid due to the non-linearly stretching sheet is examined through the revised model proposed by Kuznetsov and Nield [24]. Optimal homotopy analytic solutions of the arising boundary value problem are obtained. An implicit finite difference scheme is employed to compute the numerical solutions. The important points of present analysis are as under:
Fig. 15. Effect of Nt and n on –θ0 ð0Þ.
Table 2 Numerical values of reduced Nusselt number Nur for different values of parameters. Nt
Sc
Pr
Nur ¼ θ'ð0Þ
20
5
1.0
0.1 0.5 0.7 0.5
5
2.5
0.5
5 10 20 20
1.9112911 1.2170065 0.9815765 1.6914582 1.4740787 1.2861370 0.6619164 1.4784288 1.5758736
n
0.5
0.7 5 7
The profiles of nanoparticle volume fraction ϕ corresponding to Pr ¼ 0:7; 1; 3; 5; 10 are sketched in Fig. 9. We found that ϕ is an increasing function of Pr. The effect of Schmidt number on nanoparticle volume fraction ϕ can be perceived from Fig. 10. It is noticed earlier that increase in Sc corresponds to decrease the Brownian diffusion coefficient DB . The smaller DB gives the shorter penetration depth of nanoparticle volume fractionϕ. An increase in Nt shifts the nanoparticles away from the sheet and forms a particle free layer in the vicinity of the sheet (see Fig. 11). As a consequence the nanoparticle volume fraction distribution is formed just outside. On the other hand, the nanoparticle volume fraction boundary layer becomes thinner when strength of Brownian motion is increased as can be seen from Fig. 12. In Figs. 13–15, the reduced Nusselt number θ0 ð0Þ versus the power-law index n is plotted with the variations of Pr; ScandNt. For bigger Pr the convection (or the heat flow through unit area) is stronger in comparison to pure conduction. As a result the heat transfer rate is an increasing function of Pr and it decreases when Sc is increased. Magnitude of Nusselt number also decreases when Nt is increased. This diminution occurs due to the fact that larger thermophoretic force drives the nanoparticles of high thermal conductivity from the hot sheet to the quiescent fluid. It is notable that in contrast to the previous studies on constant wall nanoparticle volume fraction, here the reduced Nusselt number is independent of the Brownian motion parameter. This outcome is consistent with This can be easily understood by using ϕ0 ð0Þ ¼ Nt=Nb θ0 ð0Þ (from Eq. (10)) in the energy equation (8) as the wall is approached. Table 2 includes the sample of our results for reduced Nusselt number θ0 ð0Þ for different values of embedded parameters. It is found that wall temperature gradient θ0 ð0Þ is a decreasing function of n; Nt and Sc and it is an increasing function of Pr.
(a) The analytic and numerical solutions are found in excellent agreement. The solutions also agree with the exact solution in the case of regular fluid when n ¼ 3: (b) The influence of Brownian motion on the temperature and wall heat transfer rate is insignificant. However the penetration depth of nanoparticle volume fraction boundary layer is reduced when Brownian motion intensifies. (c) Thermal boundary layer becomes thicker as a result of strong thermophoretic diffusion. This in turn reduces the magnitude of reduced Nusselt number. (d) Increase in Schmidt number Sc, which is inversely proportional to the Brownian diffusion coefficient DB , corresponds to a thinner nanoparticle volume fraction boundary layer. Further there is increase in temperature function only close to the sheet when Sc is increased. (e) The equations reduce to the case of linearly stretching sheet when n ¼ 1:
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Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet M. Mustafa a,n, Junaid Ahmad Khan b, T. Hayat c,d, A. Alsaedi d a
School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan Research Centre for Modeling and Simulation (RCMS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan c Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan d Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah 21589, Saudi Arabia b
art ic l e i nf o
a b s t r a c t
Article history: Received 25 June 2014 Received in revised form 23 November 2014 Accepted 9 January 2015
This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nanofluid Axisymmetric flow Non-linear stretching sheet Optimal homotopy analysis method Keller–Box method
1. Introduction Increasing energy requirements has resulted in a massive hunt for renewable energy sources at global level. Solar energy is perhaps the most appropriate source for current energy requirements of the world. On the other hand heat transfer applications always desired fluids with high thermal conductivity. The idea of using small particles to collect solar energy was first investigated by Hunt [1] in the 1970s. Researchers concluded that heat transfer and the solar collection processes can be improved with the addition of nanoparticles in the base fluids. Masuda et al. [2] found that both viscosity and thermal conductivity can be enhanced by dispersing nanometersized metallic particles in the fluid. Choi and Eastman [3] suggested the use of nanofluids for improving heat transfer efficiency. Due to the significant thermal conductivity enhancement, nanofluids have already been used in several industrial and engineering applications including power generation, transportation, electronic and space cooling, treatment of cancer-infected tissues etc. [4–6]. Buongiorno [7] provided the mathematical model for understanding convective transport in nanofluids. Nield and Kuznetsov [8] discussed the effects of thermal convection on the horizontal layer of porous medium
n
Corresponding author Tel.: þ 92 51 90855596. E-mail addresses: [email protected], [email protected] (M. Mustafa).
saturated by a nanofluid. They also derived the constitutive equations using Buongiorno's model. In another paper, Nield and Kuznetsov [9] examined the Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Kuznetsov and Nield [10] considered the classical problem of natural convective boundary layer flow of nanofluid past a vertical plate maintained at constant temperature. The generalized stretching wall problem for nanofluids was first discussed by Khan and Pop [11]. The study of convective heat transfer on the steady boundary layer flow of nanofluid was presented by Makinde and Aziz [12]. Mustafa et al. [13] derived analytic solutions for stagnation-point flow of a nanofluid past a stretching sheet by HAM. Rana and Bhargava [14] performed a finite element analysis for boundary layer flow of nanofluid caused by a non-linearly stretching sheet. After these fundamental studies, researchers have extensively investigated the nanofluid dynamics in diverse situations [15–25]. Boundary layer development due to a moving plate in a quiescent ambient fluid was initially discussed by Sakiadis [26]. Sakiadis problem for a stretching sheet was considered by Crane [27]. He showed that the problem exhibits a closed form solution for the two-dimensional Navier–Stokes equations. Crane's problem has vital importance in numerous industrial and technological applications such as polymer extrusion in melt spinning process, annealing and tinning of copper wires, fabrication of glass and plastic, manufacturing of metallic sheets, paper production etc. In these processes the quality of desired product largely depends on the rate of heat transfer that is governed by the
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Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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structure of boundary layer near the sheet. Due to such applications, Crane's problem has been widely investigated by the researchers for different physical effects. In particular flow past a stretching surface with exponential [28], non-linear [29], quadratic [30] and even oscillatory [31] velocity distributions has also been addressed previously. The wall skin friction in case of exponentially stretching sheet reported by Magyari and Keller [28] is bigger than that computed by Vajravelu [29] for the non-linearly stretching sheet even with uw ¼ cx5 . On the other hand the wall skin friction has oscillatory behavior in case of oscillatory stretching sheet as observed by Abbas et al. [31]. The classical problem of axisymmetric flow due to radially stretching sheet was reported by Ariel [32]. He computed the numerical solution by explicit finite difference method. In another paper, Ariel [33] considered velocity slip effects on the axisymmetric flow over a stretching sheet and provided an analytic solution of the arising differential system. Series solutions for axisymmetric flow past a non-linearly stretching sheet were derived by Sajid et al. [34]. Khan and Shahzad [35] showed that an exact solution for steady axisymmetric flow due to non-linearly stretching sheet exists when the velocity of the stretching sheet is proportional to x3 : Recently Mustafa et al. [36] studied the flow of nanofluid due to plane radially stretching sheet and computed both numerical and homotopy solutions. The purpose of current study is to analyze the axisymmetric boundary layer flow of nanofluid due to non-linear stretching sheet along the radial direction. The study is important in the sense that cooling process of the extruded polymer/plastic sheet can benefit from the nanoparticle working fluid. The revised model proposed by Kuznetsov and Nield [24] is adopted. In the past, the boundary conditions for nanoparticle volume fraction analogous to temperature were employed. However in industrial applications, it is very difficult to maintain a constant value of nanoparticle volume fraction at the wall. On the other hand the model proposed in [24] suggests that mass flux of nanoparticles at the wall is zero and the particle fraction value adjusts there accordingly, which seems physically plausible. The boundary layer equations are first reduced to self-similar forms and then solved analytically by optimal homotopy analysis method (OHAM) and numerically through Keller–Box method. Error analysis of OHAM shows that 15th-order solutions are accurate correct up to nine decimal places. Analytic solutions are in a very good agreement with the numerical ones. Plots of the dimensionless quantities showing the underlying physics of the problem are discussed in detail.
2. Problem formulation Let us choose the cylindrical coordinate system ðr; θ; zÞ: Consider the laminar incompressible flow of nanofluid over a circular sheet aligned with the rθ-plane as shown in Fig. 1. The fluid resides in the half space z Z 0 of the vertical axis. The sheet is stretched in its own plane with the power-law variation of velocity along the radial direction. The sheet is maintained at constant temperature T w whereas nanoparticle mass flux at the wall is taken equal to zero. T 1 and C 1 denote the ambient values of temperature and nanoparticle volume fraction respectively. Under the usual boundary layer assumptions, the equations governing the conservation of mass, momentum, energy and nanoparticle volume fraction can be expressed as (see [33,34], etc.) ∂u u ∂w þ þ ¼ 0; ∂r r ∂z u
u
ð1Þ
∂u ∂u ∂2 u þ w ¼ νf 2 ; ∂r ∂z ∂z
ð2Þ
" # ∂T ∂T ∂2 T ∂C ∂T DT ∂T 2 þ w ¼ α 2 þ τ DB :þ ; ∂r ∂z ∂z ∂z T 1 ∂z ∂z
ð3Þ
Fig. 1. Physical configuration and coordinate system.
∂C ∂C ∂ 2 C DT ∂ 2 T u þw ¼ DB 2 þ ; ∂r ∂z T 1 ∂z2 ∂z
ð4Þ
where u and w are the velocity components along r- and z-directions respectively, p^ is the pressure, νf is the kinematic viscosity of the fluid, α is the thermal diffusivity, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, k is the thermal conductivity and τ is the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid. The boundary conditions for the considered problem are as under: u ¼ uw ðrÞ ¼ ar n ;
T ¼ Tw;
T-T 1 ;
C-C 1
u-0;
∂C DT ∂T þ ¼ 0 at y ¼ 0; ∂z T 1 ∂z as y-1; DB
ð5Þ
in which a 40 is the stretching constant and n 4 0 is the power-law index. Now we introduce the following similarity transformations rffiffiffiffiffi νf n þ 3 n1 0 0 f ðηÞ þ ηf ðηÞ ; u ¼ ar n f ðηÞ; w ¼ ar ðn 1Þ=2 2 2 a sffiffiffiffiffi T T1 C C1 a ðn 1Þ=2 r ; ϕ¼ ; η¼ z: ð6Þ θ¼ νf Tw T1 C1 It is worth mentioning to point out through the above equation that similarity transformations in Ref. [34] have been modified so that these satisfy the continuity equation (1). Obviously when n ¼ 3, Eq. (6) yields the transformations used in Ref. [35]. Through Eqs.(2)–(5) we have f ″0 þ
nþ3 02 f f ″ nf ¼ 0; 2
ð7Þ
1 nþ3 0 θ″ þ f θ þ Nbθ0 ϕ0 þ Ntθ02 ¼ 0; Pr 2 ϕ″ þ
nþ3 Nt Scf ϕ0 þ θ″ ¼ 0; 2 Nb
f ð0Þ ¼ 0; 0
f ð1Þ-0; Pr ¼
ð8Þ
νf ; α
0
f ð0Þ ¼ 1;
θð0Þ ¼ 1;
θð1Þ-0; Sc ¼
νf ; DB
ð9Þ Nbϕ0 ð0Þ þNtθ0 ð0Þ ¼ 0;
φð1Þ-0;
Nb ¼
ðρcÞp DB C 1 ; ðρcÞf νf
ð10Þ Nt ¼
ðρcÞp DT ðT w T 1 Þ : ðρcÞf T 1 νf ð11Þ
In the above equations Pr is the Prandtl number, Sc is the Schmidt number, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter. Using variables (6), the local skin
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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friction coefficient C f ¼ μð∂u=∂zÞz ¼ 0 =ρf u2w and the local Nusselt number Nu ¼ rð∂T=∂zÞz ¼ 0 =ðT w T 1 Þ can be put into the following forms: 1=2
Rer C f ¼ f ″ð0Þ;
1=2
Rer
Nu ¼ θ0 ð0Þ;
ð12Þ
where Rer ¼ uw r=νf is the local Reynolds number. The reduced Sherwood number, which is the dimensionless mass flux, is now identically zero. When n ¼ 3, the closed form exact solutions of Eqs. (7) and (8) can be sought in the case of regular fluid as (see [35] for details) pffiffiffiffi 1 f ðηÞ ¼ pffiffiffið1 e 3η Þ; ð13Þ 3 pffiffiffiffi ΓðPr; 0Þ ΓðPr; e 3η PrÞ ; θðηÞ ¼ ΓðPr; 0Þ ΓðPr; PrÞ
ð14Þ
in which Γða; xÞ is the incomplete Gamma function. 3. Optimal homotopy analysis solutions
θ0 ðηÞ ¼ expð ηÞ;
ϕ0 ðηÞ ¼
Nt expð ηÞ; Nb ð15Þ
and the auxiliary linear operators are selected as under: 3
d f df Lf ðf Þ ¼ 3 ; dη dη
2
d θ Lθ ðθÞ ¼ 2 θ; dη
2
d ϕ Lϕ ðϕÞ ¼ 2 ϕ: dη
h
i
¼ 0;
¼ 1;
Θðη; qÞ
η-1
η¼0
η¼0
Nt ∂Θðη; qÞ þ ¼ 0; Nb ∂η η ¼ 0 ∂f^ ðη; qÞ ¼ 0; Θðη; qÞη-1 ¼ 0; ∂η
f m ðηÞ ¼
;
1 X
1 ∂m Θðη; qÞ ; m m! ∂q q¼0
θm ðηÞqm ;
θm ðηÞ ¼
ϕm ðηÞqm ;
ϕm ðηÞ ¼
m¼1
Φðη; qÞ ¼ ϕ0 ðηÞ þ
1 ∂m f^ ðη; qÞ m! ∂qm
¼ 1;
m¼1
1 ∂m Φðη; qÞ : m! ∂ηm q ¼ 0
Lϕ ϕm ðηÞ χ m ϕm 1 ðηÞ ¼ ℏR ϕm ðηÞ;
h i 1 ∂2 Θðη; qÞ n þ 3 ∂Θðη; qÞ f^ ðη; pÞ þ Nθ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ¼ Pr ∂η2 2 ∂η ∂Θðη; qÞ ∂Φðη; qÞ ∂Θðη; qÞ 2 þNt ; þ Nb ∂η ∂η ∂η
Rθm ðηÞ ¼
(
1 ∂3 f m 1 mX nþ3 ∂2 f ∂f ∂f f m 1 k 2k n m 1 k k ; þ 3 2 ∂η ∂η ∂η ∂η k¼0 1 1 ∂2 θm 1 mX nþ3 ∂θk ∂θ ∂ϕ f þ Nb m 1 k k þ Pr ∂η2 2 m 1 k ∂η ∂η ∂η k¼0 ∂θ ∂θ þNt m 1 k k ; ∂η ∂η 1 ∂2 ϕm 1 mX nþ3 ∂ϕ Nt ∂2 θm 1 Scf m 1 k k þ þ ; 2 2 ∂η Nb ∂η2 ∂η k¼0
0;
m r 1;
1;
m 4 1:
h i ∂2 Φðη; qÞ nþ3 ∂Φðη; qÞ Scf^ ðη; pÞ Nϕ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ¼ þ 2 ∂η ∂η2
ð30Þ
ð31Þ
ð32Þ
ð33Þ
ð34Þ
The optimal values of the convergence control parameter ℏ can be determined by minimizing the squared residuals of the governing Eqs. (7)–(9), ζ f M ; ζ θ M and ζ ϕ M , in the domain η A ½0; 30. 2 0 132 Z 1 M X f 4 @ Nf f j ðηÞA5 dη; ð35Þ ζ M¼
ζ θM
ζ ð22Þ
ð29Þ
df m ðηÞ ¼ 0; θm ð0Þ ¼ 0; f m ð0Þ ¼ 0; dη η ¼ 0 dϕm ðηÞ Nt dθm ðηÞ þ ¼ 0; dη Nb dη η¼0 ∂f m ðηÞ ¼ 0; θm ð1Þ ¼ 0; ϕm ð1Þ ¼ 0; ∂η η- þ 1
0
in which the non-linear operators Nf ,Nθ and Nϕ can be obtained through Eqs. (7)–(9) as below: !2 h i ∂3 f^ ðη; qÞ n þ 3 ∂2 f^ ðη; qÞ ∂f^ ðη; qÞ ^ ^ f ðη; qÞ þ n ; ð21Þ Nf f ðξ; η; qÞ ¼ 2 ∂η ∂η3 ∂η2
ð26Þ
ð28Þ
χm ¼
ð20Þ
ð25Þ
The final solutions can be obtained by substituting q ¼ 1 in the above equations. The functions f m and θm can be determined from the deformation of Eqs. (7)–(10). Explicitly mth-order deformation equations corresponding to Eqs. (7)–(10) are as under Lf f m ðηÞ χ m f m 1 ðηÞ ¼ ℏRfm ðηÞ; ð27Þ
ð19Þ
η¼0
ð24Þ
p¼0
1 X
Θðη; qÞ ¼ θ0 ðηÞ þ
Rϕm ðηÞ ¼
∂Φðη; qÞ ∂η
Φðη; qÞη-1 ¼ 0;
f m ðηÞqm ;
m¼1
ð18Þ
h i ð1 qÞLϕ ½Φðη; qÞ ϕ0 ðηÞ ¼ qℏNϕ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ;
η¼0
1 X
f^ ðη; qÞ ¼ f 0 ðηÞ þ
Rfm ðηÞ ¼
ð1 qÞLθ ½Θðη; qÞ θ0 ðηÞ ¼ qℏNθ f^ ðη; qÞ; Θðη; qÞ; Φðη; qÞ ;
∂f^ ðη; qÞ ∂η
ð23Þ
By Taylor's series expansion one obtains
ð16Þ
Let q A ½0; 1 be an embedding parameter and ℏ be the non-zero auxiliary parameter, then one can express the generalized homotopic equations corresponding to (7)–(10) as h i ð17Þ ð1 qÞLf ½f^ ðη; qÞ f 0 ðηÞ ¼ qℏNf f^ ðη; qÞ ;
f^ ðη; qÞ
Nt ∂2 Θðη; qÞ : Nb ∂η2
Lθ θm ðηÞ χ m θm 1 ðηÞ ¼ ℏR θm ðηÞ;
We solve the boundary value problems given in Eqs. (7)–(10) by optimal homotopy analysis method (OHAM) [37,38]. Through the rule of solution expression and the boundary conditions (10) we select the following initial guesses f 0 ;θ0 and ϕ0 of f ðηÞ;θðηÞ and ϕðηÞ f 0 ðηÞ ¼ 1 expð ηÞ;
þ
3
ϕ
Z ¼
1 0
¼
2
0
4Nθ @
M X
f j ðηÞ;
j¼0
Z M
j¼0
1 0
2
0
4Nϕ @
M X j¼0
M X
θj ðηÞ;
j¼0
f j ðηÞ;
M X j¼0
M X
13 2 ϕj ðηÞA5 dη;
ð36Þ
j¼0
θj ðηÞ;
M X
13 2 ϕj ðηÞA5 dη:
ð37Þ
j¼0
Such kind of error has been considered in other works [39–42]. The smaller ζ M 0 s; the more accurate the Mth order approximation of the solution. A sample of the optimal values of ℏ for the functions θ and ϕ are given in Table 1.
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Table 1 Optimal values of ℏ for the functions θ and ϕ when Pr ¼ Sc ¼ 1 and Nb ¼ Nt ¼ 0:5: n
Optimal ℏ for θ
Minimum ζ θ M
Optimal ℏ for ϕ
Minimum ζ ϕ M
1 3 4 5
0.873 0.606 0.473 0.403
1.25 10 10 3.13 10 9 3.37 10 8 7.29 10 7
0.827 0.608 0.493 0.407
1.41 10 9 1.35 10 8 5.89 10 7 5.56 10 6
hj ðZ þ Z j 1 Þ; 2 j hj n þ 3 hj ðf þ f j 1 ÞðV j þ V j 1 Þ n ðSj þ Sj 1 Þ2 ¼ 0; Vj Vj1 þ 2 4 j 4 ϕj ϕj 1 ¼
ð47Þ hj n þ3 Pr ðf j þf j 1 ÞðY j þY j 1 Þ Yj Yj1 þ 2 4 hj þ Pr Nb ðY j þ Y j 1 ÞðZ j þ Z j 1 Þ 4 hj þ Pr Nt ðY j þ Y j 1 Þ2 ¼ 0; 4 Zj Zj 1 þ
Fig. 2. Discretization for difference approximations.
Eqs. (7)–(10) has been solved numerically by using implicit finite difference scheme known as Keller–Box method [43]. It is a four step method outlined as under: 1. The differential equations are reduced into first order system by using appropriate substitutions. 2. First-order differential equations are converted into difference equations using central differences. 3. The difference equations are linearized using Newton's method. 4. The resulting equations are written in matrix–vector form and then solved by block-tridiagonal-elimination procedure. Introduction the new dependent variables SðηÞ; VðηÞ; YðηÞ and ZðηÞ such that 0
S0 ¼ V ;
θ0 ¼ Y and ϕ0 ¼ Z;
ð38Þ
Eqs. (7)–(10) reduce to the following first-order system as below: V0 þ
nþ3 f V nS2 ¼ 0; 2
ð39Þ
Y0 þ
nþ3 Pr f Y þ Pr Nb YZ þ Pr Nt Y 2 ¼ 0; 2
ð40Þ
Z0 þ
nþ3 Nt Scf Z þ Y 0 ¼ 0; 2 Nb
ð41Þ
f ð0Þ ¼ 0;
Sð0Þ ¼ 1;
Sð1Þ-0;
θð0Þ ¼ 1;
θð1Þ-0;
Nb Zð0Þ þ Nt Yð0Þ ¼ 0;
ϕð1Þ-0:
ð42Þ
Let us consider a small interval along the η axis as shown in figure where the points within the interval having arbitrary step size hj are defined as below η0 ¼ 0;
ηj ¼ ηj 1 þ hj ;
j ¼ 1; 2; :::; J;
ð48Þ
hj nþ3 Nt Sc ðf j þ f j 1 ÞðZ j þ Z j 1 Þ þ ðY j Y j 1 Þ ¼ 0; 2 Nb 4 ð49Þ
4. Numerical solution by Keller–Box method
f ¼ S;
ð46Þ
ηJ ¼ η1
f 0 ¼ 0;
S0 ¼ 1;
SJ ¼ 0;
θJ ¼ 0;
θ0 ¼ 1;
Nt Z 0 ¼ Nb Y0;
ϕJ ¼ 0:
ð50Þ
Now we use Newton's method to linearize the system of Eqs. (43)– (50) by introducing following iterate: kþ1
fj
k
k
¼ f j þ δf j ;
Skj þ 1 ¼ Skj þ δSkj ;
θkj þ 1 ¼ θkj þ δθkj ;
Y kj þ 1 ¼ Y kj þ δY kj ;
ϕkj þ 1 ¼ ϕkj þδϕkj ;
Z kj þ 1 ¼ Z kj þ δZ kj ;
V kj þ 1 ¼ V kj þ δV kj ; ð51Þ
where k is iteration index. After substituting Eq. (51) into Eqs. (43)–(50) and dropping the quadratic and higher order terms of k δf j ; δSkj ; δV kj ; δθkj ; δY kj ; δϕkj and δZ kj we obtain the following linear tridiagonal system: Aδ ¼ r;
ð52Þ
in which the matrices A; δ and r can be expressed as below: 2 3 2 3 ½δ1 ½A1 ½C 1 6 ½B ½A 7 6 7 ½C 2 2 6 2 7 6 ½δ2 7 6 7 6 7 7; ½δ ¼ 6 ⋮ 7; ⋱ ⋱ ⋱ ½A ¼ 6 6 7 6 7 6 7 6 7 ½BJ 1 ½AJ 1 ½C J 1 5 4 4 ½δJ 1 5 2
3
½BJ
½AJ
½r 1 6 ½r 7 6 2 7 6 7 ⋮ 7; ½r ¼ 6 6 7 6 ½r 7 4 J 1 5 ½r J
½δJ
ð53Þ
The solution of the system (52) can be determined by employing block-tridiagonal elimination technique. The whole procedure can be found in Ibrahim and Shanker [44].
5. Numerical results and discussion
Using central differences the functions F and F 0 at point ηj 1=2 are expressed as F j 1=2 ¼ ðF j þF j 1 Þ=2 and F 0j 1=2 ¼ ðF j F j 1 Þ=hj (Fig. 2). In view of the above definitions, Eqs. (38)–(42) can be approximated within the segment P 1 P 2 as f j f j1 ¼
hj ðS þ Sj 1 Þ; 2 j
ð43Þ
Sj Sj 1 ¼
hj ðV þ V j 1 Þ; 2 j
ð44Þ
θj θj 1 ¼
hj ðY þ Y j 1 Þ; 2 j
ð45Þ
A comparison of numerical solution with the exact solution given in Eq. (14) is shown in Fig. 3. We notice that data obtained by two solutions is identical showing the validation of numerical computations. In Figs. 4 and 5, we have compared the 15th-order OHAM solutions for temperature and nanoparticle volume fraction distribution with the numerical solutions for various values of n. The two solutions yield virtually similar results in all the cases as demonstrated in these figures. Increasing values of n reduce the thickness of thermal and nanoparticle volume fraction boundary layers. This in turn augments the rate of heat transfer from the sheet. This rise in θ'ð0Þ is due to the fact that increasing n enhances the convective properties of the fluid since it increases the deformation by the shear
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Fig. 3. Temperature profiles for different values of Pr. Lines: numerical solution and circles: exact solution (15).
5
Fig. 6. Effect of Pr on θ.
Fig. 7. Effect of Sc on θ. Fig. 4. Temperature profiles for different values of n. Lines: numerical solution; circles: 15th-order analytic solution.
Fig. 8. Effect of Nt on θ Fig. 5. Nanoparticle volume fraction profiles for different values of n. Lines: numerical solution; circles: 15th-order analytic solution.
stress from the wall to the fluid. In Fig. 6, temperature θ is plotted versus the similarity variable η with the variation in Pr. Temperature decreases and becomes equal to the ambient temperature at smaller values of η as Pr is increased indicating a diminution in the penetration depth for thermal boundary layer. This results in the
enhancement of heat transfer rate from the sheet. This fact can also be understood from the fact that profiles become steeper with an augmentation of Pr signifying an increase in wall slope of temperature distribution. Fig. 7 depicts the influence of Schmidt number on the temperature distribution. Schmidt number is analogous of Pr for mass transfer and relates the viscous or hydrodynamic boundary
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Fig. 9. Effect of Pr on ϕ.
Fig. 10. Effect of Sc on ϕ.
Fig. 12. Effect of Nb on ϕ.
Fig. 13. Effect of Prand n on –θ0 ð0Þ.
Fig. 11. Effect of Nt on ϕ
layer to nanoparticle volume fraction boundary layer. Increasing Sc corresponds to stronger viscous diffusion compared to mass diffusion which enhances molecular motions and their interactions and hence temperature rises. The variation in temperature θ with increase in Sc is only accounted in the vicinity of the stretching sheet. Fig. 8 shows the impact of thermophoretic diffusion on the thermal boundary layer. A stronger thermophoretic force drives the particles from the hot sheet towards the ambient fluid and thus affects larger extent of
Fig. 14. Effect of Sc and n on –θ0 ð0Þ.
the fluid. Due to this reason, thermal boundary layer grows with an increment in Nt. In accordance with Kuznetsov and Nield [24], the outcome of increase in Brownian motion parameter on temperature is negligible. It can also be seen that profiles become less steep with an increase in Nt revealing a decrease in the magnitude of reduced Nusselt number.
Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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7
6. Concluding remarks Steady axisymmetric flow of nanofluid due to the non-linearly stretching sheet is examined through the revised model proposed by Kuznetsov and Nield [24]. Optimal homotopy analytic solutions of the arising boundary value problem are obtained. An implicit finite difference scheme is employed to compute the numerical solutions. The important points of present analysis are as under:
Fig. 15. Effect of Nt and n on –θ0 ð0Þ.
Table 2 Numerical values of reduced Nusselt number Nur for different values of parameters. Nt
Sc
Pr
Nur ¼ θ'ð0Þ
20
5
1.0
0.1 0.5 0.7 0.5
5
2.5
0.5
5 10 20 20
1.9112911 1.2170065 0.9815765 1.6914582 1.4740787 1.2861370 0.6619164 1.4784288 1.5758736
n
0.5
0.7 5 7
The profiles of nanoparticle volume fraction ϕ corresponding to Pr ¼ 0:7; 1; 3; 5; 10 are sketched in Fig. 9. We found that ϕ is an increasing function of Pr. The effect of Schmidt number on nanoparticle volume fraction ϕ can be perceived from Fig. 10. It is noticed earlier that increase in Sc corresponds to decrease the Brownian diffusion coefficient DB . The smaller DB gives the shorter penetration depth of nanoparticle volume fractionϕ. An increase in Nt shifts the nanoparticles away from the sheet and forms a particle free layer in the vicinity of the sheet (see Fig. 11). As a consequence the nanoparticle volume fraction distribution is formed just outside. On the other hand, the nanoparticle volume fraction boundary layer becomes thinner when strength of Brownian motion is increased as can be seen from Fig. 12. In Figs. 13–15, the reduced Nusselt number θ0 ð0Þ versus the power-law index n is plotted with the variations of Pr; ScandNt. For bigger Pr the convection (or the heat flow through unit area) is stronger in comparison to pure conduction. As a result the heat transfer rate is an increasing function of Pr and it decreases when Sc is increased. Magnitude of Nusselt number also decreases when Nt is increased. This diminution occurs due to the fact that larger thermophoretic force drives the nanoparticles of high thermal conductivity from the hot sheet to the quiescent fluid. It is notable that in contrast to the previous studies on constant wall nanoparticle volume fraction, here the reduced Nusselt number is independent of the Brownian motion parameter. This outcome is consistent with This can be easily understood by using ϕ0 ð0Þ ¼ Nt=Nb θ0 ð0Þ (from Eq. (10)) in the energy equation (8) as the wall is approached. Table 2 includes the sample of our results for reduced Nusselt number θ0 ð0Þ for different values of embedded parameters. It is found that wall temperature gradient θ0 ð0Þ is a decreasing function of n; Nt and Sc and it is an increasing function of Pr.
(a) The analytic and numerical solutions are found in excellent agreement. The solutions also agree with the exact solution in the case of regular fluid when n ¼ 3: (b) The influence of Brownian motion on the temperature and wall heat transfer rate is insignificant. However the penetration depth of nanoparticle volume fraction boundary layer is reduced when Brownian motion intensifies. (c) Thermal boundary layer becomes thicker as a result of strong thermophoretic diffusion. This in turn reduces the magnitude of reduced Nusselt number. (d) Increase in Schmidt number Sc, which is inversely proportional to the Brownian diffusion coefficient DB , corresponds to a thinner nanoparticle volume fraction boundary layer. Further there is increase in temperature function only close to the sheet when Sc is increased. (e) The equations reduce to the case of linearly stretching sheet when n ¼ 1:
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Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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Please cite this article as: M. Mustafa, et al., Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, International Journal of Non-Linear Mechanics (2015), http://dx.doi.org/10.1016/j.ijnonlinmec.2015.01.005i
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