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Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model
Applied Mathematics Letters 38 (2014) 87–93
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model Shihao Han a , Liancun Zheng a,∗ , Chunrui Li a,b , Xinxin Zhang b a
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
b
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
article
info
Article history: Received 11 June 2014 Accepted 11 July 2014 Available online 22 July 2014 Keywords: Upper-convected Maxwell fluid Boundary layer flow Cattaneo–Christov heat flux model Analytical solutions Slip boundary
abstract This letter presents a research for coupled flow and heat transfer of an upper-convected Maxwell fluid above a stretching plate with velocity slip boundary. Unlike most classical works, the new heat flux model, which is recently proposed by Christov, is employed. Analytical solutions are obtained by using the homotopy analysis method (HAM). The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed. A comparison of Fourier’s Law and the Cattaneo–Christov heat flux model is also presented. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Heat transfer is a widespread natural phenomenon of nature, as long as there is a temperature difference between objects or between different parts of the same object, there will be heat transfer phenomenon. Therefore, a considerable attention has been devoted to predict the heat transport behavior. In 1822, Fourier [1] firstly proposed the famous law of heat conduction law. Though this law is an empirical formula, it provides a way to study heat transfer phenomenon and becomes the basis to study heat conduction in the subsequent two centuries. Cattaneo [2] proposed a modified Fourier heat conduction law, by adding a relaxation time term, to overcome the ‘paradox of heat conduction’, which is ‘any initial disturbance is felt instantly throughout the whole of the medium under consideration’ [3,4]. Christov [3] proposed the frame-indifferent generalization of Cattaneo’ law by using the Oldroyds’ upper-convected derivative [5], who derived a single temperature governing equation. The uniqueness and structural stability of the solutions for the temperature governing equations with Cattaneo–Christov heat flux model in some initial and boundary problem, have been proved by Tibullo and Zampoli [4], Ciarletta and Straughan [6]. Straughan [7] and Haddad [8] obtained the numerical solutions for thermal convection of incompressible Newtonian fluid with Cattaneo–Christov heat flux model by using the D2 Chebyshev tau method. The Maxwell fluid is an important class of viscoelastic fluid models. In the past few decades, a large number of research achievements regarding Maxwell fluid model have been published. Among these literatures, boundary layer problems of upper-convected Maxwell fluid have obtained attention. Choi et al. [9] studied the steady suction flow, the effects of viscoelasticity and inertia in a porous channel were considered. Sadeghy et al. [10] investigated Sakiadis flow by employing three different mathematical methods and pointed out the wall skin friction coefficient decreases when increasing the value of Deborah number. MHD boundary layer flows in porous objects or space were discussed in [11–14] and stagnation-point
∗
Corresponding author. Tel.: +86 1062332002; fax: +86 1062332462. E-mail addresses: [email protected], [email protected] (L. Zheng).
http://dx.doi.org/10.1016/j.aml.2014.07.013 0893-9659/© 2014 Elsevier Ltd. All rights reserved.
88
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
flows of Maxwell fluid were studied in [15–18]. Hayat et al. [19] investigated chemical reaction effects on mass transfer and MHD boundary layer flow above a porous shrinking sheet. The convective boundary conditions and viscous dissipation heat transfer over a moving surface were considered in [20]. Aliakbar et al. [21] investigated heat transfer of an upper-convected Maxwell fluid with viscous dissipation and thermal radiation. The homotopy analysis method proposed by Liao [22,23], is an effective mathematical method to solve nonlinear differential equations. Analytical solution has important significance in the study of physical problems. According to current knowledge of the authors, no literature presented the analytical solutions for the temperature governing equation with Cattaneo–Christov heat flux model. Therefore, we will firstly obtain the analytical solutions by using homotopy analysis method. The main purpose of this letter is to study the coupled flow and heat transfer of boundary layer in viscoelastic fluid with the upper-convected Maxwell model and Cattaneo–Christov heat flux model, the similarity transformation and homotopy analysis method are used with a view to obtain analytical solutions. The effects of involved parameters on the velocity and temperature fields are analyzed and discussed. 2. The basic equations Consider a two dimensional steady boundary layer flow of the Maxwell fluid over a plate, in the absence of the pressure gradient, the governing equation for expressing conservation of mass, momentum can be written as [10,20]
∂ u ∂v + =0 ∂x ∂y u
(1)
∂u ∂u ∂ 2u ∂ 2u ∂ 2u ∂ 2u +v + λ1 u2 2 + v 2 2 + 2uv = ν 2. ∂x ∂y ∂x ∂y ∂ x∂ y ∂y
(2)
The boundary conditions with velocity slip on boundary are: u = ax + λ0
2 − συ ∂ u
y=0 , v = 0 at y = 0, u → 0 as y → ∞ (3) ∂y where a is positive constant, συ is the tangential momentum accommodation coefficient, λ0 is the molecular mean free path. συ
We use the Cattaneo–Christov heat flux model to study heat transfer of the Maxwell fluid, the heat flux model is the following form [3]
∂q + V · ∇ q − q · ∇ V + (∇ · V) q = −k∇ T (4) ∂t where q is heat flux, λ2 is relaxation time of the heat flux, T is temperature of the Maxwell fluid, k is the thermal conductivity. V = (u, v) is velocity vector of the Maxwell fluid. When λ2 = 0, Eq. (4) is simplified to Fourier’s law. Since the fluid is incompressible, which satisfies ∇ · V = 0, Eq. (4) becomes the following form [4] ∂q q + λ2 + V · ∇ q − q · ∇ V = −k∇ T . (5) ∂t
q + λ2
The energy balance equation for the steady boundary layer flow is
ρ cp V · ∇ T = −∇ · q.
(6)
Eliminating q between Eqs. (5) and (6), we can obtain the temperature governing equation for the steady flow
2 2 ∂T ∂T ∂u ∂T ∂v ∂ T ∂v ∂ T ∂u ∂T ∂ 2T ∂ 2T 2∂ T 2∂ T u +v + λ2 u +v +u +v + 2uv +u + v = α ∂x ∂y ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x ∂ x∂ y ∂ x2 ∂ y2 ∂ y2 where α = k/ρ cp is thermal diffusivity. The corresponding boundary conditions are T = Tw
at y = 0,
T → T∞
as y → ∞.
(7)
(8)
Employing the following transformations
√ a T − T∞ η=y , ψ = x ν af (η) , θ= (9) ν Tw − T∞ where ψ is the stream function and has the relationship: u = ∂ψ/∂ y, v = −∂ψ/∂ x. We obtain the following nonlinear coupled equations f ′′′ − f
′2
+ ff ′′ + β 2ff ′ f ′′ − f 2 f ′′′ = 0
1 ′′ θ + f θ ′ − γ ff ′ θ ′ + f 2 θ ′′ = 0
Pr
where β = λ1 a, γ = λ2 a, Pr = ν/α is the Prandtl number.
(10) (11)
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
89
The boundary conditions (3) and (8) become f ′ (η) = 1 + bf ′′ (η) ,
f (η) = 0, υ where b = λ0 2−σ σ
a
ν
υ
θ (η) = 1 at η = 0,
f ′ (η) → 0,
θ (η) → 0 as η → ∞
(12)
.
3. HAM solutions of the problem In this section, we employ the homotopy analysis method [22] to obtain the series solutions for the strong nonlinear coupled equations (10) and (11) with boundary conditions (12). Firstly, we chose the initial guess solutions which satisfy boundary conditions (12) as follows f0 (η) =
1
1 − e−η ,
1+b
θ0 (η) = e−η .
(13)
The auxiliary linear operators and non-linear operators are given by Lf (f ) = f ′′′ + f ′′ ,
Nf fˆ (η; p)
Lθ (θ ) = θ ′′ − θ
(14)
2
=
2ˆ ∂ 3 fˆ (η; p) ∂ fˆ (η; p) ˆ (η; p) ∂ f (η; p) − + f ∂η3 ∂η ∂η2 2 ∂ 3 fˆ (η; p) 2 ∂ fˆ (η; p) ∂ fˆ (η; p) ˆ ˆ + β 2f (η; p) − f (η; p) ∂η ∂η2 ∂η3
Nθ fˆ (η; p) ; θˆ (η; p)
=
1 ∂ 2 θˆ (η; p)
(15)
∂ θˆ (η; p) Pr ∂η ∂η 2 ∂ 2 θˆ (η; p) ∂ fˆ (η; p) ∂ θˆ (η; p) + fˆ (η; p) . − γ fˆ (η; p) ∂η ∂η ∂η2 2
+ fˆ (η; p)
(16)
The auxiliary linear operators in Eq. (14) have the property Lf (C1 + C2 η + C3 e−η ) = 0,
Lθ (C4 eη + C5 e−η ) = 0
(17)
where C1 , C2 , C3 , C4 and C5 are any constants. Like Refs. [22,23], we easily present the zeroth order problem
(1 − p) Lf fˆ (η, p) − f0 (η) = phf Nf fˆ (η, p) (1 − p) Lθ θˆ (η, p) − θ0 (η) = phθ Nθ fˆ (η, p) , θˆ (η, p) .
(18) (19)
The boundary conditions are fˆ (0, p) = 0,
fˆ ′ (0, p) = 1 + bfˆ ′′ (0, p) ,
θˆ (0, p) = 1,
fˆ ′ (∞, p) = 0,
θˆ (∞, p) = 0
(20)
where p ∈ [0, 1] is the embedding parameter, hf and hθ are the auxiliary non-zero parameters. When p = 0, fˆ (η, p) = f0 (η), θˆ (η, p) = θ0 (η), when p = 1, fˆ (η, p) = f (η), θˆ (η, p) = θ (η). Therefore, fˆ (η, p) and θˆ (η, p)can be written to the following Taylor series form fˆ (η, p) = f0 (η) +
∞
fm (η) pm ,
θˆ (η, p) = θ0 (η) +
m=1
where fm (η) =
m 1 ∂ fˆ (η,p) m! ∂ pm
∞
θm (η) pm
(21)
m=1
p=0 , θm (η) =
mˆ p) 1 ∂ θ(η, m! ∂ pm
p=0 .
As pointed by Liao [22], by selecting the appropriate hf and hθ , fˆ (η, p) and θˆ (η, p) are convergent at p = 1. Therefore, the solutions can be f (η) = f0 (η) +
∞
fm (η) ,
θ (η) = θ0 (η) +
m=1
∞
θm (η) .
(22)
m=1
Now we write high order deformation equations as Lf [fm (η) − χm fm−1 (η)] = hf Rfm (η) ,
Lθ [θm (η) − χm θm−1 (η)] = hθ Rθm (η)
(23)
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S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
(a) h-curves for velocity distribution.
(b) h-curves for temperature.
Fig. 1. The h-curves obtained by the 8-order approximation of the HAM when β = b = Pr = 1. 0.6
1.0
HAM Solution
HAM Solution
0.5
Numerical Solution
0.8
Numerical Solution 0.4
0.6
θ
f'
0.3 0.2
0.4
0.1
0.2
0.0 –0.1
0.0 0
1
2
3
η
4
0
5
(a) Velocity distribution f ′ (η).
1
2
3 η
4
5
6
(b) Temperature distribution θ(η).
Fig. 2. Comparison of the HAM solutions and numerical solutions when β = b = Pr = 1.
where
χm =
0, 1,
m≤1 m>1
(24)
m−1
Rfm (η) = fm′′′−1 −
i =0
Rθm (η) =
m−1
fm′ −i−1 fi′ +
j
m−1
fm−i−1 fi′′ + β
i=0
′ ′′ ′′′ 2fm−j−1 fj− i fi − fm−j−1 fj−i fi
(25)
j =0 i =0
j m −1 m −1 ′ ′ 1 ′′ ′′ θm−1 + fm−i−1 θi′ − γ fm−j−1 fj− . i θi + fj−i θi Pr i=0 j =0 i =0
(26)
The corresponding boundary conditions of (23) are fm (0) = 0,
fm′ (0) − bfm′′ (0) = 0,
fm′ (∞) = 0,
θm (0) = 0,
θm (∞) = 0
(27)
fm (η) and θm (η) can be obtained by solving Eq. (23) with the conditions (27). Then substituting fm (η) and θm (η) into (22), we can obtain approximate analytical solutions for the problem (10) and (11) with the boundary conditions (12). These steps can be implemented by using the Mathematical Software: Mathematica 9.0. From Ref. [22], we know that the auxiliary non-zero parameters hf and hθ strongly affect on the convergence of the HAM solutions. Therefore we must select appropriate hf and hθ to secure the HAM solutions of the problem are convergent. Fig. 1 is plotted to seek out the convergence region of the HAM solutions with the 8-order approximation. Fig. 1(a) indicates that the HAM solutions of the velocity distribution are convergent when −0.9 ≤ hf ≤ −0.1. Fig. 1(b) shows that the convergence region of temperature distribution is −1.2 ≤ hθ ≤ −0.4. Therefore we choose hf = hθ = −0.6 to secure the HAM solutions of the velocity and temperature distribution are all convergent. Comparison of the HAM solutions and numerical solutions for f ′ (η) and θ (η) is showed in Fig. 2. The numerical solutions are obtained by using finite difference method. Fig. 2 shows that the HAM solutions are in good agreement with the numerical solutions. 4. Results and discussion In this paper we employ the upper-convected Maxwell model and Cattaneo–Christov heat flux model to investigate heat transfer and boundary layer flow of a viscoelastic fluid above a stretching plate with velocity slip boundary. One pair of
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
91
0.8
0.6
f
0.4
0.2
0.0 0
1
2
3 η
4
5
6
Fig. 3. Profiles of f (η) for different values of β when b = 1. 0.6 0.5 0.4 f'
0.3 0.2 0.1 0.0 –0.1
0
1
2
3 η
4
5
6
Fig. 4. Profiles of f ′ (η) for different values of β when b = 1.
1.0 0.8
θ
0.6 0.4 0.2 0.0 0
1
2
3
η
4
5
6
7
Fig. 5. Profiles of θ(η) for different values of β when γ = b = Pr = 1.
coupled equations is obtained by applying the similarity transformation to the velocity and temperature governing equations. The approximate analytical solutions of the coupled equations (10) and (11) with the boundary conditions (12) are obtained by employing the homotopy analysis method. The upper-convected Maxwell model has one elastic term relative to the Newtonian fluid model. The elastic term indicates that the elastic force has some influence on flow and heat transfer of the viscoelastic fluid. Figs. 3–5 show the effects of elasticity number β on the velocity and temperature distribution, where β = 0 indicates that the elastic force disappears and the fluid becomes the Newtonian fluid, and the effects of elastic force increase with increasing of elasticity number β . Figs. 3 and 4 show that f (η) and f ′ (η) become smaller with the increase of β . This indicates that the velocity boundary layer is increasingly thin with the increase of β . Fig. 5 displays that fluid temperature θ (η) increases with increasing the value β . This indicates that the elastic force promotes heat transfer of the viscoelastic fluid. Figs. 6 and 7 show the slip effects on velocity distribution f (η), f ′ (η). These velocity distribution become smaller with increasing the value of slip coefficient b. The Cattaneo–Christov heat flux model is frame-indifferent generalization of Fourier’s Law [3]. In this model, the fluid velocity is introduced into the constitutive relationship of the fluid temperature and heat flux. This indicates that heat flux is not only related to temperature gradient but also related to the fluid velocity. Fig. 8 shows the γ effects on temperature distribution θ (η), where γ = 0 indicates that the heat flux model is simplified to Fourier’s Law. Fig. 8 displays that the temperature boundary layer are thinner with increasing of γ . Fig. 9 displays the effects of Prandtl number Pr on energy boundary. It is seen that the bigger the Prandtl number, the thinner the temperature boundary layer is.
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93 0.7 0.6 0.5 0.4 f
0.3 b = 1, 2, 3 0.2 0.1 0.0 –0.1
0
2
4 η
6
8
Fig. 6. Profiles of f (η) for different values of b when β = 1.
0.6 0.5 0.4 0.3 f'
b = 1, 2, 3
0.2 0.1 0.0 −0.1
0
1
2
3 η
4
5
6
Fig. 7. Profiles of f ′ (η) for different values of b when β = 1.
1.0 0.8
θ
0.6 0.4 0.2 0.0 0
2
4 η
6
8
Fig. 8. Profiles of θ(η) for different values of γ when β = b = Pr = 1.
1.0 0.8 0.6
Pr = 0.5, 1, 2, 3
θ
92
0.4 0.2 0.0 0
2
4
η
6
8
10
Fig. 9. Profiles of θ(η) for different values of Pr when β = b = γ = 1.
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
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Acknowledgment The work is supported by the National Natural Science Foundations of China (No. 51276014). References [1] J.B.J. Fourier, Théorie Analytique De La Chaleur, Paris, 1822. [2] C. Cattaneo, Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 3 (1948) 83–101. [3] C.I. Christov, On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction, Mech. Res. Commun. 36 (2009) 481–486. [4] V. Tibullo, V. Zampoli, A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids, Mech. Res. Commun. 38 (2011) 77–79. [5] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 200 (1949) 523–541. [6] M. Ciarletta, B. Straughan, Uniqueness and structural stability for the Cattaneo–Christov equations, Mech. Res. Commun. 37 (2010) 445–447. [7] B. Straughan, Thermal convection with the Cattaneo–Christov model, Int. J. Heat Mass Transfer 53 (2010) 95–98. [8] S.A.M. Haddad, Thermal instability in Brinkman porous media with Cattaneo–Christov heat flux, Int. J. Heat Mass Transfer 68 (2014) 659–668. [9] J.J. Choi, Z. Rusak, J.A. Tichy, Maxwell fluid suction flow in a channel, J. Non-Newtonian Fluid Mech. 85 (1999) 165–187. [10] Kayvan Sadeghy, Amir-Hosain Najafi, Meghdad Saffaripour, Sakiadis flow of an upper-convected Maxwell fluid, Int. J. Non-Linear Mech. 40 (2005) 1220–1228. [11] Z. Abbas, M. Sajid, T. Hayat, MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel, Theor. Comput. Fluid Dyn. 20 (2006) 229–238. [12] T. Hayat, Z. Abbas, M. Sajid, Series solution for the upper-convected Maxwell fluid over a porous stretching plate, Phys. Lett. A 358 (2006) 396–403. [13] Amir Alizadeh-Pahlavan, Vahid Aliakbar, Farzad Vakili-Farahani, Kayvan Sadeghy, MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 473–488. [14] B. Raftari, A. Yildirim, The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets, Comput. Math. Appl. 59 (2010) 3328–3337. [15] Kayvan Sadeghy, Hadi Hajibeygi, Seyed-Mohammad Taghavi, Stagnation-point flow of upper-convected Maxwell fluids, Int. J. Non-Linear Mech. 41 (2006) 1242–1247. [16] T. Hayat, Z. Abbas, M. Sajid, MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface, Chaos Solitons Fractals 39 (2009) 840–848. [17] M. Kumaria, G. Nath, Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field, Int. J. Non-Linear Mech. 44 (2009) 1048–1055. [18] T. Hayat, M. Awais, M. Qasim, A.A. Hendi, Effects of mass transfer on the stagnation point flow of an upper-convected Maxwell (UCM) fluid, Int. J. Heat Mass Transfer 54 (2011) 3777–3782. [19] T. Hayat, Z. Abbas, N. Ali, MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys. Lett. A 372 (2008) 4698–4704. [20] T. Hayat, Z. Iqbal, M. Mustafa, A. Alsaedi, Momentum and heat transfer of an upper-convected Maxwell fluid over a moving surface with convective boundary conditions, Nucl. Eng. Des. 252 (2012) 242–247. [21] V. Aliakbar, A. Alizadeh-Pahlavan, K. Sadeghy, The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 779–794. [22] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003. [23] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model Shihao Han a , Liancun Zheng a,∗ , Chunrui Li a,b , Xinxin Zhang b a
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
b
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
article
info
Article history: Received 11 June 2014 Accepted 11 July 2014 Available online 22 July 2014 Keywords: Upper-convected Maxwell fluid Boundary layer flow Cattaneo–Christov heat flux model Analytical solutions Slip boundary
abstract This letter presents a research for coupled flow and heat transfer of an upper-convected Maxwell fluid above a stretching plate with velocity slip boundary. Unlike most classical works, the new heat flux model, which is recently proposed by Christov, is employed. Analytical solutions are obtained by using the homotopy analysis method (HAM). The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed. A comparison of Fourier’s Law and the Cattaneo–Christov heat flux model is also presented. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Heat transfer is a widespread natural phenomenon of nature, as long as there is a temperature difference between objects or between different parts of the same object, there will be heat transfer phenomenon. Therefore, a considerable attention has been devoted to predict the heat transport behavior. In 1822, Fourier [1] firstly proposed the famous law of heat conduction law. Though this law is an empirical formula, it provides a way to study heat transfer phenomenon and becomes the basis to study heat conduction in the subsequent two centuries. Cattaneo [2] proposed a modified Fourier heat conduction law, by adding a relaxation time term, to overcome the ‘paradox of heat conduction’, which is ‘any initial disturbance is felt instantly throughout the whole of the medium under consideration’ [3,4]. Christov [3] proposed the frame-indifferent generalization of Cattaneo’ law by using the Oldroyds’ upper-convected derivative [5], who derived a single temperature governing equation. The uniqueness and structural stability of the solutions for the temperature governing equations with Cattaneo–Christov heat flux model in some initial and boundary problem, have been proved by Tibullo and Zampoli [4], Ciarletta and Straughan [6]. Straughan [7] and Haddad [8] obtained the numerical solutions for thermal convection of incompressible Newtonian fluid with Cattaneo–Christov heat flux model by using the D2 Chebyshev tau method. The Maxwell fluid is an important class of viscoelastic fluid models. In the past few decades, a large number of research achievements regarding Maxwell fluid model have been published. Among these literatures, boundary layer problems of upper-convected Maxwell fluid have obtained attention. Choi et al. [9] studied the steady suction flow, the effects of viscoelasticity and inertia in a porous channel were considered. Sadeghy et al. [10] investigated Sakiadis flow by employing three different mathematical methods and pointed out the wall skin friction coefficient decreases when increasing the value of Deborah number. MHD boundary layer flows in porous objects or space were discussed in [11–14] and stagnation-point
∗
Corresponding author. Tel.: +86 1062332002; fax: +86 1062332462. E-mail addresses: [email protected], [email protected] (L. Zheng).
http://dx.doi.org/10.1016/j.aml.2014.07.013 0893-9659/© 2014 Elsevier Ltd. All rights reserved.
88
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
flows of Maxwell fluid were studied in [15–18]. Hayat et al. [19] investigated chemical reaction effects on mass transfer and MHD boundary layer flow above a porous shrinking sheet. The convective boundary conditions and viscous dissipation heat transfer over a moving surface were considered in [20]. Aliakbar et al. [21] investigated heat transfer of an upper-convected Maxwell fluid with viscous dissipation and thermal radiation. The homotopy analysis method proposed by Liao [22,23], is an effective mathematical method to solve nonlinear differential equations. Analytical solution has important significance in the study of physical problems. According to current knowledge of the authors, no literature presented the analytical solutions for the temperature governing equation with Cattaneo–Christov heat flux model. Therefore, we will firstly obtain the analytical solutions by using homotopy analysis method. The main purpose of this letter is to study the coupled flow and heat transfer of boundary layer in viscoelastic fluid with the upper-convected Maxwell model and Cattaneo–Christov heat flux model, the similarity transformation and homotopy analysis method are used with a view to obtain analytical solutions. The effects of involved parameters on the velocity and temperature fields are analyzed and discussed. 2. The basic equations Consider a two dimensional steady boundary layer flow of the Maxwell fluid over a plate, in the absence of the pressure gradient, the governing equation for expressing conservation of mass, momentum can be written as [10,20]
∂ u ∂v + =0 ∂x ∂y u
(1)
∂u ∂u ∂ 2u ∂ 2u ∂ 2u ∂ 2u +v + λ1 u2 2 + v 2 2 + 2uv = ν 2. ∂x ∂y ∂x ∂y ∂ x∂ y ∂y
(2)
The boundary conditions with velocity slip on boundary are: u = ax + λ0
2 − συ ∂ u
y=0 , v = 0 at y = 0, u → 0 as y → ∞ (3) ∂y where a is positive constant, συ is the tangential momentum accommodation coefficient, λ0 is the molecular mean free path. συ
We use the Cattaneo–Christov heat flux model to study heat transfer of the Maxwell fluid, the heat flux model is the following form [3]
∂q + V · ∇ q − q · ∇ V + (∇ · V) q = −k∇ T (4) ∂t where q is heat flux, λ2 is relaxation time of the heat flux, T is temperature of the Maxwell fluid, k is the thermal conductivity. V = (u, v) is velocity vector of the Maxwell fluid. When λ2 = 0, Eq. (4) is simplified to Fourier’s law. Since the fluid is incompressible, which satisfies ∇ · V = 0, Eq. (4) becomes the following form [4] ∂q q + λ2 + V · ∇ q − q · ∇ V = −k∇ T . (5) ∂t
q + λ2
The energy balance equation for the steady boundary layer flow is
ρ cp V · ∇ T = −∇ · q.
(6)
Eliminating q between Eqs. (5) and (6), we can obtain the temperature governing equation for the steady flow
2 2 ∂T ∂T ∂u ∂T ∂v ∂ T ∂v ∂ T ∂u ∂T ∂ 2T ∂ 2T 2∂ T 2∂ T u +v + λ2 u +v +u +v + 2uv +u + v = α ∂x ∂y ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x ∂ x∂ y ∂ x2 ∂ y2 ∂ y2 where α = k/ρ cp is thermal diffusivity. The corresponding boundary conditions are T = Tw
at y = 0,
T → T∞
as y → ∞.
(7)
(8)
Employing the following transformations
√ a T − T∞ η=y , ψ = x ν af (η) , θ= (9) ν Tw − T∞ where ψ is the stream function and has the relationship: u = ∂ψ/∂ y, v = −∂ψ/∂ x. We obtain the following nonlinear coupled equations f ′′′ − f
′2
+ ff ′′ + β 2ff ′ f ′′ − f 2 f ′′′ = 0
1 ′′ θ + f θ ′ − γ ff ′ θ ′ + f 2 θ ′′ = 0
Pr
where β = λ1 a, γ = λ2 a, Pr = ν/α is the Prandtl number.
(10) (11)
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
89
The boundary conditions (3) and (8) become f ′ (η) = 1 + bf ′′ (η) ,
f (η) = 0, υ where b = λ0 2−σ σ
a
ν
υ
θ (η) = 1 at η = 0,
f ′ (η) → 0,
θ (η) → 0 as η → ∞
(12)
.
3. HAM solutions of the problem In this section, we employ the homotopy analysis method [22] to obtain the series solutions for the strong nonlinear coupled equations (10) and (11) with boundary conditions (12). Firstly, we chose the initial guess solutions which satisfy boundary conditions (12) as follows f0 (η) =
1
1 − e−η ,
1+b
θ0 (η) = e−η .
(13)
The auxiliary linear operators and non-linear operators are given by Lf (f ) = f ′′′ + f ′′ ,
Nf fˆ (η; p)
Lθ (θ ) = θ ′′ − θ
(14)
2
=
2ˆ ∂ 3 fˆ (η; p) ∂ fˆ (η; p) ˆ (η; p) ∂ f (η; p) − + f ∂η3 ∂η ∂η2 2 ∂ 3 fˆ (η; p) 2 ∂ fˆ (η; p) ∂ fˆ (η; p) ˆ ˆ + β 2f (η; p) − f (η; p) ∂η ∂η2 ∂η3
Nθ fˆ (η; p) ; θˆ (η; p)
=
1 ∂ 2 θˆ (η; p)
(15)
∂ θˆ (η; p) Pr ∂η ∂η 2 ∂ 2 θˆ (η; p) ∂ fˆ (η; p) ∂ θˆ (η; p) + fˆ (η; p) . − γ fˆ (η; p) ∂η ∂η ∂η2 2
+ fˆ (η; p)
(16)
The auxiliary linear operators in Eq. (14) have the property Lf (C1 + C2 η + C3 e−η ) = 0,
Lθ (C4 eη + C5 e−η ) = 0
(17)
where C1 , C2 , C3 , C4 and C5 are any constants. Like Refs. [22,23], we easily present the zeroth order problem
(1 − p) Lf fˆ (η, p) − f0 (η) = phf Nf fˆ (η, p) (1 − p) Lθ θˆ (η, p) − θ0 (η) = phθ Nθ fˆ (η, p) , θˆ (η, p) .
(18) (19)
The boundary conditions are fˆ (0, p) = 0,
fˆ ′ (0, p) = 1 + bfˆ ′′ (0, p) ,
θˆ (0, p) = 1,
fˆ ′ (∞, p) = 0,
θˆ (∞, p) = 0
(20)
where p ∈ [0, 1] is the embedding parameter, hf and hθ are the auxiliary non-zero parameters. When p = 0, fˆ (η, p) = f0 (η), θˆ (η, p) = θ0 (η), when p = 1, fˆ (η, p) = f (η), θˆ (η, p) = θ (η). Therefore, fˆ (η, p) and θˆ (η, p)can be written to the following Taylor series form fˆ (η, p) = f0 (η) +
∞
fm (η) pm ,
θˆ (η, p) = θ0 (η) +
m=1
where fm (η) =
m 1 ∂ fˆ (η,p) m! ∂ pm
∞
θm (η) pm
(21)
m=1
p=0 , θm (η) =
mˆ p) 1 ∂ θ(η, m! ∂ pm
p=0 .
As pointed by Liao [22], by selecting the appropriate hf and hθ , fˆ (η, p) and θˆ (η, p) are convergent at p = 1. Therefore, the solutions can be f (η) = f0 (η) +
∞
fm (η) ,
θ (η) = θ0 (η) +
m=1
∞
θm (η) .
(22)
m=1
Now we write high order deformation equations as Lf [fm (η) − χm fm−1 (η)] = hf Rfm (η) ,
Lθ [θm (η) − χm θm−1 (η)] = hθ Rθm (η)
(23)
90
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
(a) h-curves for velocity distribution.
(b) h-curves for temperature.
Fig. 1. The h-curves obtained by the 8-order approximation of the HAM when β = b = Pr = 1. 0.6
1.0
HAM Solution
HAM Solution
0.5
Numerical Solution
0.8
Numerical Solution 0.4
0.6
θ
f'
0.3 0.2
0.4
0.1
0.2
0.0 –0.1
0.0 0
1
2
3
η
4
0
5
(a) Velocity distribution f ′ (η).
1
2
3 η
4
5
6
(b) Temperature distribution θ(η).
Fig. 2. Comparison of the HAM solutions and numerical solutions when β = b = Pr = 1.
where
χm =
0, 1,
m≤1 m>1
(24)
m−1
Rfm (η) = fm′′′−1 −
i =0
Rθm (η) =
m−1
fm′ −i−1 fi′ +
j
m−1
fm−i−1 fi′′ + β
i=0
′ ′′ ′′′ 2fm−j−1 fj− i fi − fm−j−1 fj−i fi
(25)
j =0 i =0
j m −1 m −1 ′ ′ 1 ′′ ′′ θm−1 + fm−i−1 θi′ − γ fm−j−1 fj− . i θi + fj−i θi Pr i=0 j =0 i =0
(26)
The corresponding boundary conditions of (23) are fm (0) = 0,
fm′ (0) − bfm′′ (0) = 0,
fm′ (∞) = 0,
θm (0) = 0,
θm (∞) = 0
(27)
fm (η) and θm (η) can be obtained by solving Eq. (23) with the conditions (27). Then substituting fm (η) and θm (η) into (22), we can obtain approximate analytical solutions for the problem (10) and (11) with the boundary conditions (12). These steps can be implemented by using the Mathematical Software: Mathematica 9.0. From Ref. [22], we know that the auxiliary non-zero parameters hf and hθ strongly affect on the convergence of the HAM solutions. Therefore we must select appropriate hf and hθ to secure the HAM solutions of the problem are convergent. Fig. 1 is plotted to seek out the convergence region of the HAM solutions with the 8-order approximation. Fig. 1(a) indicates that the HAM solutions of the velocity distribution are convergent when −0.9 ≤ hf ≤ −0.1. Fig. 1(b) shows that the convergence region of temperature distribution is −1.2 ≤ hθ ≤ −0.4. Therefore we choose hf = hθ = −0.6 to secure the HAM solutions of the velocity and temperature distribution are all convergent. Comparison of the HAM solutions and numerical solutions for f ′ (η) and θ (η) is showed in Fig. 2. The numerical solutions are obtained by using finite difference method. Fig. 2 shows that the HAM solutions are in good agreement with the numerical solutions. 4. Results and discussion In this paper we employ the upper-convected Maxwell model and Cattaneo–Christov heat flux model to investigate heat transfer and boundary layer flow of a viscoelastic fluid above a stretching plate with velocity slip boundary. One pair of
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93
91
0.8
0.6
f
0.4
0.2
0.0 0
1
2
3 η
4
5
6
Fig. 3. Profiles of f (η) for different values of β when b = 1. 0.6 0.5 0.4 f'
0.3 0.2 0.1 0.0 –0.1
0
1
2
3 η
4
5
6
Fig. 4. Profiles of f ′ (η) for different values of β when b = 1.
1.0 0.8
θ
0.6 0.4 0.2 0.0 0
1
2
3
η
4
5
6
7
Fig. 5. Profiles of θ(η) for different values of β when γ = b = Pr = 1.
coupled equations is obtained by applying the similarity transformation to the velocity and temperature governing equations. The approximate analytical solutions of the coupled equations (10) and (11) with the boundary conditions (12) are obtained by employing the homotopy analysis method. The upper-convected Maxwell model has one elastic term relative to the Newtonian fluid model. The elastic term indicates that the elastic force has some influence on flow and heat transfer of the viscoelastic fluid. Figs. 3–5 show the effects of elasticity number β on the velocity and temperature distribution, where β = 0 indicates that the elastic force disappears and the fluid becomes the Newtonian fluid, and the effects of elastic force increase with increasing of elasticity number β . Figs. 3 and 4 show that f (η) and f ′ (η) become smaller with the increase of β . This indicates that the velocity boundary layer is increasingly thin with the increase of β . Fig. 5 displays that fluid temperature θ (η) increases with increasing the value β . This indicates that the elastic force promotes heat transfer of the viscoelastic fluid. Figs. 6 and 7 show the slip effects on velocity distribution f (η), f ′ (η). These velocity distribution become smaller with increasing the value of slip coefficient b. The Cattaneo–Christov heat flux model is frame-indifferent generalization of Fourier’s Law [3]. In this model, the fluid velocity is introduced into the constitutive relationship of the fluid temperature and heat flux. This indicates that heat flux is not only related to temperature gradient but also related to the fluid velocity. Fig. 8 shows the γ effects on temperature distribution θ (η), where γ = 0 indicates that the heat flux model is simplified to Fourier’s Law. Fig. 8 displays that the temperature boundary layer are thinner with increasing of γ . Fig. 9 displays the effects of Prandtl number Pr on energy boundary. It is seen that the bigger the Prandtl number, the thinner the temperature boundary layer is.
S. Han et al. / Applied Mathematics Letters 38 (2014) 87–93 0.7 0.6 0.5 0.4 f
0.3 b = 1, 2, 3 0.2 0.1 0.0 –0.1
0
2
4 η
6
8
Fig. 6. Profiles of f (η) for different values of b when β = 1.
0.6 0.5 0.4 0.3 f'
b = 1, 2, 3
0.2 0.1 0.0 −0.1
0
1
2
3 η
4
5
6
Fig. 7. Profiles of f ′ (η) for different values of b when β = 1.
1.0 0.8
θ
0.6 0.4 0.2 0.0 0
2
4 η
6
8
Fig. 8. Profiles of θ(η) for different values of γ when β = b = Pr = 1.
1.0 0.8 0.6
Pr = 0.5, 1, 2, 3
θ
92
0.4 0.2 0.0 0
2
4
η
6
8
10
Fig. 9. Profiles of θ(η) for different values of Pr when β = b = γ = 1.
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Acknowledgment The work is supported by the National Natural Science Foundations of China (No. 51276014). References [1] J.B.J. Fourier, Théorie Analytique De La Chaleur, Paris, 1822. [2] C. Cattaneo, Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 3 (1948) 83–101. [3] C.I. Christov, On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction, Mech. Res. Commun. 36 (2009) 481–486. [4] V. Tibullo, V. Zampoli, A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids, Mech. Res. Commun. 38 (2011) 77–79. [5] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 200 (1949) 523–541. [6] M. Ciarletta, B. Straughan, Uniqueness and structural stability for the Cattaneo–Christov equations, Mech. Res. Commun. 37 (2010) 445–447. [7] B. Straughan, Thermal convection with the Cattaneo–Christov model, Int. J. Heat Mass Transfer 53 (2010) 95–98. [8] S.A.M. Haddad, Thermal instability in Brinkman porous media with Cattaneo–Christov heat flux, Int. J. Heat Mass Transfer 68 (2014) 659–668. [9] J.J. Choi, Z. Rusak, J.A. Tichy, Maxwell fluid suction flow in a channel, J. Non-Newtonian Fluid Mech. 85 (1999) 165–187. [10] Kayvan Sadeghy, Amir-Hosain Najafi, Meghdad Saffaripour, Sakiadis flow of an upper-convected Maxwell fluid, Int. J. Non-Linear Mech. 40 (2005) 1220–1228. [11] Z. Abbas, M. Sajid, T. Hayat, MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel, Theor. Comput. Fluid Dyn. 20 (2006) 229–238. [12] T. Hayat, Z. Abbas, M. Sajid, Series solution for the upper-convected Maxwell fluid over a porous stretching plate, Phys. Lett. A 358 (2006) 396–403. [13] Amir Alizadeh-Pahlavan, Vahid Aliakbar, Farzad Vakili-Farahani, Kayvan Sadeghy, MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 473–488. [14] B. Raftari, A. Yildirim, The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets, Comput. Math. Appl. 59 (2010) 3328–3337. [15] Kayvan Sadeghy, Hadi Hajibeygi, Seyed-Mohammad Taghavi, Stagnation-point flow of upper-convected Maxwell fluids, Int. J. Non-Linear Mech. 41 (2006) 1242–1247. [16] T. Hayat, Z. Abbas, M. Sajid, MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface, Chaos Solitons Fractals 39 (2009) 840–848. [17] M. Kumaria, G. Nath, Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field, Int. J. Non-Linear Mech. 44 (2009) 1048–1055. [18] T. Hayat, M. Awais, M. Qasim, A.A. Hendi, Effects of mass transfer on the stagnation point flow of an upper-convected Maxwell (UCM) fluid, Int. J. Heat Mass Transfer 54 (2011) 3777–3782. [19] T. Hayat, Z. Abbas, N. Ali, MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys. Lett. A 372 (2008) 4698–4704. [20] T. Hayat, Z. Iqbal, M. Mustafa, A. Alsaedi, Momentum and heat transfer of an upper-convected Maxwell fluid over a moving surface with convective boundary conditions, Nucl. Eng. Des. 252 (2012) 242–247. [21] V. Aliakbar, A. Alizadeh-Pahlavan, K. Sadeghy, The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 779–794. [22] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003. [23] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513.