divergent channel considering thermal radiation

Journal of Molecular Liquids 220 (2016) 592–603

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Investigation of MHD nanofluid flow and heat transfer in a stretching/ shrinking convergent/divergent channel considering thermal radiation A.S. Dogonchi, D.D. Ganji ⁎ Mechanical Engineering Department, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran

a r t i c l e

i n f o

Article history: Received 23 January 2016 Received in revised form 14 April 2016 Accepted 9 May 2016 Available online xxxx Keywords: Stretchable/shrinkable walls Nanofluid MHD Duan–Rach Approach (DRA) Thermal radiation

a b s t r a c t In this paper, heat transfer of a steady, viscous incompressible water based MHD nanofluid flow from a source or sink between two stretchable or shrinkable walls with thermal radiation effect is investigated. A similarity transformation is used to convert the governing radial momentum and energy equations into nonlinear ordinary differential equations with the appropriate boundary conditions. These nonlinear ordinary differential equations are solved analytically by Duan–Rach Approach (DRA). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method (ADM) by evaluating the inverse operators at the boundary conditions directly. The approximate analytical investigation is carried out for different values of the embedding parameters, namely: stretching/shrinking parameter, radiation parameter, Reynolds number, Hartmann number, opening angle and solid volume fraction. The results show that the fluid velocity and temperature distribution increase with the increasing of stretching parameter. The results were also compared with numerical solution in order to verify the accuracy of the proposed method. It has been seen that the current results in comparison with the numerical ones are in excellent agreement. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The fluid flow between two plates is one of the most applicable cases in the mechanical engineering and industrial applications such as cold drawing operation in the polymer industry, extrusion of molten polymers through converging dies, pressure driven transport of particles through a symmetric converging/diverging channel. This type of fluid flow was first analyzed by Jeffery [1] and Hamel [2]. The attendance of a magneto-hydrodynamic (MHD) field can affect this type of fluid flow [3–5]. The theoretical study of magneto-hydrodynamic (MHD) channel has been a subject of many applications such as in designing cooling systems with liquid metals, MHD power generation, accelerators, pumps and flow meters [6,7]. There are many techniques and newly developed methods such as an Adomian decomposition method (ADM) [8,9], Differential Transformation Method (DTM) [10–18], and Homotopy Perturbation Method (HPM) [19–21] to solve the nonlinear differential equation of Jeffery–Hamel fluid flow and many problems in the engineering field. The motion of a spherical particle in a plane Couette Newtonian fluid flow was investigated by Dogonchi et al. [12]. They applied the differential transformation method (DTM) and Padé approximation to solve governing equations. Their results indicate that the horizontal and vertical velocities of spherical solid particle in ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (A.S. Dogonchi), [email protected] (D.D. Ganji).

http://dx.doi.org/10.1016/j.molliq.2016.05.022 0167-7322/© 2016 Elsevier B.V. All rights reserved.

water fluid are higher than the glycerin and ethylene-glycol fluids. Yaghoobi and Torabi [13] found an analytical solution for falling nonspherical particle by differential transformation method (DTM). They investigated the influence of the sphericity parameter on velocity and acceleration profiles. The unsteady motion of a vertically falling nonspherical particle in incompressible Newtonian media was investigated by Dogonchi et al. [14]. They applied the differential transformation method (DTM) and Padé approximation to solve governing equations. They concluded that the velocity of the gold particle is higher than the copper and aluminum particles. The Jeffery–Hamel flow in a stretchable convergent/divergent channels was investigated by Turkyilmazoglu [22]. His results show that the temperature distribution increases with the increasing of stretching parameter for the convergent channel. Heat transfer of a steady, incompressible water based nanofluid flow over a stretching sheet in the presence of transverse magnetic field with thermal radiation and buoyancy effects was investigated by Rashidi et al. [23]. They found that the rising of buoyancy parameter increases the velocity and decreases the temperature of the nanofluid. Abolbashari et al. [24] analyzed the entropy for an unsteady MHD nanofluid flow past a stretching permeable surface. They applied the Homotopy Analysis Method (HAM) to solve governing equations. They reported that the increasing of the nanoparticle volume fraction parameter, unsteadiness parameter, magnetic parameter, suction parameter, Reynolds number, Brinkman number, and Hartmann number causes an increase of the entropy generation number. A steady magnetohydrodynamic (MHD) convective and slip flow due to a rotating disk with

A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

593

Table 2 Comparison between DRA and numerical results for F′(1) when α= −5∘ ,Re=50 for different values of stretching/shrinking parameter. C

DRA results φ = 0% (Present)

Numerical results (M. Turkyilmazoglu) [22]

Error (%)

−2 −1 0 1 2

−5.130921689 −4.652183982 −2.833915413 0 3.654305033

−5.1309222926 −4.6521591354 −2.8339514330 0 3.6697111853

0.00001177 0.00053412 0.00127102 – 0.41981919

Table 3 Comparison between DRA and numerical results for F′(1) when α=5∘ ,Re=50 for different values of stretching/shrinking parameter. Fig. 1. The geometry of the MHD nanofluid flow between two stretchable or shrinkable walls with thermal radiation effect.

viscous dissipation and Ohmic heating was investigated by Rashidi and Erfani [25]. They applied Differential Transformation Method (DTM) to solve governing equations. Their results indicate that the temperature increases with the increasing Eckert number (Ec) and Schmidt number (Sc). Abbasbandy et al. [26] applied the Homotopy Analysis Method (HAM) on Falkner–Skan flow of MHD Oldroyd-B fluid and achieved comparable results to the numerical ones. Convective Heat Transfer for MHD Viscoelastic Fluid Flow over a Porous Wedge with Thermal Radiation was investigated by Rashidi et al. [27]. They applied Homotopy Analysis Method (HAM) to solve governing equations. Their results indicate that as the wedge angle increases the heat transfer to the fluid increases for the other constant specified parameter. Recently several authors investigated nanofluid flow and heat transfer [28–34]. In 1986 Adomian [8] published an analytical method to solve nonlinear equations. Esmaili et al. [35] applied this technique to solve the governing equation of Jeffery–Hamel fluid flow. Comparison of the results illustrates that the analytical method and numerical data are in a good agreement with each other. Subsequently, different methods were used to solve the governing equations. Motsa et al. [4] employed novel hybrid spectral-homotopy analysis method to solve the nonlinear equation for the MHD Jeffery–Hamel problem. Their results show that the spectral-homotopy analysis technique converges at least twice as fast as the standard homotopy analysis method. Moghmi et al. [5] applied the homotopy analysis method (HAM) for the nonlinear magnetohydrodynamic (MHD) Jeffery–Hamel problem. They studied the influence of the Hartman number on velocity profile. The influence of MHD field and nanoparticle on the Jeffery–Hamel fluid flow was studied by Sheikholeslami et al. [3]. They applied the Adomian decomposition method (ADM) to solve governing equations. Their results indicate that when the Hartmann number increases, backflow will reduce. All these methods need to find the unknown initial value (f″(0)) of the problem and the final solution depends on the accuracy of the initial value which is determined by numerical method. Duan et al. [36] have presented a new modification of the ADM to solve a wide class of multi-order and multi-point nonlinear boundary value problems (BVP). Dogonchi and Ganji [37] applied this modified method (DRA) to solve the equation of non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications. They found that the Nusselt number has a direct relationship with Reynolds number, Prandtl number and power law index. This analytical method was successfully applied to various engineering problems [38,39]. Table 1 Thermo-physical properties of water and nanoparticle [23].

Copper (Cu) Pure water

ρ(kg/m3)

Cp(J/kg K)

k(W/m K)

σ((Ω . m)−1)

μ(kg/m s)

8933 997.1

385 4179

401 0.613

5.96 × 107 0.05

– 0.001

C

DRA results φ = 0% (Present)

Numerical results (M. Turkyilmazoglu) [22]

Error (%)

−1.5 −1 0 1 1.5

−5.082942399 −3.508090102 −1.109360533 0 −0.1423152590

−5.0829256568 −3.5081031667 −1.1093265266 0 −0.1464298835

0.00032941 0.00037242 0.0030655 – 2.80996228

In the present work, we have applied this modified method to solve the equation of magneto-hydrodynamic (MHD) nanofluid flow from a source or sink between two stretchable or shrinkable walls with thermal radiation effect and we have compared the proposed method with the numerical solution in order to verify the accuracy of it. The effects of the stretching/shrinking parameter, the Hartmann number, the volume fraction of nanofluid, the radiation parameter, etc. on flow and heat transfer characteristics are investigated. 2. Problem description Consider the steady two-dimensional fully developed flow of an incompressible conductive viscous fluid from a source or sink between two stretchable or shrinkable walls that meet at an angle 2α as shown in Fig. 1. The walls are supposed to radially stretch or shrink in accordance with u ¼ uw ¼

s r

ð1Þ

Table 4 Comparison between DRA and numerical results for − Θ′(1) when α = − 5 ∘ , Re = 50 , Pr = 1 , Ec = 0 for different values of stretching/shrinking parameter. C

DRA results the absence of N and φ = 0% (Present)

Numerical results (M. Turkyilmazoglu) [22]

Error (%)

−2 −1 0 1 2

0.03157845854 0.03734696604 0.04214811723 0.04640127099 0.05052578617

0.0315761821 0.0373226368 0.0421517243 0.0464015106 0.0502423154

0.0072094 0.06518632 0.00855734 0.00051638 0.5642072

Table 5 Comparison between DRA and numerical results for − Θ′(1) when α = 5∘ , Re=50, Pr =1,Ec=0 for different values of stretching/shrinking parameter. C

DRA results the absence of N and φ = 0% (Present)

Numerical results (M. Turkyilmazoglu) [22]

Error (%)

−1.5 −1 0 1 1.5

0.03244718254 0.03475109986 0.03998210836 0.04640127099 0.05019038094

0.0324575970 0.0347758169 0.0399820121 0.0464015106 0.0504570638

0.03208635 0.07107537 0.0002408 0.00051638 0.52853424

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Fig. 2. Velocity profile (f) and Temperature distribution (Θ) for different values of stretching(a)/shrinking(b) parameter in convergent channel when Ha = 100, α = −2.5°, Re = 50, Pr = 6.2, N = 0.1, Ec = 0.01 and φ = 10%.

With s being the stretching/shrinking rate. The walls are considered to be divergent if α N 0 and convergent if α b 0. We assume that the velocity is purely radial and depends on r and θ and further there is no magnetic field in the z-direction. The reduced forms of continuity, Navier–Stokes and energy equations in polar coordinates are:

Using Rosseland approximation for radiation (see Rashidi et al. [23]) we have   ∂T 4  qr;rad: ¼ − 4σ  =3knf ∂r

ð6Þ

1 ∂ðruÞ ρ ðruÞ ¼ 0; r nf ∂r

  ∂T 4  qθ;rad: ¼ − 4σ  =3knf ∂θ

ð7Þ

∂u 1 ∂P μ nf u ¼− þ ρnf ∂r ρnf ∂r 1 ∂P 2μ nf ∂u − ¼ 0: ρnf r ∂θ ρnf r2 ∂θ

ð2Þ ! 2 2 ∂ u 1 ∂u 1 ∂ u u σ B20 þ u; þ 2 2− 2 − 2 r ∂r r ∂θ ρnf r 2 r ∂r

ð3Þ

ð4Þ

! 2 2  ∂ T 1 ∂T 1 ∂ T 1 ∂  − þ rqr;rad: þ ∂r2 r ∂r r2 ∂θ2 ρC p nf r ∂r  2  2 !  2 !  μ nf 1 ∂  ∂u u 1 ∂u    − þ þ q 2 þ r r ∂θ ∂r ρC p nf r2 ∂θ θ;rad: ρC p nf

  knf ∂T  ¼ u ∂r ρC p nf

ð5Þ Where u = u(r,θ) is the velocity, P is the fluid pressure, σ is the conductivity of the fluid, B0 is the electromagnetic induction, T is the fluid temperature and qrad. is the radiative heat flux.

Here σ* is the Stefan–Boltzmann constant and knf⁎ is the mean absorption coefficient of the nanofluid. Further we assume that the temperature difference within the flow is such that T4 may be expanded in a Taylor series. Hence, expanding T4 about T∞ and neglecting higher order terms we get T 4 ≅ 4T 3∞ T−3T 4∞

ð8Þ

Therefore Eq. (5) is simplified to ! ! 2 2 2 ∂ T 1 ∂T 1 ∂ T 16σ  T 3∞ ∂ T 1 ∂T   þ þ þ þ  ∂r2 r ∂r r2 ∂θ2 3knf ρC p nf ∂r2 r ∂r !  2  2 !  2 ! 2  3 μ 16σ T ∂ T ∂u u 1 ∂u nf   þ þ þ   ∞ 2 þ r r ∂θ ∂r 3knf ρC p nf ∂2 θ ρC p nf

  knf ∂T  ¼ u ∂r ρC p nf

ð9Þ

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595

Fig. 3. Velocity profile (f) and Temperature distribution (Θ) for different values of stretching(a)/shrinking(b) parameter in divergent channel when Ha = 100, α = 2.5°, Re = 50, Pr = 6.2, N = 0.1, Ec = 0.01 and φ = 10%.

Where ρnf is the effective density of the nanofluid, μnf is the effective dynamic viscosity of the nanofluid, (ρCp)nf is the heat capacitance and knf is the thermal conductivity of the nanofluid are given as in Rashidi et al. [23]     ρnf ¼ ð1−ϕÞρ f þ ϕρs ; μ nf ¼ μ f =ð1−ϕÞ2:5 ; ρC p nf ¼ ð1−ϕÞ ρC p f   ð10Þ   ks þ 2k f −2ϕ k f −ks   þ ϕ ρC p s ; knf =k f ¼ ks þ 2k f þ 2ϕ k f −ks where μf is the dynamic viscosity of the basic fluid, ρf is the density of the pure fluid, (ρCp)f is the specific heat parameter of the base fluid, kf is the thermal conductivity of the base fluid, ρs is the density of the nanoparticles, (ρCp)s is the specific heat parameter of the nanoparticles and ks is the thermal conductivity of the nanoparticles. The governing equations are accompanied with the boundary conditions, due to the symmetry assumption at the channel centerline (θ = 0) ∂u ∂T ¼0¼ ; ∂θ ∂θ

uc u¼ r

ð11Þ

With uc being the centerline rate of movement and Tw is the constant wall temperature. We should remark that in general conditions where asymmetry is also allowed, infinitely many solutions exist [22]. To preserve the symmetry with respect to the centerline, we impose the same stretching/shrinking rates on both walls. Considering only radial flow, Eq. (1) implies that f ðθÞ ¼ ruðr; θÞ

ð13Þ

Using dimensionless parameters f ðηÞ ¼

f ðθÞ ; f max

ΘðηÞ ¼ r 2

T θ where η ¼ Tw α

ð14Þ

Substituting these into the governing equations and eliminating the pressure term yield the nonlinear third-order ordinary differential equation for the flow from the radial momentum equation:   ‴ 0 0 f ðηÞ þ 2αReAð1−ϕÞ2:5 f ðηÞf ðηÞ þ 4−ð1−ϕÞ2:5 Ha α 2 f ðηÞ ¼ 0 ð15Þ And second-order differential equation for the heat

And due to the stretching/shrinking convergent/divergent wall condition at the plates (θ = ±α) s u ¼ uw ¼ ; r

T¼

Tw r2

ð12Þ

 B Prf ðηÞ þ 2 þ 2N ΘðηÞ B1  1 EcPr  2 0 þ 4α f ðηÞ2 þ f ðηÞ2 ¼ 0 2:5 Re B1 ð1−ϕÞ

ð1 þ NÞΘ″ ðηÞ þ 2α 2



ð16Þ

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A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

Fig. 4. Velocity profile (f) for different values of Ha in stretching (a,c: C = 0.5)/shrinking (b,d: C = −0.5) convergent (c,d)/divergent (a,b) channel when Re = 50 and φ = 10%.

Here A, B and B1 are constants given by: A ¼ ð1−ϕÞ þ

ρs ϕ; ρf

  ρC p s  ϕ; B ¼ ð1−ϕÞ þ  ρC p f

B1 ¼

  ks þ 2k f −2ϕ k f −ks   ks þ 2k f þ 2ϕ k f −ks

Subject to the boundary conditions f ð0Þ ¼ 1;

0

f ð0Þ ¼ 0;

f ð1Þ ¼ C;

Θ0 ð0Þ ¼ 0;

Θð1Þ ¼ 1

ð17Þ

Where C= s/uc is the stretching (C N 0) or shrinking (C b 0) parameter, Re = αuc/νf is the Reynolds number, Pr = ucρCp/kf is the Prandtl number, Ec = αu2c /CpfTw is the Eckert number and N = 16σ⁎T ∞ 3/3knf⁎knf is the radiation parameter. It is noted that setting C to zero at this stage leads to stationary wall condition for the traditional Jeffery–Hamel flow. The elliptic function solutions as given in the literature may be fulfilled here for nonzero C, but, it is thought that this is not very convenient. Physical quantities of interest are the skin fraction coefficient and Nusselt number which are defined as: !    μ nf 1 ∂u  16σ  T 3∞ 1 ∂T   Nu ¼ r k þ C f ¼ nf  kf Tw r ∂θ θ¼α ρ f u2c r ∂θ θ¼α 3knf After simplification, we obtain:

Fig. 5. Temperature distribution (Θ) for different values of Ha in stretching/shrinking divergent/convergent channel when Re = 50, Pr = 6.2, N = 0.1, Ec = 0.01 and φ = 10%.

     μ knf ð1 þ NÞ 0   nf 0  Cf ¼  f ð1ÞNu ¼  Θ ð1Þ   ρf kf α

ð18Þ

ð19Þ

A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

Fig. 6. Velocity profile (f) for different values of α in stretching (a: C = 0.5)/shrinking (b: C = −0.5) divergent/convergent channel when Ha = 100, Re = 50 and φ = 10%.

597

Fig. 7. Temperature distribution (Θ) for different values of α in stretching/shrinking divergent (a)/convergent (b) channel when Ha = 100, Re = 50, Pr = 6.2, Ec = 0.01, N = 0.1 and φ = 10%.

3. Description of the Duan–Rach Approach (DRA) Consider a third-order nonlinear differential equation: Lu ¼ Nu þ g ðxÞ;

ð20Þ

subject to the mixed set of Dirichlet and Neumann boundary conditions uðx1 Þ ¼ α 0 ; u0 ðx1 Þ ¼ α 1 ; u0 ðx2 Þ ¼ α 2 ; x2 ≠x1

ð21Þ

Applying the inverse operator L−1 to both sides of Eq. (20) yields

1 L−1 ½Nu þ g  ¼ uðxÞ−uðx0 Þ−u0 ðx1 Þðx−x0 Þ− u″ ðξÞ 2 h i  ðx−x1 Þ2 −ðx0 −x1 Þ2

ð24Þ

3

d where L ¼ dx 3 is the linear differential operator to be inverted, Nu is an analytic nonlinear operator, and g(x) is the system input. We take the inverse linear operator as

L−1 ð•Þ ¼

Z

x x0

Z

x x1

Z

x ξ

ð•Þdxdxdx;

ð22Þ

where ξ is a prescribed value in the specified interval. Thus we have: h i 1 L−1 Lu ¼ uðxÞ−uðx0 Þ−u0 ðx1 Þðx−x0 Þ− u″ ðξÞ ðx−x1 Þ2 −ðx0 −x1 Þ2 ð23Þ 2

We differentiate Eq. (23), then let x = x2 and solve for u″(ξ), hence

u″ ðξÞ ¼

u0 ðx2 Þ−u0 ðx1 Þ 1 − x2 −x1 x2 −x1

Z

x2 Z x x1

ξ

½Nu þ g dxdx:

ð25Þ

598

A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

Fig. 8. Velocity profile (f) for different values of Re in stretching (a: C = 0.5)/shrinking (b: C = −0.5) divergent/convergent channel when Ha = 100 and φ = 10%.

Substituting Eq. (25) into Eq. (24) yields, i u0 ðx Þ−u0 ðx Þ 1h 2 1 ðx−x1 Þ2 −ðx0 −x1 Þ2 2 x2 −x1 Z Z 1 ðx−x1 Þ2 −ðx0 −x1 Þ2 x2 x þ L−1 g þ L−1 Nu− gdxdx ð26Þ x2 −x1 2 ξ x1 Z Z 1 ðx−x1 Þ2 −ðx0 −x1 Þ2 x2 x Nudxdx: − x2 −x1 2 ξ x1

uðxÞ ¼ uðx0 Þ þ u0 ðx1 Þðx−x0 Þ þ

Thus in Eq. (26) the three known boundary values u(x 0 ), u′(x 1 ) and u′(x2 ) are included and the undetermined coefficient was replaced. Next, the solution is decomposed and the nonline∞ ∞ arity uðxÞ ¼ ∑m¼0 um ðxÞ; NuðxÞ ¼ ∑m¼0 Am ðxÞ where A m (x) = Am(u0(x), u1(x), … , um(x)) are the Adomian polynomials. From Eq. (25), the solution components are determined by the modified recursion scheme:

Fig. 9. Velocity profile (f) for different values of φ in stretching (a: C = 0.5)/shrinking (b: C = −0.5) divergent/convergent channel when Ha = 100 and Re = 50.

umþ1 ¼ L−1 Am −

1 ðx−x1 Þ2 −ðx0 −x1 Þ2 x2 −x1 2

Z

x2 Z x x1

ξ

ð28Þ

Am dxdx:

4. Implementation of the method In our study, the Duan–Rach Approach must be modified. We do not use the prescribed value ξ. According to Eq. (20), Eqs. (15) and (16) can be written as follows:   0 0 L3 f ðηÞ ¼ −2αReAð1−ϕÞ2:5 f ðηÞf ðηÞ− 4−ð1−ϕÞ2:5 Ha α 2 f ðηÞ

ð29Þ

  1 0 B   2α 2 Pr f ðηÞ þ 2 þ 2N ΘðηÞ C B 1 B 1 B L2 ΘðηÞ ¼ −  C ð30Þ 1 Ec Pr  2 0 2 2 A N þ 1 @þ 4α f ðηÞ þ f ðηÞ B1 ð1−ϕÞ2:5 Re 3

d Where the differential operator L3 and L2 are given by L3 ¼ dη 3 and 2

i u0 ðx Þ−u0 ðx Þ 1h 2 1 u0 ¼ uðx0 Þ þ u0 ðx1 Þðx−x0 Þ þ ðx−x1 Þ2 −ðx0 −x1 Þ2 2 x2 −x1 ð27Þ Z Z 1 ðx−x1 Þ2 −ðx0 −x1 Þ2 x2 x þ L−1 g − gdxdx; x2 −x1 2 ξ x1

−1 1 d L2 ¼ dη and L− 2 2 respectively. Assume that the inverse operator L 3

exist, then we have: L−1 3 ð•Þ ¼

Z ηZ ηZ 0

0

η 0

ð•Þdηdηdη; L−1 2 ð•Þ ¼

Z ηZ

η

0

0

ð•Þdηdη

ð31Þ

A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

Fig. 10. Temperature distribution (Θ) for different values of φ in stretching/shrinking divergent (a: α = 2.5°)/convergent (b: α = −2.5°) channel when Ha = 100, Re = 50, Pr = 6.2, Ec = 0.01 and N = 0.1. 1 Operating with L− in Eq. (29) and after exerting boundary 3 conditions on it:

599

Fig. 11. Temperature distribution (Θ) for different values of radiation parameter (N) in stretching/shrinking divergent (a: α = 2.5°)/convergent (b: α = −2.5°) channel when Ha = 100, Re = 50, Pr = 6.2, Ec = 0.01 and φ = 10%.

1 Operating with L− in Eq. (30) and after exerting boundary 2 conditions on it:

Obviously, we do not have the values of f″(0)and Θ(0). In the standard Adomian Decomposition Method (ADM), we need to evaluate those unknown conditions with numerical methods. Consequently, the boundary value problem (BVP) is turned into an initial value problem (IVP). The accuracy of the solution depends on the accuracy of the two unknown parameters. In our study, we use the Duan–Rach Approach [36] to find a totally analytical solution. By putting η = 1 in Eq. (32), we have:

ΘðηÞ ¼ Θð0Þ þ Θ0 ð0Þη þ L−1 2 ðN 2 uÞ:

f ð0Þ ¼ 2C−2

0

″

f ðηÞ ¼ f ð0Þ þ f ð0Þη þ f ð0Þ

η2 þ L−1 3 ðN 1 uÞ: 2

ð32Þ

ð33Þ

″

Where N1u and N2u are introduced as: 2:5

N1 u ¼ −2αReAð1−ϕÞ

  0 f ðηÞf ðηÞ− 4−ð1−ϕÞ2:5 Ha α 2 f ðηÞ 0

  1 0 B   2α 2 Prf ð η Þ þ 2 þ 2N Θ ð η Þ C B 1 B1 B N2 u ¼ − C 1 EcPr  2 0 2 2 A N þ1 @þ f ð η Þ þ f ð η Þ 4α B1 ð1−ϕÞ2:5 Re

h

i L−1 3 N1 u

 η¼1

þ1

ð36Þ

Where ð34Þ

h

i L−1 3 N1 u

η¼1

¼

Z 1Z ηZ 0

0

η 0

ðNuÞdηdηdη

ð37Þ

Substituting Eq. (36) into Eq. (32) yields, ð35Þ

h i h i 2 −1 f ðηÞ ¼ 1−η2 þ Cη2 þ L−1 3 N 1 u −η L3 N 1 u

η¼1

ð38Þ

600

A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

Fig. 13. Skin friction coefficient (Cf) for different values of stretching/shrinking parameter (C) and solid volume fraction (φ) when Ha = 100, α = 2.5°, Re = 50.

Applying Eq. (40), we obtain the terms of the Adomian polynomials and put them in Eq. (39), and we determine fm(η) as follows: f 0 ðηÞ ¼ 1−η2 þ Cη2 f 1 ðηÞ ¼

1 Að1−ϕÞ2:5 C 2 Reαη2 þ −… 30

ð41Þ

The functions f2(η), f3(η), ⋯ can be determined in a similar way from Eq. (39). For convenience, we do not represent all terms of fm(η). ∞ Using f ðηÞ ¼ ∑m¼0 f m ðηÞ ¼ f 0 ðηÞ þ f 1 ðηÞ þ f 2 ðηÞ þ …, thus f ðηÞ ¼ 1−η2 þ Cη2 þ

Fig. 12. Nusselt number (Nu) for different values of stretching/shrinking parameter (C), radiation parameter (N) and solid volume fraction when Ha = 100, α = 2.5°, Re = 50, Pr = 6.2 and Ec = 0.01.

1 Að1−ϕÞ2:5 C 2 Reαη2 þ −… 30

According to Eq. (42), the accuracy increases by increasing the number of solution terms (m). For Θ(η), we proceed in the same manner. We get the following recursive scheme: h i Θ0 ðηÞ ¼ 1Θmþ1 ðηÞ ¼ − L−1 2 N2 u

η¼1

1

Thus the right hand side of Eq. (38) does not contain the undetermined parameter f″(0). Finally, we have the modified recursive scheme:

2

h i þ L−1 2 N2 u

ð43Þ

η

Where ½L−1 2 N 2 uη¼1 ¼ ∫0 ∫0 ðN 2 uÞdηdη ∞

Using ΘðηÞ ¼ ∑m¼0 Θm ðηÞ ¼ Θ0 ðηÞ þ Θ1 ðηÞ þ Θ2 ðηÞ þ …, thus ΘðηÞ ¼ 1−



1 2:5

ðN þ 1ÞB1 ð1−ϕÞ

 

2:5 9 2 4 2  10−11 8:333  10 Bð1−ϕÞ CPrReα η þ−… Re

ð44Þ

2

f 0 ðηÞ ¼ 1−η h þ Cη i h i 2 −1 f mþ1 ðηÞ ¼ L−1 3 Am ðηÞ −η L3 Am ðηÞ

ð42Þ

ð39Þ

η¼1

Where the Am(η) are the Adomian polynomials, which can be determined by the formula

Obviously, the temperature field Θ(η) depends on the number of terms of the velocity field f(η) found in Eq. (42). It is noted that in this paper, the results for f(η) and Θ(η) at 6th-order of approximations are presented. 5. Results and discussion

"

Am ðηÞ ¼

!#

m m X 1 d N λi F i ðηÞ m! dλm i¼0

ð40Þ λ¼0

that was first published by Adomian and Rach [40].

The transformed radial momentum and energy Eqs. (15) and (16) subjected to the boundary conditions Eq. (17) were analytically solved by using the modified standard Adomian Decomposition Method (ADM) called Duan–Rach Approach (DRA). The effects of various parameters such as the stretching/shrinking parameter (C), the Reynolds

A.S. Dogonchi, D.D. Ganji / Journal of Molecular Liquids 220 (2016) 592–603

601

Fig. 14. Temperature distribution (Θ) for different values of Prandtl number (Pr) in stretching/shrinking divergent/convergent channel when Ha = 100, Re = 50, N = 0.1, Ec = 0.01 and φ = 0%.

number (Re), the Hartmann number (Ha) and so on are investigated on the velocity and temperature. The thermo-physical properties of the nanofluid have been summarized in Table 1. To validate the analytical results, we compared the analytical results with those obtained by numerical solution [22] for different values of the embedding parameters. The results are well matched with the results carried out by numerical solution as they have been shown in Tables 2–5. In these tables, an error is introduced as follows:   DRA results−Numericalresults   100: %Error ¼   Numericalresults A low maximum error (%) in these tables emphasizes on accuracy and efficiency of the Duan–Rach Approach (DRA). This accuracy shows high confidence in the validity of this problem, and it reveals an excellent agreement in engineering accuracy. Figs. 2 and 3 show the effect of the stretching/shrinking parameter on the fluid velocity and temperature distribution in convergent and divergent channel, respectively. It is observed that both the velocity profile and temperature distribution increase with the increasing of stretching parameter and their variation more gradual than when the stretching parameter at a higher value, but these treatments of velocity and temperature are completely vice versa for shrinking parameter.

Also, it is clear that due to stretching an overheating process happens, leading to growth of thermal layer and to an increasing heat transfer rate. On the other hand, shrinking cools down the system by decreasing the thickness of thermal layer. So, where cooling is required, a wall shrinking is desirable, and vice versa, where heating is required, wall stretching is desirable. The effect of the Hartmann number (Ha) on velocity profile and the temperature distribution of Cu–water for a stretching/shrinking divergent/convergent channel is demonstrated in Figs. 4 and 5, respectively. The velocity curves show that the rate of transport is considerably reduced with an increase of Hartmann number. This clearly indicates that the transverse magnetic field opposes the transport phenomena. Because the variation of Ha leads to the variation of the Lorentz force due to the magnetic field and the Lorentz force produces more resistance to transport phenomena. As seen in this figure increasing in the Ha makes an increase in velocity profile so, by the increasing of Hartmann number, the flow reversal disappears. On the other hand, the magnetic field enhances the temperature at all points leading to increase in thermal boundary-layer thickness. Figs. 6 and 7 show the variation of the fluid velocity and temperature distribution with an opening angle (α), respectively, when the other parameters are constant, for both stretching/shrinking divergent/convergent channels. In the stretching/shrinking convergent channel, the

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fluid velocity becomes flat and thickness of boundary layer decreases. In the stretching/shrinking divergent channel, as the opening angle (α) increases, the magnitude of fluid velocity reduces. On the other hand, temperature distribution for both cases increases. Also, according to these figures, by increasing the opening angle in stretching/shrinking divergent channel, the probability of the backflow phenomenon increases. But this probability is more in the stretching divergent channel. Fig. 8 shows the effect of the Reynolds number (Re) on the fluid velocity in stretching/shrinking divergent/convergent channel. It is observed that the increase in Reynolds number causes a decrease in fluid velocity in stretching/shrinking divergent channels, also for higher Reynolds number, the flow moves reversely and the backflow is observed. On the other hand, for stretching/shrinking convergent channels, the results were reversed and by increasing in Reynolds number, the fluid velocity was increased and no backflow was observed. Figs. 9 and 10 show the effect of the solid volume fraction (φ) on the fluid velocity and temperature distribution in stretching/shrinking divergent/convergent channel, respectively. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. From these figures, it can be concluded that increasing solid volume fraction in stretching/shrinking divergent channel lead to decrease in fluid velocity and temperature distribution. Physically speaking, as the nanoparticles add to the pure fluid, the density of the fluid increases and then the fluid become denser so that it can have more difficulty moving through the channel. Fig. 11 shows the effect of the radiation parameter (N) on the temperature distribution in stretching/shrinking divergent/convergent channel. The increasing values of radiation parameter lead to decrease in the temperature of the nanofluid and their variation more gradual than when the radiation parameter at a lower value. The presence of radiation parameter leads to thinning of the thermal layer. Figs. 12 and 13 show the effect of the stretching/shrinking parameter (C), the radiation parameter (N) and solid volume fraction (φ) on the Nusselt number (Nu) and skin friction coefficient (Cf), respectively. It is observed that the magnitude of the Nusselt number increases with the increasing of stretching/shrinking parameter, radiation parameter and solid volume fraction. On the other hand, while the solid volume fraction increases, the magnitude of the skin friction coefficient is increased. We finally explore the effect of the Prandtl number (Pr) on the temperature distribution for a stretching/shrinking divergent/convergent channel in Fig. 14. It is observed that the temperature distribution increases with the increasing of Pr. Also, from Fig. 14 it is clear that for shrinking divergent channel, we have the lowest temperature distribution in a given Prandtl number. For instance, when the Prandtl number is 0.7 in Fig. 14, the maximum value of the temperature for shrinking divergent channel is almost 1.004798. However, this value for shrinking convergent channel is 1.004866. As previously mentioned, Tables 2–5 illustrate that the approach used has high accuracy for different values of embedded parameters. In the ADM, for given Re, Ha, α, N and φ, we have to solve f (1) = 0 and Θ'(0) = 0 to find f″(0) and Θ(0). If we change the value of Re, we have to again evaluate the values of f″(0) and Θ(0). In our work (Duan–Rach Approach), the obtained solutions for f(η) (Eq. (42)) and Θ(η) (Eq. (44)) are purely analytical approach and we do not need any other calculations if we change any of the parameters of the flow and the final solution does not contain undetermined coefficients. 6. Conclusions In this paper, the steady two-dimensional, viscous incompressible water based MHD nanofluid flow from a source or sink between two stretchable or shrinkable walls with thermal radiation effect is investigated analytically using Duan–Rach Approach (DRA). The Duan–Rach Approach allows us to find an analytical solution without using a numerical method to evaluate the missing parameters f″(0) and Θ(0).

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Nomenclature A,B,B1: Constant parameter C: Stretching/shrinking parameter Ec: Eckert number Cp: Specific heat (j/kgK) f: Dimensionless velocity Θ: Dimensionless temperature P: Pressure term Re: Reynolds number

Pr: Prandtl number s: Stretching/shrinking rate (m2/s) B0: Magnetic field qrad.: Radiative heat flux T: Temperature (K) Tw: Surface temperature of channel (K) T∞: Free stream temperature r,θ: Cylindrical coordinates u: Velocity component in the radial direction (m/s) uc: Rate of movement in the radial direction (m2/s) uw: Wall velocity component in the radial direction (m/s) Ha: Hartman number N: Radiation parameter Nu: Nusselt number Cf: Skin fraction coefficient DRA: Duan–Rach Approach ADM: Adomian Decomposition Method k: Thermal conductivity k⁎: Mean absorption coefficient Greek symbols

α: Angle of the channel η: Dimensionless angle σ: Electrical conductivity θ: Any angle ρ: Density μ: Dynamic viscosity ν: Kinematic viscosity φ: Solid volume fraction σ⁎: Stefan–Boltzmann constant Subscripts

nf: Nanofluid f: Base fluid s: Nano-solid-particles

603