An analytical solution of the MHD Jeffery–Hamel flow by the modified Adomian decomposition method

Accepted Manuscript An analytical solution of the MHD Jeffery-Hamel flow by the modified Adomian decomposition method A. Dib, A. Haiahem, B. Bou-said PII: DOI: Reference:

S0045-7930(14)00273-4 http://dx.doi.org/10.1016/j.compfluid.2014.06.026 CAF 2608

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Computers & Fluids

Received Date: Revised Date: Accepted Date:

10 July 2013 22 June 2014 25 June 2014

Please cite this article as: Dib, A., Haiahem, A., Bou-said, B., An analytical solution of the MHD Jeffery-Hamel flow by the modified Adomian decomposition method, Computers & Fluids (2014), doi: http://dx.doi.org/10.1016/ j.compfluid.2014.06.026

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An analytical solution of the MHD Jeffery-Hamel flow by the modified Adomian decomposition method A. Diba,∗, A. Haiahema , B. Bou-saidb a

Laboratory of Industrial Mechanics, University of Annaba, B.O.12,23000 Algeria b Universit´e de Lyon, CNRS INSA-Lyon, LaMCoS, UMR5259, F-69621, France

Abstract In this paper, we apply a new approach of the Adomian decomposition method developed by Duan-Rach(DRA) to solve the MHD Jeffery-Hamel flow. A purely analytical solution can be obtained by this approach. This method modifies the Adomian decomposition method (ADM) by evaluating the inverse operator at the boundary conditions directly. The results show a good agreement with numerical method (4th-order Runge-Kutta algorithm) and homotopy analysis method (HAM).The algorithm derived from this approach can be easily implemented. Keywords: Jeffery-Hamel flow, Magnetohydrodynamics, Duan-Rach approach, Adomian decomposition method (ADM), Nonlinear equations. 1. Introduction The flow between two inclined plates is one of the most widely applied cases in the mechanical engineering applications. Jeffery [1] and Hamel [2] ∗

corresponding author. Tel: +213 0 55 555 7037;Fax:+213 0 38 861070 Email addresses: [email protected] (A. Dib), [email protected] (A. Haiahem), [email protected] (B. Bou-said)

Preprint submitted to Computers & Fluids

June 28, 2014

set the mathematical foundation of this kind of flow. A wealth of information about Jeffery-Hamel flow can be found in Batchelor [3].Under certain assumptions; Jeffery-Hamel flow can be described by ban exact similarity solution of the Navier-Stokes equations in the special case of two-dimensional flow. The presence of a magnetohydrodynamic (MHD) field can affect this kind of flow [4, 5].The theoretical study of magnetohydrodynamic (MHD) channel has been a subject of many applications in the design of cooling systems with liquid metals, accelerators, pumps and flow meters [6, 7, 8]. There are many techniques to solve the nonlinear differential equation of Jeffery-Hamel flow.The equation can be solved by numerical methods such as Runge-Kutta. In 1986 Adomian [9] published an analytical method to solve nonlinear equations. Esmaili et al. [10] applied this technique to resolve the convergent-divergent flow. The results obtained show a good agreement with the numerical methods. Subsequently, several methods were used to solve the governing equations. Esmaeilpour et al.

[11] employed optimal

homotopy asymptotic method (OHAM). Moghmi et al. [5] applied the homotopy analysis method for the nonlinear MHD Jeffery-Hamel problem. A novel hybrid spectral-homotopy analysis technique developed by Motsa et al. [4] was used to obtain a rapid convergence. The Adomian decomposition method (ADM) was widely used by many authors to solve the Ordinary Differential Equation (ODE) which illustrates this flow. Cherruault et al. [12, 13] studied the convergence of the Adomian decomposition method. An advantage of this method is that it can provide an analytical approximation to a wide class of nonlinear equations without the use of linearization, perturbation or discretization methods. The study conducted in Ref. [10] for

2

solving the Jeffery-Hamel flow shows a good agreement of this method with a boundary layer theory and slender channel theory. Many authors have tried to modify the ADM. Jin [14] modified ADM for solving a kind of evolution equation. All these methods need to evaluate the second derivative at the starting point. For finding F  (0), numerical methods have been used. The fact that we use numerical methods to evaluate the second derivative results in a semi analytic solution. Duan et al.

[15] have presented a new mod-

ification of the ADM to solve a wide class of multi-order and multi-point nonlinear boundary value problems (BVP). In the present work, we have applied this modified method to solve MHD Jeffery-Hamel flow and have made a comparison with a numerical solution. The numerical results of this problem have been performed using Maple13. 2. Mathematical formulation of Jeffery-Hamel flow Consider the steady two-dimensional flow of an incompressible conductive viscous fluid between two rigid plane walls that meet at an angle 2α as shown in fig. 1.We assume that the velocity is purely radial and depends on r and θ. The continuity equation and Navier-Stokes equation in polar coordinates are [16]: ρ ∂ (ru(r, θ)) = 0, r ∂r

u(r, θ)

(1)

∂u(r, θ) 1 ∂p ∂ 2 u(r, θ) 1 ∂u(r, θ) = − + ν[ + ∂r ρ ∂r ∂r2 r ∂r 2 σB02 1 ∂ u(r, θ) u(r, θ) − ] − u(r, θ), + 2 r ∂θ2 r2 ρr2 3

(2)

Fig. 1 The Geometry of the MHD Jeffery-Hamel flow.

1 ∂p 2ν ∂u(r, θ) − 2 = 0, ρr ∂θ r ∂θ

(3)

where P is the fluid pressure, B0 the electromagnetic induction, σ the conductivity of the fluid, ρ the fluid density, and ν is the coefficient of kinematic viscosity. From Eq. (1)

f (θ) ≡ ru(r, θ).

(4)

Using dimensionless parameters

F (η) =

f (θ) θ where η = . fmax α

(5)

Eliminating P between Eqs. ( 2) and ( 3), we obtain a third-order ordinary differential equation for the normalized function profile F (η):

F  (η) + 2αReF (η)F  (η) + (4 − Ha)α2 F  (η) = 0, 4

(6)

since we have a symmetric geometry, the boundary conditions will be

F (0) = 1, F  (0) = 0, F (1) = 0.

(7)

The Reynolds number is

Re =

Umax rα  divergent channel : α > 0, Umax > 0  , ν convergent channel : α < 0, Umax < 0

(8)

the Hartmann number is  Ha =

σB02 . ρν

(9)

3. Description of the Duan-Rach approach for solving BVPs Consider a third-order non linear differential equation

Lu = N u + g(x),

(10)

subject to the mixed set of Dirichlet and Neumann boundary conditions

u(x0 ) = α0 , u (x1 ) = α1 , u (x2 ) = α2 , x1 = x2 , where L =

d3 dx3

(11)

is the linear differential operator to be inverted, N u is an

analytic nonlinear operator, and g(x) is the system input. 5

We take the inverse linear operator as

−1



x



x



L (•) =

x

(•)dxdxdx, x0

x1

(12)

ξ

where ξ is a prescribed value in the specified interval. Thus we have

1 L−1 Lu = u(x) − u(x0 ) − u (x1 )(x − x0 ) − u (ξ)[(x − x1 )2 2 2 − (x0 − x1 ) ].

(13)

Applying the inverse operator L−1 to both sides of Eq. ( 9) yields

L−1 [N u + g] = u(x) − u(x0 ) − u (x1 )(x − x0 ) 1  u (ξ)[(x − x1 )2 − (x0 − x1 )2 ]. − 2

(14)

We differentiate Eq. ( 13), then let x = x2 and solve for u (ξ) , hence,

u (x2 ) − u (x1 ) 1 u (ξ) = − x2 − x1 x2 − x1 



x2



x

[N u + g]dxdx. x1

ξ

Substituting Eq. ( 15) into Eq. ( 14) yields,

6

(15)

1 u(x) = u(x0 ) + u (x1 )(x − x0 ) + [(x − x1 )2 2   (x ) − u (x ) u 2 1 − (x0 − x1 )2 ] + L−1 g + L−1 N u x2 − x1   1 (x − x1 )2 − (x0 − x1 )2 x2 x gdxdx − 2 x2 − x1 x1 ξ   1 (x − x1 )2 − (x0 − x1 )2 x2 x − N udxdx. 2 x2 − x1 x1 ξ

(16)

Thus in Eq. ( 16) the three known boundary values u(x0 ), u (x1 ) and u (x2 ) are included and the undetermined coefficient was replaced. Next, the  solution is decomposed and the nonlinearity u(x) = ∞ n=0 un (x), N u(x) = ∞ n=0 An (x) where An (x) = An (u0 (x), u1 (x), ...., un (x)) are the Adomian polynomials. From Eq. (15), the solution components are determined by the modified recursion scheme

u0 = u(x0 ) + u (x1 )(x − x0 ) 1 u (x2 ) − u (x1 ) + [(x − x1 )2 − (x0 − x1 )2 ] 2 x2 − x1   2 1 (x − x1 ) − (x0 − x1 )2 x2 x −1 + L g− gdxdx, 2 x2 − x1 x1 ξ

un+1

1 (x − x1 )2 − (x0 − x1 )2 = L An − 2 x 2 − x1 −1

7



x2



x

An dxdx. x1

ξ

(17)

(18)

4. Application of the Duan-Rach approach to Jeffery-Hamel flow In our study, the Duan-Rach approach must be modified. We do not use the prescribed value ξ From Eq. ( 6), we have,

F  (η) = −2αReF (η)F  (η) − (4 − Ha)α2 F  (η).

(19)

We take the inverse linear operator as

−1



η



η



η

L (•) =

dηdηdη. 0

0

(20)

0

Operating with L−1 on Eq. 20 and after exerting boundary conditions of Eq. 7 on it, we have:

F (η) = F (0) + F  (0)η + F  (0)

η2 + L−1 (N F (η)), 2

(21)

where N F (η) = −2αReF (η)F  (η) − (4 − Ha)α2 F  (η).

(22)

Obviously, we do not have the value of F  (0) = β. In previously works [10, 14], the authors used a numerical method to evaluate β. Consequently, the boundary value problem (BVP) was turned into an initial value problem (IVP).The accuracy of the solution depends on the accuracy of the parameter β. 8

In this study, we put η = 1 in Eq. (21) to obtain

F  (0) = −2([L−1 N F (η)]η=1 − 1),

(23)

where −1



[L (•)]η=1 =

1



η



η

dηdηdη. 0

0

(24)

0

Substituting Eq. ( 23) into Eq. ( 21) yields,

F (η) = 1 − η 2 + L−1 (N F (η)) − η 2 [L−1 N F (η)]η=1 .

(25)

Thus the right hand side of Eq. (25) does not contain the undetermined parameterβ = F  (0). Finally, we have the modified recursive scheme:

F0 (η) = 1 − η 2 ,

(26)

Fn+1 (η) = F0 (η) + L−1 An (η) − η 2 [L−1 An (η)]n=1 , n ≥ 0,

(27)

where the An (η) are the Adomian polynomials, which can be determined by the formula

An (η) =

n  1 dn [N ( λi Fi (η))]λ=0 , n! dλn i=0

(28)

9

was first published by Adomian and Rach [17] in 1983. Applying Eq. ( 28), we obtain the following terms of the Adomian polynomials:

A0 (η) = −2αReF0 (η)F0 (η) − (4 − Ha)α2 F0 (η) A1 (η) = −2αReF1 (η)F0 (η) − 2αReF0 (η)F1 (η) − (4 − Ha)α2 F1 (η) A2 (η) = −2αReF2 (η)F0 (η) − 2αReF1 (η)F1 (η) − 2αReF0 (η)F2 (η) − (4 − Ha)α2 F2 (η) + · · ·.

(29)

To determineFn (η), we have

F0 (η) = 1 − η 2 1 1 2 1 1 F1 (η) = − αReη 6 + ( αRe + (4 − Ha)α2 )η 4 − ( αRe 30 4 3 3 3 1 2 2 + (4 − Ha)α )η 4 1 2 2 10 1 1 α Re η + ( α2 Re2 + α3 Re F2 (η) = − 1350 40 70 1 3 1 2 2 1 3 8 − α ReHa)η + (− α Re − α Re 208 30 9 1 3 2 4 1 4 1 4 4 4 + α ReHa − α + α Ha − α Ha )η . 36 45 45 360 (30) The functionsF3 (η), F4 (η), · · ·can be determined in similar way from Eq. (27). 10

For convenience, we do not represent all terms of Fn (η).  Using F (η) = ∞ n=0 Fn (η) = F0 (η) + F1 (η) + F3 (η) + · · ·,thus

1 1 1 2 αReη 6 + ( αRe + (4 − Ha)α2 )η 4 30 4 3 3 1 1 1 α2 Re2 η 10 ( αRe + (4 − Ha)α2 )η 2 − 3 4 1350 1 1 1 3 α ReHa)η 8 ( α2 Re2 + α3 Re − 40 70 208 1 1 1 2 (− α2 Re2 − α3 Re + α3 ReHa − α4 30 9 36 45 1 4 1 4 4 6 α Ha − α Ha )η + · · ·. 45 360

F (η) = 1 − η 2 − − + + +

(31)

According to Eq. ( 31), the accuracy increases by increasing the number of solution terms (n). 5. Results and discussion The objective of the present study is to apply the Duan-Rach approach (DRA) to obtain a purely analytical solution of the MHD Jeffery-Hamel problem. To achieve this goal, a few cases of the Duan-Rach approach solution are compared with the homotopy analysis method (HAM) and the numerical results. For the divergent channel, the comparison between the numerical, HAM and Duan-Rach approach when Re = 50 and Ha = 1000 are shown in table 1. The results of the Duan-Rach approach show a good agreement with the numerical values using Runge-kutta method. The error bar shows a high accuracy of this approach as in the case of HAM. In this table the

11

errors are introduced as fallow:

Err1.(%) = |

Duan − Rach approach − N umerical |, N umerical

(32)

Err2.(%) = |

HAM − N umerical |. N umerical

(33)

For the convergent channel, a comparison between the numerical method, the homotopy analysis method, and the Duan-Rach approach when Re = 50 and Ha = 1000 are shown in table 2. The obtained results show a good agreement between these three methods. The errors are calculated in the same manner of Eqs. ( 32) and ( 33). The influence of magnetic field in Jeffery-Hamel flow is shown in fig. 2 for divergent and convergent channels, respectively. The results obtained by the Duan-Rach approach, and those obtained by the Runge-Kutta method are in good agreement. We do not observe a backflow in presence of a magnetic field. The influence of Reynolds number (Re) in the flow is shown in Fig. 3. The results of our work and the numerical method (Runge-Kutta method) match perfectly. The backflow was observed for high Re in divergent channel. The opening angle α affects the flow, especially in the divergent channel case, Fig 4.In the convergent channel, the velocity profile becomes flat and thickness of boundary layer decreases, fig. 4. Backflow in the convergent channel is excluded when varying the parameters of the flow. The plots confirm that the approach used is of a high accuracy for different Re,α, and Ha. In the ADM, for given Re,α and Ha we have to solve F (1) = 0 to find β = F  (0). If we change the value of Re, we have to again evaluate the value of β. In DRA, Eq. ( 31) gives us the velocity profile at any Re,α, and Ha. We have not done a comparison of 12

Table 1 The comparison between HAM [4] and DRA solution in divergent channel when Re = 50 and Ha = 1000. α = 5◦

η DRA

HAM

Numerical

Err1.(%)

Err2.(%)

0.00

1.0000000000

1.0000000000

1.0000000000

0.00E+00

0.00E+00

0.05

0.9976051267

0.9976051267

0.9976051266

1.00E-10

1.00E-10

0.10

0.9904272153

0.9904272156

0.9904272154

1.01E-10

2.02E-10

0.15

0.9784856247

0.9784856266

0.9784856255

8.18E-10

1.12E-09

0.20

0.9618100734

0.9618100746

0.9618100752

1.87E-09

6.24E-10

0.25

0.9404368605

0.9404368643

0.9404368631

2.76E-09

1.28E-09

0.30

0.9144036477

0.9144036501

0.9144036551

8.09E-09

5.47E-09

0.35

0.8837428532

0.8837428568

0.8837428679

1.66E-08

1.26E-08

0.40

0.8484737049

0.8484737066

0.8484737175

1.49E-08

1.28E-08

0.45

0.8085929567

0.8085929619

0.8085929634

8.29E-09

1.86E-09

0.50

0.7640642402

0.7640642412

0.7640642437

4.58E-09

3.27E-09

0.55

0.7148059233

0.7148059133

0.7148059216

2.38E-09

1.16E-08

0.60

0.6606772604

0.6606772666

0.6606772549

8.32E-09

1.77E-08

0.65

0.6014624581

0.6014624674

0.6014624512

1.15E-08

2.69E-08

0.70

0.5368520924

0.5368520875

0.5368520846

1.45E-08

5.40E-09

0.75

0.4664210599

0.4664210783

0.4664210525

1.59E-08

5.53E-08

0.80

0.3896018966

0.3896019056

0.3896018881

2.18E-08

4.49E-08

0.85

0.3056518140

0.3056518011

0.3056518077

2.06E-08

2.16E-08

0.90

0.2136112091

0.2136111723

0.2136112049

1.97E-08

1.53E-07

0.95

0.1122503862

0.1122503245

0.1122503834

2.49E-08

5.25E-07

1.00

0.0000000000

0.0000000000

0.0000000000

0.00E+00

0.00E+00

13

Table 2 The comparison between HAM [4] and DRA solution in convergent channel when Re = 50 and Ha = 1000. α = −5◦

η DRA

HAM

Numerical

Err1.(%)

Err2.(%)

0.00

1.0000000000

1.0000000000

1.0000000000

0.00E+00

0.00E+00

0.05

0.9976051267

0.9976051267

0.9976051266

1.00E-10

1.00E-10

0.10

0.9904272153

0.9904272156

0.9904272154

1.01E-10

2.02E-10

0.15

0.9784856247

0.9784856266

0.9784856255

8.18E-10

1.12E-09

0.20

0.9618100734

0.9618100746

0.9618100752

1.87E-09

6.24E-10

0.25

0.9404368605

0.9404368643

0.9404368631

2.76E-09

1.28E-09

0.30

0.9144036477

0.9144036501

0.9144036551

8.09E-09

5.47E-09

0.35

0.8837428532

0.8837428568

0.8837428679

1.66E-08

1.28E-08

0.45

0.8085929567

0.8085929619

0.8085929634

8.29E-09

1.86E-09

0.50

0.7640642402

0.7640642412

0.7640642437

4.58E-09

3.27E-09

0.55

0.7148059233

0.7148059133

0.7148059216

2.38E-09

1.16E-08

0.60

0.6606772604

0.6606772666

0.6606772549

8.32E-09

1.77E-08

0.65

0.6014624581

0.6014624674

0.6014624512

1.15E-08

2.69E-08

0.70

0.5368520924

0.5368520875

0.5368520846

1.45E-08

5.40E-09

0.75

0.4664210599

0.4664210783

0.4664210525

1.59E-08

5.53E-08

0.80

0.3896018966

0.3896019056

0.3896018881

2.18E-08

4.49E-08

0.85

0.3056518140

0.3056518011

0.3056518077

2.06E-08

2.16E-08

0.90

0.2136112091

0.2136111723

0.2136112049

1.97E-08

1.53E-07

0.95

0.1122503862

0.1122503245

0.1122503834

2.49E-08

5.25E-07

1.00

0.0000000000

0.0000000000

0.0000000000

0.00E+00

0.00E+00

14

Fig. 2 A comparison of numerical results (circles) and Duan-Rach approach (solid lines) for the velocity profile using Re = 50 when Ha is varied.

the convergence of this approach with standard ADM because we have found the same rate of convergence.

6. Conclusion In this paper, the Duan-Rach approach (DRA) was used to obtain a purely analytical solution of MHD Jeffery-Hamel. The Duan-Rach approach allows us to find a solution without using numerical methods to evaluate the coefficient β. This undetermined coefficient in standard ADM is inserted due to the nature of the boundary condition F (1) = 0.As a consequence, we need to solve the nonlinear algebraic equation that produces more than one root (the number of roots is equal to the number of the terms in ADM). The solution found with this approach is compared with that obtained by 15

Fig. 3 A comparison of numerical results (circles) and Duan-Rach approach (solid lines) for the velocity profile using Ha = 50 when Re is varied.

Fig. 4 A comparison of numerical results (circles) and Duan-Rach approach (solid lines) for the velocity profile using Re = 50 and Ha = 0 when α is varied.

16

the homotopy analysis method (HAM). The comparison confirms the validity of this approach. Thus, we do not need to evaluate any parameters at each stage of approximation. The final solution does not contain an undetermined coefficient. This approach allows us to automate the algorithm obtained. These findings establish the Duan-Rach approach (DRA) as an accurate and efficient alternative to the standard Adomian decomposition method (ADM). References References [1] Jeffery GB. The two-dimensional steady motion of a viscous fluid. Phil Mag 1915;29:455–65. [2] Hamel G. Spiralf¨ormige bewgungen z¨aher fl¨ u ssigkeiten. Jahresber Deutsch Math-Verein 1916;25:34–60. [3] Batchelor GK. An introduction to fluid dynamics. Cambridge university press; 2000. [4] Motsa SS, Sibanda P, Awad FG, Shateyi S. A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem. Comput Fluids 2010;39:1219–25. [5] Moghimi SM, Domairry G, Soleimani S, Ghasemi E, Bararnia H. Application of homotopy analysis method to solve MHD Jeffery-Hamel flows in non-parallel walls. Adv Eng Software 2011;42:108–13.

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[6] Cha JE, Ahn YC, Kim MH. Flow measurement with an electromagnetic flowmeter in two-phase bubbly and slug flow regimes. Flow meas instrum 2002;12:329–39. [7] Mossimo J. Some nonlinear proplems involving a free boundary plasma physics. J Differ Equ 1979;34:114–38. [8] Nijsing R, Eifler W. A computational analysis of transient heat transfer in fuel rod bundles with single phase liquid metal cooling. Nucl Eng Des 1980;62:39–68. [9] Adomian G. Nonlinear stochastic operator equations. Academic Press, New York; 1986. [10] Esmaili Q, Ramiar A, Alizadeh E, Ganji DD. An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method. Phys Lett A 2008;372:3434–9. [11] Esmaeilpour M, Ganji DD. Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method. Comput Math App 2010;59:3405–11. [12] Cherruault Y.

Convergence of adomian’s method.

Kybernetes

1989;18:31–8. [13] Cherruault Y, Adomian G. Decomposition methods: a new proof of convergence. Math Comput Model 1993;18:103–6. [14] Jin C, Liu M. A new modification of adomian decomposition method for

18

solving a kind of evolution equation. App Math Comput 2005;169:953– 62. [15] Duan JS, Rach R. A new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations. App Math Comput 2011;218:4090–118. [16] Schlichting H. Boundary-layer theory. McGraw-Hill Press,New York; 2000. [17] Adomian G, Rach R. Inversion of nonlinear stochastic operators. J Math Anal and Appl 1983;91:39–46.

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Highlights

1. 2. 3. 4.

We study the MHD Jeffery-Hamel flow.

Duan-Rach approach (DRA) was used to solve the governing equations. The final solution does not contain an undetermined coefficient. This approach allows us to automate the algorithm obtained.