Int. J. Appl. Comput. Math DOI 10.1007/s40819-017-0384-7 ORIGINAL PAPER

Heat Transfer Analysis of MHD Three Dimensional Casson Fluid Flow Over a Porous Stretching Sheet by DTM-Padé Majeed A. Yousif1 · M. Hatami2 · Hajar F. Ismael1

© Springer (India) Private Ltd. 2017

Abstract In this paper, three dimensional incompressible Casson fluid flow past a linear stretching porous plate under the magnetic field effect is analyzed using differential transformation method (DTM) and numerical method. For increasing the accuracy of DTM, Padé approximation is applied. Comparison between DTM- Padé and numerical method shows that Padé with order [15,15] can be an exact and high efficiency procedure for solving these kinds of problems. The influence of the Casson fluid parameter (β), Prandtl number (Pr), magnetic parameter (M), stretching parameter (c) and porous parameter (λ) on non-dimensional temperature and velocity profiles are investigated. The results indicated for the Casson fluid flow that increasing the Hartmann number make a decrease in velocity boundary layer thicknesses. Keywords DTM-Padé method · Casson fluid flow · MHD · Heat transfer

List of Symbols x, y, and z u, v, and w U w and V w v β B K Pr Rex

B

Direction of axes Velocities along x-,y-, and z- axes Stretching velocities Kinematic viscosity Casson fluid parameter Nanoparticle volume fraction Porous medium permeability Prandtl number Local Reynolds number

M. Hatami [email protected]

1

Department of Mathematics, University of Zakho, P.O. Box 12, Duhok, Kurdistan Region, Iraq

2

Mechanical Engineering Department, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

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τwx and τwy f  (η) and g  (η) θ (η) λ M C a, b, α, δ and γ

Wall shear stresses Velocity profiles Temperature profile Porosity parameter Hartmann parameter Stretching parameter Constants

Introduction Casson fluid can be defined as a shear thinning liquid which is assumed to have an infinite velocity at zero rate of shear, a yield stress below which no flow occurs and a zero velocity at an infinite rate of shear. Actually Casson fluid flow is a category of non-Newtonian fluids presented by researchers [1–5]. Dash et al. [3] investigated the flow behavior of Casson fluid in a tube filled by a homogeneous porous medium. Nadeem et al. [4] discussed about the magneto-hydrodynamic (MHD) effect on the Casson fluid flow over an exponentially shrinking sheet by an analytical method called Adomian decomposition method (ADM), they just solved the problem for velocity profiles. Maleque [5] examined the unsteady MHD Non-Newtonain incompressible Casson fluid flow due to a porous rotating disk with a uniform angular velocity in the presence of an axial uniform magnetic field and a uniform electric field. They found that Increasing values of electric field lead to decrease the boundary layer thickness. Differential transformation method is one of the powerful analytical methods which does not need to any small parametel like p in homotopy perturbation method (HPM) for discretization, perturbation or linearization. This method constructs an analytical solution in the form of a polynomial. It is different from the traditional higher-order Taylor series method. The Taylor series method is computationally expensive for large orders. The differential transform method is an alternative procedure for obtaining an analytic Taylor series solution of differential equations. The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization and therefore, it is not affected by errors associated to discretization. The concept of DTM was first introduced by Zhou [6], who solved linear and nonlinear problems in electrical circuits. Jang et al. [7] applied the two-dimensional DTM to partial differential equations. This method such as other analytical methods was successfully applied to various application problems [8–25]. All of these successful applications verified the validity, effectiveness and flexibility of the DTM. Keimanesh et al. [26] used the multi-step differential transform method (MsDTM) to find the analytical solution of the resulting ordinary differential equation. Steady hydromagnetic convective heat and mass transfer with slip flow from a spinning disk with viscous dissipation and Ohmic heating was investigated by Rashidi et al. [27] using DTMPadé. DTM constructs for differential equations an analytical solution in the form of a power series. Furthermore, power series are not useful for large values of η, say η → ∞. It is now well-known that the Padé approximants [28] and [29] have the advantage of manipulating the polynomial approximation into rational functions of polynomials. It is therefore essential to combination of the series solution, obtained by the DTM with the Padé approximant to provide an effective tool to handle boundary value problems at infinite domains. One of the first successful applications of DTM to boundary-layer flows was presented by Rashidi and Domairry [30]. Rashidi et al. [31] studied the new analytical method (DTM-Padé) for solv-

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Fig. 1 Physical model and coordinate system

ing magnetohydrodynamic boundary-layer equations. He showed that differential transform method (DTM) solutions are only valid for small values of independent variable. According the above descriptions, the main objective of this study is to apply DTM- Padé to find the approximate solution of nonlinear differential equations governing the problem of flow and heat transfer of Casson fluid flow over a porous stretching plate under the magnetic field effect. The main difference and novelty of this work is solving the energy equation and find the thermal boundary layer by the present analytical method.

Mathematical Model We assume three directional incompressible Cassson fluid flows past a linear stretching plate under the effect of energy equation (see Fig. 1). The sheet is induced by stretching in xyplane while fluid is set along z-axis. Moreover, constant magnetic field under the effect of heat equation is applied to the Casson fluid. The equations which govern (3D) Casson with heat equation in incompressible fluid are states as [4]: u x + v y + wz = 0



uu x + vu y + wu z = υ 1 +

1 β

(1)

 u zz −

σ B2 ρ

u−

υ u K

(2)

  υ 1 σ B2 vzz − v− v uvx + vv y + wvz = υ 1 + β ρ K uTx + vTy + wTz = αTzz

(3) (4)

here u, v, w and υ are velocities in the x, y, z directions and kinematic viscosity of the fluid, respectively. β is the Casson fluid induction„ B is the magnetic parameter and K is the porous medium permeability, finally α is thermal diffusivity parameter of the fluid. The relevant boundary conditions of Eqs. (2), (3), and (4), for the problems are given by: u = U (x) = ax, u → 0,

v → 0,

v = V (x) = by, T = Tw (x) T → T∞

as

z=0 as

(5)

z→∞

(6)

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in the above equations a and b are constants greater than zero, and Tw (x, z) = T∞ + cx 2 T0 /2v, where T0 is heating or cooling temperature. In order to reduce the governing Eqs. (1–4) with associated boundary conditions into non-dimensional, we apply [1] √ u = ax f η , v = bygη , w = − aυ ( f + cg) ,   √ cx 2 T0 θ (η) η = aυ z (7) Tw (x, z) = T∞ + 2υ into Eqs. (1–6), continuity equation is identically satisfy, and Eqs. (2–6), finally changes to:    2 1 f ηηη − f η + ( f + g) f ηη − (M + λ) f η = 0, 1+ (8) β    2 1 gηηη − gη + ( f + g) gηη − (M + λ) gη = 0, (9) 1+ β 1 θηη − 2θ f η + ( f + g) θη = 0, (10) Pr (11) f (η) = 0, f η (η) = 1, g (η) = 0, gη (η) = c, θ = 1 as η = 0 f η (η) → 0, gη (η) → 0 θ → 0 as η → ∞ σ B2

In above expressions, M = ρa , λ = Prandtl number, respectively.

υ ak

and Pr =

ϑ α

(12)

are Magnetic, porosity parameter and

Fundamental of Differential Transformation Method Let x(t) be analytic in a domain D and let t = ti represent any point in D. The function x(t) is then represented by one power series whose center is located at ti . The Taylor series expansion function of x(t)is in form of:   ∞  (t − ti )k d k x (t) ∀t ∈ D (13) x (t) = k! dt k t=ti k=0

The particular case of Eq. (13) when ti = 0 is referred to as the Maclaurin series of x(t) and is expressed as:  ∞ k  k  t d x (t) x (t) = ∀t ∈ D (14) k! dt k t=0 k=0

As explained in [6] the differential transformation of the function x(t) is defined as follows:   ∞  H k d k x (t) X (k) = (15) k! dt k t=0 k=0

where x(t) is the original function and X (k) is the transformed function. The differential spectrum of X (k) is confined within the interval t ∈ [0, H ], where H is a constant. The differential inverse transform of X (k) is defined as follows:  ∞   t k X (k) (16) x (t) = H k=0

It is clear that the concept of the differential transformation is based upon the Taylor series expansion. The values of function X (k) at values of argument k are referred to as discrete,

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Int. J. Appl. Comput. Math Table 1 The operations for the one-dimensional differential transform method () Original function

Transformed function

w (x) = u (x) ± v (x)

W (k) = U (k) ± V (k)

w (x) = λu (x)

W (k) = λU (k) , λ is constant

w (x) = du(x) dx

W (k) = (k + 1) U (k)

r w (x) = d du(x) xr

W (k) = (k + 1) (k + 2) . . . (k + r ) U (k + r )

w (x) = u (x) v (x)

W (k) =

dv(x) w (x) = du(x) dx dx

W (k) =

w (x) = u (x) dv(x) dx

W (k) =

2 w (x) = u (x) d v(x) 2

W (k) =

dx

k r =0 k r =0 k r =0 k r =0

U (r ) V (k − r ) (r + 1) (k − r + 1) U (r + 1) V (k − r + 1) (k − r + 1) U (r ) V (k − r + 1) (k − r + 2) (k − r + 1) U (r ) V (k − r + 2)

i.e. X (0) is known as the zero discrete, X (1) as the first discrete, etc. the more discrete available, the more precise it is possible to restore the unknown function. The function x(t) consists of T function X (k), and its value is given by the sum of the T -function with (t/H )k as its coefficient. In real applications, at the right choice of constant H , the larger values of argument k the discrete of spectrum reduce rapidly. The function x(t) is expressed by a finite series and Eq. (16) can be written as: x (t) =

 n   t k X (k) H

(17)

k=0

Mathematical operations performed by differential transform method are listed in Table 1.

Padé Approximants A Padé approximate is the ratio of two polynomials constructed from the coefficients of the Taylor series expansion of a function u(x). The [L/M] Padé approximants to a function y(x) are given by [28]   L PL (x) (18) = M Q M (x) where PL (x) is polynomial of the degree of at most L and Q M (x) is a polynomial of the degree of at most M. The formal power series y (x) =

∞ 

ai x i

(19)

i=0

and y (x) −



PL (x) = O x L+M+1 Q M (x)

(20)

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determine the coefficients of PL (x) and Q M (x) by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave [L/M] unchanged, we imposed the normalization condition Q M (0) = 1.0 (21) Finally, PL (x) and Q M (x) need to include non-common factors. If the coefficient of PL (x) and Q M (x) is written as ⎧ ⎨ PL (x) = p0 + p1 x + p2 x 2 + · · · + p L x L , ⎩

(22) Q M (x) = q0 + q1 x + q2 x 2 + · · · + q M x M ,

and using Eqs. (21) and (22), we may multiply (15) by Q M (x), which linearizes the coefficient equations. Eq. (20) can be presented in more details as ⎧ a L+1 + a L q1 + · · · + a L−M+1 q M = 0 , ⎪ ⎪ ⎪ ⎨ a L+2 + a L+1 q1 + · · · + a L−M+2 q M = 0 , .. ⎪ ⎪ . ⎪ ⎩ a L+M + a L+M−1 q1 + · · · + a L q M = 0 , ⎧ a 0 = p0 , ⎪ ⎪ ⎪ ⎪ a ⎪ ⎨ 1 + a 0 q 1 = p1 , a 2 + a 1 q 1 + a 0 q 2 = p2 , ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎩ a L + a L−1 q1 + . . . + a0 q L = p L

(23)

(24)

To solve these equations, we start with (23), which is a set of linear equations for all the unknown q  s. Once the q  s are known, then (24) gives an explicit formula for the unknown p  s, which complete the solution. If (23) and (24) are nonsingular, then we can solve them directly and obtain (25), where (25) holds, and if the lower index on a sum exceeds the upper, the sum is replaced by zero: ⎡

⎤ a L−M+2 · · · a L+1 a L−M+1 .. ⎢ .. ⎥ . . .. ⎢ ⎥ .. . det ⎢ . ⎥ ⎣ aL ⎦ a L+1 · · · a L+M   L L L j j ··· j a x a x a x j−M j−M+1 j L j=M j=M−1 j=0 = ⎡ ⎤ M a L−M+1 a L−M+2 · · · a L+1 ⎢ .. ⎥ .. . . .. ⎢ ⎥ .. . det ⎢ . ⎥ ⎣ aL a L+1 · · · a L+M ⎦ xM x M−1 · · · 1

(25)

To obtain a diagonal Padé approximants of a different order such as [2/2], [4/4], or [6/6], the symbolic calculus software Maple is used.

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Analytical Approximations by Means of DTM-Padé The following transforms are obtained, from Table 1 to each term of Eqs. (8–10) to taking the one-dimensional differential transform f ηηη → (k + 3) (k + 2) (k + 1) F (k + 3) , f f ηη →

k 

(26)

(k + 2 − r ) (k + 1 − r ) F (r ) F (k + 2 − r ) ,

(27)

(k + 1 − r ) (r + 1) F (r + 1) F (k + 1 − r ) ,

(28)

r =0

f η2 →

k  r =0

G ηηη → (k + 3) (k + 2) (k + 1) G (k + 3) , ggηη →

k 

(29)

(k + 2 − r ) (k + 1 − r ) G (r ) G (k + 2 − r ) ,

(30)

(k + 1 − r ) (r + 1) G (r + 1) G (k + 1 − r ) ,

(31)

r =0

gη2 →

k  r =0

θηη → (k + 2) (k + 1) Θ (k + 2) , f gηη →

k 

(32)

(k + 1 − r ) (k + 2 − r ) F (r ) G (k + 2 − r ) ,

(33)

(k + 1 − r ) (k + 2 − r ) G (r ) F (k + 2 − r ) ,

(34)

(k + 1 − r ) F (r ) Θ (k + 1 − r ) ,

(35)

(k + 1 − r ) G (r ) Θ (k + 1 − r ) ,

(36)

(k + 1 − r ) Θ (r ) F (k + 1 − r ) ,

(37)

r =0

g f ηη →

k  r =0

f θη →

k  r =0

gθη →

k  r =0

θ fη →

k  r =0

where F (k) ,  (k) , and  (k) are the transformed functions of f (k) , θ (k) , and φ (k) respectively and are given by

f (η) = θ (η) = φ (η) =

∞  k=0 ∞  k=0 ∞ 

F (k) ηk ,

(38)

 (k) ηk ,

(39)

 (k) ηk ,

(40)

k=0

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Substituting Eqs. (26–37) into Eqs. (8–10) and using boundary conditions (11) and (12) we have 

1 1+ β =

 (k + 3) (k + 2) (k + 1) F (k + 3)

k 

(r + 1) F (r + 1) (k − r + 1) F (k − r + 1)

r =0

−

k 

(k + 2 − r ) (k + 1 − r ) F (r ) F (k + 2 − r )

r =0

−

k 

(k + 2 − r ) (k + 1 − r ) G (r ) F (k + 2 − r )

r =0

  + λ + M 2 (k + 1) F (k + 1)   1 1+ (k + 3) (k + 2) (k + 1) G (k + 3) β =

k 

(41)

(r + 1) G (r + 1) (k − r + 1) G (k − r + 1)

r =0

−

k 

(k + 2 − r ) (k + 1 − r ) G (r ) G (k + 2 − r )

r =0

−

k 

(k + 2 − r ) (k + 1 − r ) F (r ) G (k + 2 − r )

r =0

  + λ + M 2 (k + 1) G (k + 1)   1 (k + 2) (k + 1)  (k + 2) Pr =2

k 

(k − r + 1)  (r ) F (k − r + 1) −

r =0

−

k 

(42)

k 

(k − r + 1) F (r )  (k − r + 1)

r =0

(k − r + 1) G (r )  (k − r + 1) ,

r =0

F (0) = 0,

F (1) = 1,

F (2) = α,

G (0) = 0,

G (1) = c,

G (2) = δ,

 (0) = 1,

 (1) = γ .

(43) (44) (45) (46)

Moreover, substituting Eqs. (44–46) into Eqs. (41–43), we can calculate the values of F (k), G(k) and  (k) by a recursive method. Hence, substituting all F (k), G(k) and  (k) into Eqs. (38–40), we have the series solutions as below:  f (η) ∼ = η + αη2 +

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    1 + M2 + λ 3 2α − 2cα + 2α M 2 + λ 4  η + 



η 6 1 + β1 24 1 + β1

(47)

Int. J. Appl. Comput. Math

Fig. 2 Velocity f  (η), when λ = M = c = 0.5 and β = 1

Fig. 3 Temperature θ (η), when λ = M = c = 0.5 and β = 1

g (η) ∼ = cη + δη2 +

    2   c + c M2 + λ 3 −2δ + 2cδ + 2δ M 2 + λ 4  



η + η 6 1 + β1 24 1 + β1

(48)

1 θ (η) ∼ = 1 + γ η + Pr η2 + Pr (−γ − cγ + 2 (2α + γ )) η3 6 ⎞⎞ ⎛ ⎛ 2+λ 1 1 + M  ⎠⎠ η4 + Pr ⎝−2Pr − 2c Pr − αγ − γ δ + 2 ⎝ Pr + 2αγ +

12 2 1 + β1 (49)

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Fig. 4 Velocity g  (η), when λ = M = c = 0.5 and β = 1 Table 2 Comparison between the results of DTM- Padé [15,15] and numerical shooting method obtained by MATHEMATICA, when λ = M = c = 0.5 and β = 1 η

f  (η)

g  (η)

θ (η)

DTM-Padé

Numerical

DTM-Padé

Numerical

DTM-Padé

Numerical

0.0

1.000000

1.000000

0.500000

0.500000

1.000000

1.00000

1.0

0.360965

0.360965

0.190919

0.190917

0.415264

0.415264

2.0

0.124921

0.124921

0.067412

0.067409

0.183444

0.183907

3.0

0.042410

0.042409

0.023047

0.023042

0.080609

0.081505

4.0

0.014088

0.014088

0.007679

0.007671

0.032588

0.033703

Table 3 Comparison between the results of DTM- Padé [15,15] and numerical solution obtained by fourthorder Runge–Kutta–Fehlberg (see [1]) for skin friction coefficients, when M = 0, c = 0.5 and β = 1 λ

Numerical

[1] − 1 + β1 f  (0)

Present work

 − 1 + β1 f  (0)

0.0

1.5459

1.5472

0.6579

0.6589

0.5

1.8361

1.83633

0.8228

0.8230

1.0

2.0884

2.0885

0.9614

0.9614

Numerical

[1] − 1 + β1 g  (0)

Present work

 − 1 + β1 g  (0)

The analytical solution obtained by the DTM cannot satisfy at infinity boundary conditions, so we use the Padé approximant to converge the series solution. It is therefore apply the Padé approximation to Eqs. (47–49) under boundary conditions (11) and (12) at η = ∞, we can obtain α, δ and γ . For example, the values of α = −0.491568, δ = −0.223689 and γ = −0.93335, after applying DTM- Padé, when M = λ = c = 0.5 and β = 1.

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Fig. 5 Effect of Casson fluid parameter (β) on Velocity f  (η), when λ = M = c = 0.5

Fig. 6 Effect of Casson fluid parameter (β) on Velocity g  (η), when λ = M = c = 0.5

Results and Discussions As described above, the main purpose of this study is to introduce DTM and DTM-Padé as powerful analytical method for solving the governing equation of fluid flow and heat transfer of Casson fluid over a porous stretching plate. A sample result is presented in Figs. 2, 3 and 4 for f ’(η), g’(η) (as velocity profiles) and θ (η) (as temperature profile) when Pr = λ = M = c = 0.5 and β = 1. This figures confirm that DTM and DTM-Padé have aa good agreement in the initial distance of fluid entrance on plate (Low η), but after passing a distance DTM

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Fig. 7 Effect of porosity parameter (λ) on velocity f  (η) , when M = c = 0.5 and β = 1

Fig. 8 Effect of porosity parameter (λ) on velocity f  (η), when M = c = 0.5 and β = 1

cannot predict a correct solution and its predicted values reaches to infinite in all velocity and temperature profiles. While by applying the DTM-Padé with the accuracy order of [15,15] an excellent agreement between the numerical and analytical method is observed. So this method can be applied in the next steps by this order. Tables 2 and 3 are presented to show the accuracy by exact values for profiles values and skin friction coefficients, respectively. As seen, for both profile values and calculated skin friction coefficient DTM-Padé demonstrates an acceptable values with minimum possible errors.

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Fig. 9 Effect of Hartmann number (M) on velocity f  (η), when λ = c = 0.5 and β = 1

Fig. 10 Effect of Hartmann number (M) on velocity g  (η), when λ = c = 0.5 and β = 1

Figures 5 and 6 is depicted to show the effect of Casson fluid parameter (β) on the velocity profiles f ’(η) and g’(η), respectively. It is obvious that increasing the β causes a decrease in the velocity boundary layer. The influence of the porosity parameter (λ) on the velocity profiles are discussed on the Figs. 7 and 8. These figures also approve that increasing the porosity makes decrease in the velocity boundary layer thicknesses due to more fluid penetration in larger porosity numbers. From Figs. 9 and 10 it can be observed that for higher values of Magnetic parameter (M), it reduces both boundary layer thickness and the magnitude of the velocity. Physically, present phenomena occur when magnetic field

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Fig. 11 Effect of stretching parameter (c) on Velocity f  (η), when λ = M = 0.5 and β = 1

Fig. 12 Effect of stretching parameter (c) on Velocity g  (η), when λ = M = 0.5 and β = 1

can induced current in the conductive fluid, then it create a resistive-type force on the fluid in the boundary layer, which slow down the motion of the fluid. So finally, it is conclude that magnetic field is used to control boundary layer separation. The same treatment also was observed by Nadeem et al. [1]. To show the consequence of stretching parameter (c) on velocity profiles Figs.11 and 12 are depicted when λ = M = 0.5 and β = 1. This parameter has different effect of f ’(η) and g’(η), it means that its increase makes increase in the velocity values for g’(η) but decrease in f ’(η) which is due to stretching direction. Finally the effect

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Int. J. Appl. Comput. Math

Fig. 13 Effect of Prandtl number (Pr) on Temperature θ (η), when λ = M = 0.5 and β = 1

of Prandtl number on thermal boundary layer is illustrated via Fig. 13 which reveals that Casson fluids with larger Pr number have a thinner thermal boundary layer.

Conclusion In this paper, differential transformation method (DTM) with Padé approximation has been successfully applied to find the solution of Casson MHD fluid flow and heat transfer over a stretching plate. The proper order of Padé accuracy were selected as [15,15] to ensure the convergence of the DTM. The obtained solutions revealed that DTM and DTM-Padé can be simple, powerful and efficient techniques for finding analytical solutions of these kinds of physical problems with nonlinear differential equations. The results show that Casson fluids with larger Prandtl number have a thinner thermal boundary layer. Also it was found that increasing the Casson fluid parameter (β) causes a decrease in the velocity boundary layer thicknesses.

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