Appl. Math. Mech. -Engl. Ed., 32(2), 167–178 (2011) DOI 10.1007/s10483-011-1403-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid over unsteady stretching sheet∗ T. HAYAT1,2 , M. MUSTAFA1,2 , A. A. HENDI2 (1. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan; 2. Department of Physics, Faculty of Science, King Saud University, Riyadh 11320, Saudi Arabia)

Abstract This article studies the three-dimensional boundary layer flow of an elasticoviscous fluid over a stretching surface. Velocity of the stretching sheet is assumed to be time-dependent. Effect of mass transfer with higher order chemical reaction is further considered. Computations are made by the homptopy analysis method (HAM). Convergence of the obtained series solutions is explicitly analyzed. Variations of embedding parameters on the velocity and concentration are graphically discussed. Numerical computations of surface mass transfer are reported. Comparison of the present results with the numerical solutions is also given. Key words solution

unsteady flow, mass transfer, viscoelastic fluid, homptopy analysis method

Chinese Library Classification O357 2010 Mathematics Subject Classification

1

80A20

Introduction

It is known that non-Newtonian fluids frequently appear in industry and engineering[1–7] . In particular, the boundary layer flows of viscous and non-Newtonian fluids over a stretching surface are of considerable interest in recent years because of two reasons. Firstly, the governing equations for such flows are non-linear which provide interesting challenges to the researchers from different quarters. Secondly, such flows have particular relevance in the extrusion of polymer sheets, glass blowing, manufacturing plasatic films, crystal growing, hot rolling and many others. Much attention in the past has been given to the steady two-dimensional stretching flows. However, little efforts are devoted to analyze time-dependent stretching flow problems[8–11] . The boundary layer flows of an elastico-viscous fluid are not widely discussed. Beard and Walters[12] reported the seminal work on the flow of an elastico-viscous fluid. They considered the stagnation-point flow of an elastico-viscous fluid and obtained the pertubation solution for small values of viscoelastic parameter. Partial slip effects on the stagnation-point flow of an elastico-viscous fluid were considered by Ariel[13] . Very recently, Hayat et al.[14] investigated the steady two-dimensional flow of an elastico-viscous fluid in the presence of free stream velocity. The solution of the resulting differential equations was derived by the homotopy analysis method (HAM). ∗ Received Sept. 19, 2010 / Revised Nov. 26, 2010 Corresponding author M. MUSTAFA, E-mail: meraj [email protected]

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T. HAYAT, M. MUSTAFA, and A. A. HENDI

Cortell[15–16] analyzed the mass transfer effect on the two-dimensional flow of a viscoelastic fluid. Nazar and Latip[17] considered the three-dimensional flow of a second grade fluid and provided the numerical solution. Recently, Mukhopadhyay[18] examined the effect of thermal radiation on the unsteady mixed convection flow and heat transfer over a porous stretching surface. The object of the current investigation is to study the unsteady three-dimensional flow and mass transfer characteristics over an unsteady stretching sheet. Analytic solutions for the momentum and mass transfer are developed by the homotopy analysis method suggested by Liao[19] . This method has been successfully applied by several researchers to various interesting problems[20–34] . Graphical results for various values of emerging parameters are discussed in detail. Numerical values of surface mass transfer are computed.

2

Mathematical formulation

Let us consider the unsteady three-dimensional flow and mass transfer characteristics of an elastico-viscous fluid over an unsteady stretching surface of concentration Cw . The surrounding stationary fluid has concentration C∞ which is sufficiently small in comparison to Cw (see Fig. 1). The unsteady three-dimensional boundary layer flow and mass transfer are governed by the following equations[17,30] :

Fig. 1

Geometry of the problem

∂u ∂v ∂w + + = 0, (1) ∂x ∂y ∂z ∂u ∂u ∂u ∂u +u +v +w ∂t ∂x ∂y ∂z  2 3 ∂3u ∂w ∂ 2 u  ∂ u ∂ 3 u ∂u ∂ 2 u ∂u ∂ 2 w ∂u ∂ 2 u ∂ u +u −2 , (2) +w 3 − − −2 = ν 2 − k0 2 2 2 2 ∂z ∂z ∂t ∂x∂z ∂z ∂x ∂z ∂z ∂z ∂z ∂x∂z ∂z ∂z 2 ∂v ∂v ∂v ∂v +u +v +w ∂t ∂x ∂y ∂z  ∂3v ∂3v ∂w ∂ 2 v  ∂3v ∂v ∂ 2 v ∂v ∂ 2 w ∂v ∂ 2 v ∂2v + v − 2 , + w − − − 2 = ν 2 − k0 ∂z ∂z 2 ∂t ∂y∂z 2 ∂z 3 ∂y ∂z 2 ∂z ∂z 2 ∂z ∂y∂z ∂z ∂z 2 ∂C ∂C ∂C ∂C ∂2C +u +v +w = D 2 − Kn (t)C n , ∂t ∂x ∂y ∂z ∂z

(3) (4)

where u, v, and w are the velocity components, ρ is the fluid density, ν is the kinematic viscosity, μ is the dynamic viscosity, k0 is the material fluid parameter, C is the concentration of species of the fluid, D is the diffusion coefficient of the diffusion species in the fluid, and Kn (t) = kn (1−αt) is the reaction rate, in which αt < 1.

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid

The boundary conditions for the problem under consideration are ⎧ ⎨u = u (x) = ax , v = v (y) = by , w = 0, C = C at z = 0, w w w 1 − αt 1 − αt ⎩ u → 0, v → 0, C → C∞ as z → ∞,

169

(5)

in which a and b are the positive constants. In order to make the problems dimensionless, we define  ⎧ a ax ay ⎪   ⎪ ⎪ ⎨ η = ν(1 − αt) z, u = 1 − αt f (η), v = 1 − αt g (η), (6)  ⎪ aν C − C∞ ⎪ ⎪ ⎩w = − . (f (η) + g(η)), φ = 1 − αt Cw − C∞ Since the incompressibility condition is now automatically satisfied, and (2)–(5) take the forms  η f  − f 2 − ζ f  + f  + (f + g)f     η 2 (7) + K − ζ f iv + 2f  + (f + g)f iv + (f  − g  )f  − 2(f  + g  )f  = 0, 2  η g  − g 2 − ζ f  + f  + (f + g)g     η 2 (8) + K − ζ f iv + 2f  + (f + g)g iv + (g  − f  )g  − 2(f  + g  )g  = 0, 2 ζ φ + Sc(f + g)φ − Scγφn − Sc ηφ = 0, 2  f (0) = 0, g(0) = 0, f (0) = 1, g  (0) = c, φ = 1 at η = 0, f  (∞) = 0, g  (∞) = 0, φ = 0 as η → ∞,

(9)

(10)

where ζ is the unsteadiness parameter, K is the viscoelastic parameter, Sc is the Schimdt number, c is the stretching ratio, and γ is the chemical reaction parameter. These are defined as n−1 α k0 a b υ kn Cw ζ= , K= , c = , Sc = , γ = . (11) a υ(1 − αt) a D a It should be pointed out that when c = 0, the resulting two-dimensional problem (g = 0) is    η  η f  − f 2 − ζ f  + f  + f f  + K − ζ f iv + 2f  + f f iv + f 2 − 2f  f  = 0, (12) 2 2 ζ φ + Scf φ − Scγφn − Sc ηφ = 0, (13) 2 f (0) = 0, f  (0) = 1, f  (∞) = 0, (14) φ = 1 at η = 0, φ = 0 as η → ∞. For the steady-state case (ζ = 0), (12)–(14) admit the exact solutions of the forms 1 (0 < K  1), f (η) = 1 − exp(−mη), m = √ 1−K

 Sc

 |F | α1 + α2 ; 1 + 2α2 ; − m 2 exp(−mη) 

, φ(η) = exp − (α1 + α2 ) Sc |F | α1 + α2 ; 1 + 2α2 ; − m 2

(15) (16)

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4Scγm2 + Sc2 Sc α1 = , α = , 2 2m2 2m2 and |F | are the confluent hypergeometric functions. For c = 1 and f = g, the corresponding axisymmetric flow is    η  η f  − f 2 − ζ f  + f  + 2f f  + K − ζ f iv + 2f  + 2f f iv − 4f  f  = 0, 2 2 ζ φ + 2Scf φ − Scγφn − Sc ηφ = 0 2

where

with the boundary conditions (14). The skin friction coefficients and the surface mass transfer are defined as   ζ 1 Rex2 Cf x = f  + K (3f  + ηf  ) + 3(f  + g  )f  − (f + g)f  , 2 η=0    1 ζ , Rey2 Cf y = g  + K (3g  + ηg  ) + 3(f  + g  )g  − (f + g)g  2 η=0 ∂φ

, φ (0) =

∂z z = 0 where Rex =

3

uw x ν

vw y ν

and Rey =

(17)

(18) (19)

(20) (21) (22)

are the local Reynolds numbers.

Solutions by homotopy analysis method

3.1 Zeroth-order deformation problems To solve (7)–(10) by means of the HAM, the velocity distributions f (η), g(η), and the concenteration field φ(η) are expressed by the set of base functions  k  η exp(−rη) | k  0, r  0 (23) as follows:

⎧ ∞ ∞   ⎪ ⎪ 0 ⎪ f (η) = a + akm,r η k exp(−rη), ⎪ 0,0 ⎪ ⎪ ⎪ r=0 k=0 ⎪ ⎪ ⎪ ∞  ∞ ⎨  Akm,r η k exp(−rη), g(η) = A00,0 + ⎪ ⎪ r=0 k=0 ⎪ ⎪ ⎪ ∞ ∞  ⎪  ⎪ ⎪ ⎪ bkm,r η k exp(−rη), ⎪ ⎩ φ(η) =

(24)

r=0 k=0

where akm,r , Akm,r , and bkm,r are the coefficients. Employing rule of solution expressions for f (η), g(η), and φ(η) and (7)–(10), the initial guesses f0 (η), g0 (η), and φ0 (η) and the linear operators L1 and L2 are f0 (η) = 1 − exp(−η), L1 (f ) = f





−f ,

g0 (η) = c(1 − exp(−η)),

φ0 (η) = exp(−η),



L2 (f ) = f − f

(25) (26)

with the properties

L1 [C1 + C2 exp(η) + C3 exp(−η)] = 0, L2 [C4 exp(η) + C5 exp(−3η)] = 0,

(27)

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid

171

in which Ci (i = 1, · · · , 5) are the constants, and (7)–(10) further suggest the following definition of the non-linear operators: N1 [f (η, p), g(η, p)] =

 η ∂ 2 f (η, p) ∂f (η, p)  ∂ 3 f (η, p)  ∂f (η, p) 2 − − ζ + ∂η 3 ∂η 2 ∂η 2 ∂η  2 4

 ∂ f (η, p) η ∂ f (η, p) ∂ 3 f (η, p)  + f (η, p) + g(η, p) − Kζ + 2 ∂η 2 2 ∂η 4 ∂η 3  

4 2 2  ∂ f (η, p) ∂ f (η, p) ∂ g(η, p)  ∂ 2 f (η, p) + − + K f (η, p) + g(η, p) ∂η 4 ∂η 2 ∂η 2 ∂η 2  ∂f (η, p) ∂g(η, p)  ∂ 3 f (η, p)  , −2 + ∂η ∂η ∂η 3

(28)

N2 [f (η, p), g(η, p)] =

 η ∂ 2 g(η, p) ∂g(η, p)  ∂ 3 g(η, p)  ∂g(η, p) 2 − − ζ + ∂η 3 ∂η 2 ∂η 2 ∂η  2 4

 ∂ g(η, p) η ∂ g(η, p) ∂ 3 g(η, p)  + f (η, p) + g(η, p) − Kζ + ∂η 2 2 ∂η 4 ∂η 3  

4 2 2  ∂ g(η, p) ∂ f (η, p) ∂ g(η, p)  ∂ 2 g(η, p) − − + K f (η, p) + g(η, p) ∂η 4 ∂η 2 ∂η 2 ∂η 2  ∂f (η, p) ∂g(η, p)  ∂ 3 g(η, p)  + , −2 ∂η ∂η ∂η 3

(29)

N3 [f (η, p), g(η, p), φ(η, p)] =

  ∂φ(η, p) η ∂φ(η, p) ∂2φ n − Scγφ (η, p) − Scζ . + Sc f (η, p) + g(η, p) 2 ∂η ∂η 2 ∂η

(30)

The problems corresponding to the zeroth order are (1 − p)L1 [f (η, p) − f0 (η)] = pf N1 [f (η, p), g(η, p)],

(31)

(1 − p)L1 [g(η, p) − g0 (η)] = pg N2 [f (η, p), g(η, p)],

(32)

(1 − p)L2 [φ(η, p) − φ0 (η)] = pφ N3 [f (η, p), g(η, p), φ(η, p)],  f (0, p) = 0, f (0, p) = 1, g(0, p) = 0, g (0, p) = 1,

(33)

φ(0, p) = 1,



f (∞, p) = 0,

g (∞, p) = 0,

φ(∞, p) = 1.

(34)

Here,  is the auxiliary non-zero parameter, and p ∈ [0, 1] is an embedding parameter. For p = 0 and p = 1, one can define ⎧ ⎪ ⎨ f (η, 0) = f0 (η), g(η, 0) = g0 (η), ⎪ ⎩ φ(η, 0) = φ0 (η),

f (η, 1) = f (η), g(η, 1) = g(η),

(35)

φ(η, 1) = φ(η).

When p varies from 0 to 1, the initial guesses f0 (η), g0 (η), and φ0 (η) approach to f (η), g(η),

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T. HAYAT, M. MUSTAFA, and A. A. HENDI

and φ(η), respectively, and by the Taylor’s series expansion we have ⎧ ∞  ⎪ ⎪ ⎪ f (η, p) = f0 (η) + fm (η)pm , ⎪ ⎪ ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ∞ ⎨  g(η, p) = g0 (η) + gm (η)pm , ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ∞ ⎪  ⎪ ⎪ ⎪ φ(η, p) = φ (η) + φm (η)pm , ⎪ 0 ⎩

(36)

m=1

⎧ 1 ∂ m f (η, p)

⎪ ⎪ ⎪ fm (η) =

, ⎪ ⎪ m! ∂pm p=0 ⎪ ⎪ ⎨ 1 ∂ m g(η, p)

gm (η) =

, ⎪ m! ∂pm p=0 ⎪ ⎪ ⎪

m ⎪ ⎪ 1 ∂ φ(η, p) ⎪ ⎩ φm (η) =

. m! ∂pm p=0

(37)

The convergence of the series (36) strongly depends upon f , g , and φ . The values of f , g , and φ are chosen in such a way that the series (36) converges when p = 1. Hence, (36) becomes ⎧ ∞  ⎪ ⎪ ⎪ f (η) = f (η) + fm (η), ⎪ 0 ⎪ ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ∞ ⎨  g(η) = g0 (η) + gm (η), ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ∞ ⎪  ⎪ ⎪ ⎪ φ(η) = φ (η) + φm (η). ⎪ 0 ⎩

(38)

m=1

3.2 mth-order deformation problems The mathematical problems at this order are   L1 fm (η, p) − χm fm−1 (η) = f R1,m (η),   L1 gm (η, p) − χm gm−1 (η) = g R2,m (η),   L2 φm (η, p) − χm φm−1 (η) = φ R3,m (η),    fm (0) = fm (∞) = fm (∞) = gm (0)  gm (∞) = φm (0) = φm (∞) = 0,

fm (0) = = 0, m  1, χm = 1, m > 1,

(39) (40) (41) =

 gm (∞)

(42) (43)

 η      fm−1 − Kζ fm−1 + fm−1 + 2fm−1 2 2 m−1 

 (fm−1−k + gm−1−k )fk − fm−1−k fk + K (fm−1−k + gm−1−k )fkiv +

 −ζ R1,m (η) = fm−1

η

k=0

     + (fm−1−k − gm−1−k )fk − 2(fm−1−k + gm−1−k )fk ,

(44)

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid  R2,m (η) = gm−1 −ζ

+

m−1 



η

2



  gm−1 − Kζ + gm−1

η

2

  gm−1 + 2f gm−1



173

 (fm−1−k + gm−1−k )gk − gm−1−k gk + K (fm−1−k + gm−1−k )gkiv

k=0

     + (fm−1−k − gm−1−k )gk − 2(fm−1−k + gm−1−k )gk .

(45)

For n = 1, one obtains η R3,m (η) = φm−1 − Scγφm−1 − Scζ φm−1 2 m−1  + Scφm−1−k (fm−1−k + gm−1−k ).

(46)

k=0

For n = 2, we have m−1  η R3,m (η) = φm−1 − Scζ φm−1 − Scγ φm−1−k φk 2 k=0

+

m−1 

Scφm−1−k (fm−1−k + gm−1−k ).

(47)

k=0

For n = 3, we can write m−1 k   η R3,m (η) = φm−1 − Scζ φm−1 − Scγ φm−1−k φk−l φl 2 k=0

+

m−1 

Scφm−1−k (fm−1−k + gm−1−k ).

l=0

(48)

k=0

4

Convergence of the series solutions

The series given by (38) are the solutions of the present flow if one ensures the convergence of these series. The convergence region and rate of approximation of these solutions strongly depend upon f , g , and φ . For the admissible values of auxilliary parameters, the -curves are displayed for 15th-order of approximation. Figure 2 indicates that respective admissible values of f , g , and φ are −1.6  f  −0.5, −1.4  g  −0.5, −1.5  φ  −0.75, respectively, when K = 0.1, ζ = 0.1, Sc = γ = 1.0, and c = 0.5. Figure 3 plots the -curves, for the residual error of the functions f and g. We have seen that through these ranges of -curves we obtain the correct results upto eight decimal places. The performed computations indicate that the solution series (38) converge in the whole region of η when f = g = −0.9 and φ = −1.2.

5

Results and discussion

This section examines the effect of pertinent parameters on the velocity profiles f  , g  , and the concentration field φ. A comparative study between the present HAM solution and the previously obtained numerical solution for the steady case is made. Figure 4 indicates the effect of viscoelastic parameter K on f  and g  . Obviously, the velocity fields and the boundary layer thickness decrease when K increases. The influence of stretching ratio c on the velocity profiles is displayed in Fig. 5. It is observed that the x-component of the velocity and boundary layer thickness decrease. However, y-component of the velocity and the boundary layer thickness are increasing functions of c. Figure 6 indicates the influence of unsteadiness parameter on velocity profiles f  and g  . There is a noticeable decrease in f  and g  when the unsteadiness parameter

174

T. HAYAT, M. MUSTAFA, and A. A. HENDI

Fig. 2

-curves for the functions f , g, and φ at the 15th-order of approximations

Fig. 3

Residual errors for the functions f and g at η = 0 when K = ζ = 0.1

Fig. 4

Influence of K on f  and g 

ζ increases. The influences of parameters K and c on the component w which is represented by f and g are observed in Fig. 7. There is a decrease in the z-component of velocity for large values of K. However, the z-component of velocity is an increasing function of c. Figs. 8–10 plot the variations of the unsteadiness parameter ζ, the stretching ratio c, the Schmidt number Sc and the chemical reaction parameter γ on the concentration field φ. The influence of c and ζ on φ is shown in Fig. 8. We observe that φ decreases when c increases. However, φ is an increasing function of ζ. The effect of the Schimdt number Sc on the concentration field is discussed in Fig. 9(a). For the same thermal diffusivity, as Sc gradually increases, this corresponds to the weaker molecular diffusivity and thinner boundary layer thickness. φ decreases when the magnitude of generative chemical reaction parameter γ (> 0) increases which is obvious from Fig. 9(b). However, it increases for large values of the destructive chemical reaction parameter γ (> 0) (see Fig. 10(a)). Figure. 10(b) displays the effects of the chemical reaction parameter

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid

γ on the gradient of mass transfer φ (η).

Fig. 5

Influence of c on f  and g 

Fig. 6

Influence of ζ on f  and g 

Fig. 7

Fig. 8

Influence of K and c on f + g

Influence of c and ζ on φ

175

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T. HAYAT, M. MUSTAFA, and A. A. HENDI

Fig. 9

Fig. 10

Influence of Sc and γ (> 0) on φ

Influence of γ (< 0) on φ and φ (η)

Table 1 is presented for the analysis of series solution. It is obvious from the table that convergence for f and g is achieved at 15th-order of approximations while such convergence for φ is obtained later, i.e., at 25th-order of approximations. Table 2 is prepared to see the comparison of values of skin friction coefficients for the steady state case. The variation of surface mass transfer for different values of embedding parameters are computed in Table 3. It is observed that surface mass transfer is a decreasing function of K and ζ. Furthermore, the magnitude of φ (0) is greater for large values of Sc and γ. Table 1

Convergence of the HAM solutions for different orders of approximations when K = 0.1, ζ = 1.0, Sc = γ = 1, and c = 0.5

Orders of approximations

−f  (0)

−g  (0)

−φ (0)

1 5 10 15 20

1.237 917 1.272 742 1.272 797 1.272 796 1.272 796

0.552 291 0.566 459 0.566 604 0.566 603 0.566 603

1.070 833 1.057 425 1.056 421 1.056 340 1.056 319

Table 2

−K

0.0 0.1 0.2 0.4 0.6 0.8

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

Orders of approximations

−f  (0)

−g  (0)

−φ (0)

25 30 35 40

1.272 796 1.272 796 1.272 796 1.272 796

0.566 603 0.566 603 0.566 603 0.566 603

1.056 310 1.056 310 1.056 310 1.056 310

Comparison of values of f  (0) and g  (0) with those of Nazar and Latip state case (ζ = 0) −f  (0), c = 0.5

−g  (0), c = 0.5

[17]

for the steady

−f  (0), c = 0.0

HAM

Numerical[17]

HAM

Numerical[17]

HAM

Numerical[17]

1.093 095 1.000 699 0.929 309 0.823 869 0.748 149 0.690 238

1.093 5 1.001 2 0.930 6 0.827 8 0.755 9 0.702 3

0.465 204 0.418 367 0.348 360 0.336 581 0.303 570 0.278 862

0.465 6 0.433 7 0.407 6 0.367 8 0.338 9 0.317 1

1.000 000 0.953 463 0.912 870 0.845 154 0.790 569 0.745 356

1.001 3 0.955 7 0.916 3 0.851 5 0.800 6 0.759 7

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid Table 3

177

Values of surface mass transfer −φ (0) for different values of K, ζ, Sc, and γ

K

ζ

Sc

γ

−φ (0)

0.0 0.1 0.2 0.3 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1 0.0 0.1 0.2 0.3

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.072 76 1.056 31 1.032 25 0.968 92 1.078 85 1.056 31 1.032 13 1.012 44

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

K

ζ

Sc

γ

−φ (0)

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

1.2 1.4 1.6 2.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.2 1.4 1.6

1.171 23 1.277 52 1.376 85 1.559 08 1.056 31 1.118 31 1.177 03 1.232 93

Acknowledgements We are thankful to the reviewer for his/her valuable comments and suggestions regarding an earlier version of this paper. The first author as a visiting professor is grateful to King Saud University for the support (No. Ksu-Vpp-103).

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