Applied Mathematical Sciences, Vol. 5, 2011, no. 12, 557 - 571

Mass Transfer over Unsteady Stretching

Surface Embedded in Porous Medium in the

Presence of Variable Chemical Reaction and

Suction/Injection

Elsayed M.A. Elbashbeshy¹ , T.G. Emam², M.S. Abdel-wahed³

¹Mathematics Department, Faculty of Science Ain Shams University, Abbassia, Cairo, Egypt

²Mathematics Department, The German University in Cairo-GUC

New Cairo City, Cairo, Egypt tarek.emam@guc.edu.eg

³Engineering Mathematics and Physics Department

Faculty of Engineering, Benha University, Cairo, Egypt

Abstract Problem statement: The effects of suction/injection and variable chemical reaction on mass transfer characteristics over unsteady stretching surface embedded in porous medium are studied. Approach: The governing time dependent boundary layer equations are transformed to ordinary differential equations containing chemical reaction parameter, unsteadiness parameter, and Schmidt number. These equations are solved through applying symbolic computation to Runge-Kutta technique in order to solve a two point boundary value problem. Results: The velocity and the Concentration profiles are computed and discussed in details for various values of the different parameters. Comparison of the obtained numerical results is made with previously published results. Conclusion: it found that, the velocity as well as concentration decreases with increasing the permeability parameter (λ), suction/injection parameter (fw), and unsteadiness parameter (A). and the concentration decreases with increasing the chemical reaction parameter (L) while the velocity has negligible change specially at low rate of chemical reaction .

Keywords: stretching surface, chemical reaction, porous medium, suction / injection.

558 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed Introduction The study of mass transfer over the stretching surface is useful for improving a number of chemical technologies such as Aerodynamic extrusion of plastic sheets, polymer production, cooling of nuclear reactors, and manufacturing of ceramic etc. Apelblat [1] studied analytical solution for mass transfer with chemical reaction of the first order. Das, et al. [2] have studied the effect of chemical reaction on the flow past an impulsively started infinite vertical plate with uniform heat flux. Muthucumaraswamy et al. [3] have studied the effect of homogenous chemical reaction of first order and free convection on the oscillating infinite vertical plate with variable temperature and mass diffusion. Anjalidevi et al. [4] have examined the effect of chemical reaction on the flow in the presence of heat transfer and magnetic field. Muthucumaraswamy et al. [5] have investigated the effect of thermal radiation effects on flow past an impulsively started infinite isothermal vertical plate in the presence of first order chemical reaction. Al-Odat et al. [6] have analyzed the effects of magnetic field and the chemical reaction on the velocity, temperature and concentration profiles as well as the local heat and mass transfer rates are presented. Muthucumaraswamy [7] studied the effects of suction on heat and mass transfer along a moving vertical surface in the presence of chemical reaction. EL-Arabawy [8] studied the effects of suction/injection and chemical reaction on mass transfer over a stretching surface. Rashed [9] studied the effect of radiation on the heat transfer from a stretching surface in a porous medium. Elbashbeshy et al. [10] have analyzed the effect of internal heat generation and suction or injection on the heat transfer in a porous medium over a stretching surface. Sultana, et al. [11] have analyzed the effects of internal heat generation, radiation and suction or injection on the heat transfer in a porous medium over a stretching surface. Rajeswari, et al. [12] have studied the effect of chemical reaction, heat and mass transfer on nonlinear MHD boundary layer flow through vertical porous surface with heat source in the presence of suction. Mahmoud [13] studied the effect of slip velocity at the wall on flow and mass transfer of an electrically conduction visco-elastic fluid past a stretching sheet embedded in a porous medium in the presence of chemical reaction and concentration dependent viscosity. Kandasamy, et al. [14] have studied the effect of variable viscosity, heat and mass transfer on nonlinear mixed convection flow over a porous wedge with chemical reaction in the presence of heat radiation. The purpose of this investigation is to study the effects of variable chemical reaction and suction/injection on mass transfer over unsteady stretching surface embedded in porous medium. Formulation of the problem

Mass transfer over unsteady stretching surface 559 Consider a unsteady, laminar, incompressible and viscous fluid through a porous medium of permeability k' over stretching surface with variable chemical reaction and suction/injection. The fluid properties are assumed to be constant in the a limited temperature range. The concentration of diffusing species is very small in comparison to the other chemical species, the concentration of species far from the surface, C∞ is infinitesimally very small [15].The chemical reactions are taking place in the flow and all physical properties are assumed to be constant. The x-axis runs along the continuous surface in the direction of the motion and the y-axis is perpendicular to it. The conservation equations for the unsteady flow are

(1)

(2)

(3)

Subjected to The Boundary Conditions

(4)

Where u and v are velocity components in the x and y directions, respectively , t is the time, υ is the viscosity , k' is the permeability of the porous medium , C is the concentration of the fluid, D is the effective diffusion coefficient. We assume that the stretching velocity Uw (x,t) , the fluid Concentration Cw (x,t) , and the chemical reaction coefficient K1 are of the form

(5)

Where a, b, and c are constants with a > 0 , b ≥ 0 , c ≥ 0 (with ct < 1), and k is the chemical reaction rate . We now introduce the following dimensional functions f and θ , and the similarity variables η

(6)

CKyCD

yCv

xCu

tC

uky

uyuv

xuu

tu

yv

xu

12

2

2

2

'

0

−∂∂

=∂∂

+∂∂

+∂∂

−∂∂

=∂∂

+∂∂

+∂∂

=∂∂

+∂∂

υυ

∞→→→

====

∞ yatCCu

yatCCVvUu www

0

0,,

( ) ( )∞

∞

−−

===CC

CCfxUyx

U

ww

w ηθηυψυ

η ,,

ctaxtxUw −

=1

),(ct

bxCtxCw −+= ∞ 1

),(,xkK =1,

560 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed Where ψ(x,y,t) is a stream function defined as u = ∂ψ/∂y and v = - ∂ψ/∂x , which indentically satisfies the mass conservation equation (1) substituting into eqs. (2) and (3) we obtain

(7)

(8)

where (A = c/a ) is the unsteadiness parameter ,(Sc = υ/D) is the schmidt number, (λ = υ x /k' Uw) is the permeability parameter, and (L = k Sc /Uw) is the chemical reaction parameter. The boundary condition (4) become

(9) Not that fw > 0 corresponds to suction, fw < 0 corresponds to injection and fw = 0 impermeable surface Numerical Solutions and Results We first convert the two Equations (7),(8) into a system of linear equations of first order , by using

(10)

Subjected to The initial Conditions

(11)

( ) 0221 =′−′−′′+′′+′−′′′ ffffffAf λη

( ) ( ) 021 =−+′−′−′+′′ θθθηθθθ LASffS CC

( ) ( ) ( ) 10,10,0 ==′= θfff w

( ) ( ) ∞→→→′ ηηθη asf ,0,0

θθ ′==′′=′== 54321 ,,,, ggfgfgfg

( )

( ) ( ) 44251521

45

54

2321

2312

23

32

21

LgggggSggASggg

gggAgggg

gggg

CC +−−+=′=′

+++−=′

=′=′

η

λη

( ) ( ) ( ) ( ) ( ) ngmgggfg w ===== 0,0,10,10,0 53421

Mass transfer over unsteady stretching surface 561 where m and n are unknown to be determined as a part of the numerical solution, Symbolic computation is then applied to Rung-Kutta technique to solve the equations [17]. The unknown initial values m, n are considered as symbolic variables therefore they will appear in a system of algebraic equations, after the integration of the ordinary differential equations. Then this algebraic equation system can be solved for the unknown initial values and substituted into the solution. consequently, only one integration pass is enough to solve the problem instead of an iteration technique like shooting method. the computations have been carried out for a various values of the permeability parameter (λ), porosity parameter (fw), schmidt number (Sc), chemical reaction parameter (L) ,and the unsteadiness parameter (A). To validate the numerical method used in this study , the steady flow (A= 0) and impermeable medium (λ = 0) case was considered in table (1) and the results for -f''(0) are compared with the exact solution which reported in Magyari and Keller [16] Also we have compared our results for the case of steady flow (A= 0) and the impermeable surface (fw = 0) in table (2) with the exact solution which reported in Mostafa Mahmoud [13]. The quantitative comparison is shown in tables (1-2) are founded to be in a very good agreement Table (1) : Values of - f ''(0) for a various values of (fw) at A = 0, λ = 0

Table (2) : Values of - f ''(0) for a various values of (λ) at A= 0, fw = 0

fw Exact solution [16] present result λ Exact

solution [13] present result

-1.5 0.5000000 0.5000000 0.1 1.048808848 1.048808848 -1.0 0.6180339 0.6180341 0.3 1.140175425 1.140175425 0.0 1.0000000 1.0000000 0.5 1.224744871 1.224744871 1.0 1.6180339 1.6180339 0.7 1.303840481 1.303840481 1.5 2.0000000 1.9999999 1.0 1.414213562 1.414213562

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +−−=−= −

24

,)(21 fwfw

fwf e αα

ηαη

λδδ

ηδη

+==−− 1,)( 1 ef

�=0.1,0.3,0.5,0.7,1

Sc=0.024, A=0, L=0

fw = 0

fw= +1

fw= +1.5

0 1 2 3 4 5 6 7�

0.2

0.4

0.6

0.8

1

1.2f ����

�=0.1,0.3,0.5,0.7,1

Sc=0.024, A=0, L=0

fw = -1.5

fw= -1

0 1 2 3 4 5 6 7 8 9 10�

0.2

0.4

0.6

0.8

1

1.2f ����

562 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed Also the results for concentration gradient at the surface –θ'(0) are compared in tables (3-4) with the exact solution which reported in EL-Arabawy [8] at the steady state flow (A= 0). Table (3) : Values of -θ'(0) for a various values of (fw) at Sc = 0.7, A = 0 and L = 0

Table (4) : Values of -θ'(0) for a various values of (fw) at Sc = 0.7, A = 0 and L= 0.8

fw Exact solution [8]

present result fw Exact

solution [8] present result

-1.5 0.45312 0.45311 -1.5 0.83831 0.83830 -1.0 0.53880 0.53879 -1.0 0.95568 0.95567 0.0 0.79366 0.79366 0.0 1.24670 1.24670 1.0 1.20087 1.20087 1.0 1.62421 1.62420 1.5 1.45734 1.45733 1.5 1.84849 1.84849

Fig(1): velocity profiles for various values of suction parameter fw and the permeability parameter (λ). Fig(2): velocity profiles for various values of injection parameter fw and the permeability parameter (λ).

Sc = 0.024

Sc = 0.7Sc=7

A=0, L=0, �=0.1,0.3,0.5,0.7,1 fw=0fw=1fw=1.5

0 1 2 3 4 5 6 7 8 9 10�

0.2

0.4

0.6

0.8

1

1.2����

A=0,2,4,6,8

Sc=0.024, L=0, �=1

fw=0

fw= +1

fw= +1.5

0 1 2 3 4�

0.2

0.4

0.6

0.8

1

1.2f ����

A=0,2,4,6,8

Sc= 0.024, L=0, �=1

fw= -1.5

fw= -1

0 1 2 3 4 5 6 7�

0.2

0.4

0.6

0.8

1

1.2f ����

Mass transfer over unsteady stretching surface 563 Fig(3): velocity profiles for various values of suction parameter fw and the unsteadiness parameter (A). Fig(4): velocity profiles for various values of injection parameter fw and the unsteadiness parameter (A). Fig(5): concentration profiles for various values of suction parameter fw and the permeability parameter (λ).

fw=0, L=0, �=1, A=2,4,6

Sc=0.024Sc = 0.7

Sc= 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

fw=0, L=0.2, �=1, A=2,4,6

Sc= 0.024

Sc = 0.7Sc = 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

fw=0, L=0.8, �=1, A=2,4,6

Sc= 0.024

Sc= 0.7

Sc = 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

564 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed Fig(6): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc). Fig(7): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc). Fig(8): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

fw=0, L=1.2, �=1, A=2,4,6

Sc= 0.024

Sc= 0.7

Sc =7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

fw= +1.5, L=0, �=1, A=2,4,6

Sc = 0.024

Sc = 0.7

Sc = 7

0 1 2 3 4 5�

0.2

0.4

0.6

0.8

1

1.2����

fw= +1.5, L = 0.2, �=1, A =2,4,6

Sc = 0.024

Sc = 0.7

Sc = 7

0 1 2 3 4 5�

0.2

0.4

0.6

0.8

1

1.2����

Mass transfer over unsteady stretching surface 565 Fig(9): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc). Fig(10): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

Fig(11): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

fw= +1.5, L=0.8, �=1 , A =2,4,6

Sc = 0.024

Sc = 0.7Sc = 7

0 1 2 3 4 5�

0.2

0.4

0.6

0.8

1

1.2����

fw= +1.5, L=1.2, �=1, A=2,4,6

Sc = 0.024Sc = 0.7

Sc = 7

0 1 2 3 4 5�

0.2

0.4

0.6

0.8

1

1.2����

fw= -1.5, L=0, �=1, A=2,4,6

Sc = 0.024

Sc = 0.7

Sc = 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

566 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed Fig(12): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

Fig(13): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

Fig(14): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

fw= -1.5, L= 0.2, �=1, A=2,4,6

Sc= 0.024Sc = 0.7

Sc= 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

fw= -1.5, L= 0.8, �=1, A=2,4,6

Sc = 0.024

Sc = 0.7Sc = 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

fw= - 1.5, L=1.2, �=1, A=2,4,6

Sc= 0.024Sc = 0.7Sc = 7

0 1 2 3�

0.2

0.4

0.6

0.8

1

1.2����

Mass transfer over unsteady stretching surface 567 Fig(15): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc).

Fig(16): concentration profiles for various values of unsteadiness parameter (A) and schmidt number (Sc). Fig(17): concentration profiles for various values of unsteadiness parameter (A) and Schmidt number (Sc).

568 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed Discussions The influence of the mass transfer, permeability parameter (λ), chemical reaction parameter (L) and unsteadiness parameter (A) on the velocity gradient and dimensionless concentration are shown in fig. 1-17 with a various values of Schmidt number . Figure (1) shows the effect of suction parameter (fw) and permeability parameter (λ) at the steady state (A=0) and no chemical reaction (L=0) on the velocity profiles. We observe that the velocity gradient at the surface decrease with the increase of permeability parameter and suction parameter. Figure (2) shows the effect of injection parameter (fw) and permeability parameter (λ) at the steady state (A=0) and no chemical reaction (L=0) on the velocity profiles. We observe that the velocity gradient at the surface decrease with the increase of permeability parameter and injection parameter. Figures (3,4) show the effect of suction/injection parameter (fw) and unsteadiness parameter (A) at the case of no chemical reaction (L=0) on the velocity profiles. We observe that the velocity gradient at the surface decrease with the increase of unsteadiness parameter and suction/injection parameter. Figure (5) shows the effect of suction parameter (fw) and permeability parameter (λ) at the steady state (A=0) and no chemical reaction (L=0) on the concentration profiles at a various values of Schmidt number (Sc). We observe that the concentration decrease with the increase of permeability parameter, suction parameter, and Schmidt number. Also the effect of the permeability parameter on the concentration decreasing by increasing the suction parameter and schmidt number. Figures (6-9) show the effect of chemical reaction parameter (L) and unsteadiness parameter (A) on the concentration profiles at a various values of Schmidt number (Sc) at the case of impermeable surface (fw=0). We observe that the concentration decrease with the increase of unsteadiness parameter and Schmidt number. Also the effect of the unsteadiness parameter on the concentration decreasing by increasing of Schmidt number, and by increasing of chemical reaction parameter the effect of the unsteadiness parameter on the concentration decreasing. Also We observe that the concentration boundary layer thickness decreasing by increasing of unsteadiness parameter for each Schmidt number and this effect may be vanishes by increasing of Schmidt number. Also the concentration boundary layer thickness remains constant by increasing of chemical reaction parameter for each Schmidt number.

Mass transfer over unsteady stretching surface 569 Figures(10-13)show the effect of chemical reaction parameter (L) and unsteadiness parameter (A) on the concentration profiles at a various values of Schmidt number (Sc) at the case of suction (fw = +1.5). We observe that the concentration decrease with the increase of unsteadiness parameter and Schmidt number. Also the effect of the unsteadiness parameter on the concentration decreasing by increasing of Schmidt number, by increasing of chemical reaction parameter the effect of the unsteadiness parameter on the concentration decreasing. Also We observe that the concentration boundary layer thickness decreasing by increasing of unsteadiness parameter for each Schmidt number and this effect may be vanishes by increasing of Schmidt number. Also the concentration boundary layer thickness remains constant by increasing of chemical reaction parameter for each Schmidt number. Also the concentration boundary layer thickness in the case of impermeable surface (fw = 0) is thin than the concentration boundary layer in the case of suction (fw = + 1.5). Figures (14-17)show the effect of unsteadiness parameter (A) on the concentration profiles at various values of chemical reaction parameter (L) and Schmidt number (Sc) for injection parameter(fw=-1.5).We observethat the concentration decreasing with increasing unsteadiness parameter (A) and Schmidt number (Sc) for each chemical reaction parameter (L), but at (Sc =7) the concentration suddenly dropped to zero for each chemical reaction parameter (L). Conclusion Numerical solution has been obtained for the effects of suction/injection and chemical reaction on mass transfer characteristics over unsteady stretching surface. Numerical computations show that the present values of concentration surface gradient are in close agreement with those obtained by previous investigation in the absence of unsteadiness parameter, and variable chemical reaction. The following results are obtained :

• velocity gradient at the surface decreases with the increase of permeability parameter (λ), suction/injection parameter and unsteadiness parameter (A).

• the concentration decreases with the increase of permeability parameter (λ) ,suction/injection parameter.

• the concentration decreases with increasing of chemical reaction parameter (L), unsteadiness parameter (A) and Schmidt number .

• the concentration suddenly dropped to zero for a various values of chemical reaction parameter (L) and unsteadiness parameter (A) at Sc = 7.

• the effect of chemical reaction parameter (L) and unsteadiness parameter (A) on the concentration decreasing by increasing of Schmidt number.

570 E. M. A. Elbashbeshy, T. G. Emam, M. S. Abdel-wahed

• concentration gradient at the surface increases as Schmidt number (Sc) increases.

• concentration gradient at the surface is higher for suction compared to injection.

• increasing of Schmidt number (Sc) reduce the concentration boundary layer thickness.

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