International Journal of Scientific & Engineering Research, Volume 6, Issue 2, February-2015 1310

ISSN 2229-5518

Analytic Approach in Solving Steady Laminar

Flow of Fluid over a Stretching Sheet

Jaionto Karmokar1*, Sk. Abdulla-Al-Faisul2, Mst. Ayrin Aktar3

1Department of Applied Mathematics, Faculty of Science, University of Rajshahi, Rajshahi-6205, Bangladesh

2Lecturer. Dept. of Textile Eng. Southeast University, Dhaka, Bangladesh

3Department of Applied Mathematics, Faculty of Science, University of Rajshahi, Rajshahi-6205, Bangladesh

Abstract— W e have considered the steady laminar flow over a linearly stretching sheet subjected to an order of chemical reaction. A simi- larity transformation is utilized to convert the governing nonlinear partial differential equations into ordinary differential equations. The local skin friction, rate of heat transfer and rate of mass transfer on the wall may calculate with the help of boundary conditions.

Keywords— Laminar flow, Chemical reaction, Similarity transformation, Heat and mass transfer, Stretching sheet

—————————— ——————————

1 INTRODUCTION

The heat and mass transfer in laminar boundary layer flow over a linearly stretching sheet have important applications in many fields of engineering. In addition, the diffusing species can be generated or absorbed due to some kind of chemical reaction with the ambient fluid. Mahapatra and Gupta (2002) studied the heat transfer in stagnation-point flow towards a stretching sheet. On the other hand, Afify (2004) analyzed the MHD free convective flow and mass transfer over a stretching sheet with homogeneous chemical reaction. Further, Alam and Ahammad (2011) investigated the effects of variable chemical reaction and variable electric conductivity on free convective heat and mass transfer flow over an inclined stretching sheet with variable heat and mass fluxes under the influence of Dufour and Soret effects. Ferdows and Qasem Al-Mdallal (2012) studied effects of order of chemical reaction on a boundary layer flow with heat and mass transfer over a linearly stretching sheet. Hayata et al. (2012) studied the unsteady three dimensional flow of couple stress fluid over a stretching surface with chemical reaction based on using homotopy analysis method. Singh (2012) investigated the MHD Flow with Viscous Dissipation and Chemical Reaction over a Stretching Porous Plate in Porous Medium. Makinde and Sibanda (2012) investigated the effects of chemical re- action on boundary layer flow past a vertical stretching sur- face in the presence of internal heat generation.

We have considered steady, laminar flow of a fluid over a stretching sheet. Again we have considered the stretched with a velocity proportional to x axis as shown below. We have assumed that the fluid far away from the sheet is at rest and at

temperature 𝑇∞ and concentration 𝐶∞ . Further, the stretched sheet is kept at fixed temperature 𝑇𝑤 (< 𝑇∞ ) and concentra- tion 𝐶𝑤 (< 𝐶∞ ).

𝑇∞ 𝐶∞

Tw 𝐶𝑤

The equation governing the motion are :

We have concerned with two dimensional steady, laminar

flow of a fluid over a linearly stretching sheet. In this paper we

𝜕u

𝜕𝑥

+ 𝜕v = 0 (1)

𝜕𝑦

have investigated analytically the effects of chemical reaction

𝑢 𝜕u

+ 𝑣 𝜕u = 𝜐 𝜕 𝑢

(2)

on the steady laminar two dimensional boundary layer flow

𝜕𝑥

𝜕𝑦

𝜕𝑦2

2 2

and heat and mass transfer over a stretching sheet. The

𝑢 𝜕𝑇 + 𝑣 𝜕𝑇

= 𝛼 𝜕 𝑇

+ 𝜐

� 𝜕u �

(3)

method of solution is based on the well-known similarity

transformations.

𝜕𝑥

𝜕𝑦

𝜕𝑦2

𝐶𝑝

𝜕𝑦

IJSER © 2015 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 6, Issue 2, February-2015 1311

ISSN 2229-5518

𝑢 𝜕𝐶 + 𝑣 𝜕𝐶

= 𝐷 𝜕 𝐶 𝑛

𝜕𝑥

𝜕𝑦

𝜕𝑦2 − 𝑘1 (𝐶 − 𝐶∞ )

(4)

3 BOUNDARY CONDITIONSS

u = 𝑢0x, v = 0, T = Tw , C = 𝐶𝑤 at y = 0

u→ 0, T→ Tw , C→ 𝐶𝑤 , as y→ ∞ (5)

u → velocity component in the x direction

v → velocity component in the y direction

T → the fluid temperature in the boundary layer

C concentration of the fluid

kinematic viscosity thermal diffusivity

specific heat at constant pressure

constant of first-order chemical reaction rate

D effective diffusion coefficient similarity variable

f dimensionless stream function

θ dimensionless temperature

φ dimensionless concentration

ψ stream function

In order to solve Equations (1)-(5), we introduce the following similarity transformation:

Also

And

.

Putting these values in equation (2), we get

.

(11)

We define the stream function defined as

, (10)

Now we get

And

Again we get

Now

{ }

and

(12)

IJSER © 2015 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 6, Issue 2, February-2015 1312

ISSN 2229-5518

And

.

Now, putting these values in equation (3), we get

.

Putting these values in equation (4), we get

.

.

where,

.

= 0

(13)

where

.

And

.

(15)

And

.

Consequently, equations (2)-(4) and the boundary conditions

(5) can be written in the following form,

Again we get

(16)

(17)

()

(14)

subject to the boundary conditions

(18)

Now

0

where ,

represents Prandtl number,

represents Eckert number,

(19)

IJSER © 2015 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 6, Issue 2, February-2015 1313

ISSN 2229-5518

action parameter.

represents Schmidt number,

represents the chemical re-

tion-point flow towards a stretching sheet, Heat and Mass

Transfer, 38(6), 517-521, 2002.

[7] Singh P.K. and Singh J., MHD Flow with Viscous Dissipa- tion and Chemical Reaction over a Stretching Porous Plate in Porous Medium, International Journal of Engineering

From the equations (16), (17) and (18) we can find out the local

skin-friction coefficient, , rate of heat transfers, and rate of mass transfers, as

(20) (21) (22)

where, Re = is the Reynolds number.

We have considered steady laminar flow over a stretching sheet in the present of chemical reaction. By using some suitable transformations we have reduced the partial differential equation into ordinary differential equation. Final- ly we have derived a set of equation from which we can find local skin-friction coefficient, rate of heat transfer and rate of mass transfer in terms of Reynolds number.

[1] Afify A. A., MHD free convective flow and mass transfer over a stretching sheet with chemical reaction, Heat and Mass Transfer, 40(6-7), 495-500, 2004.

[2] Alam M. S., Ahammad. M.U.,Effects of variable chemical reaction and variable electric conductivity on free convec- tive heat and mass transfer flow along an inclined stretching sheet with variable heat and mass fluxes under the influence of Dufour and Soret effects, Nonlinear Anal- ysis: Modeling and Control, 16(1), 1-16, 2011.

[3] Ferdows M., Qasem M. Al-Mdallal, Effects of Order of Chemical Reaction on a Boundary Layer Flow with Heat and Mass Transfer over a Linearly Stretching Sheet, Amer- ican Journal of Fluid Dynamics, 2(6), 89-94, 2012.

[4] Hayata T., Awais M., Safdar A. and Hendi A. A., Unsteady three dimensional flow of couple stress fluid over a stretching surface with chemical reaction, Nonlinear Analysis: Modeling and Control, 17(1), 47–59, 2012.

[5] Makinde O.D. and Sibanda P., Effects of chemical reaction on boundary layer flow past a vertical stretching surface in the presence of internal heat generation, Int. J. of Nu- merical Fluid Flows, In press, 2012.

[6] Mahapatra T. R. and Gupta A. S., Heat transfer in stagna-

Research and Applications, 2 (2), 1556-1564, 2012.

[8] Kandasamy R., Hayat T., Obaidat S., Group theory trans- formation for Soret and Dufour effects on free convective heat and mass transfer with thermo phoresis and chemical reaction over a porous stretching surface in the presence of heat source/sink, Nuclear Engineering and Design, 241 (6), 2155–2161, 2011.

IJSER © 2015 http://www.ijser.org