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Open Journal of Fluid Dynamics, 2017, 7, 152-168

http://www.scirp.org/journal/ojfd

ISSN Online: 2165-3860

ISSN Print: 2165-3852

MHD Free Convection Flow past an Inclined

Stretching Sheet with Considering Viscous

Dissipation and Radiation

Maruf Hasan

, Enamul Karim

, Abdus Samad

Department of Mathematics, Comilla University, Comilla, Bangladesh

Department of Applied Mathematics, University of Dhaka, Dhaka, Bangladesh

Email: marufek@yahoo.com

Abstract

The present study concentrates on the analysis of MHD free convection flow

past an inclined stretching sheet. The viscous dissipation and radiation effects

are assumed in the he

at equation. Approximation solutions have been derived

for velocity, temperature, concentration, Nusselt number, skin friction and

Sherwood number using Nachtsheim-

Swigert shooting iteration technique

along with the six-order Runge-Kutta iteration scheme.

Graphs are plotted to

find out the characteristics of different physical parameters. The variations of

physical parameters on skin friction coefficient, Nusselt number and She

wood number are displayed via table.

Keywords

Viscous Dissipation, Magnetic Field, Skin Friction, Eckert Number

1. Introduction

In recent years, considerable interest has been shown in investigating radiation

interaction with natural convection flow commonly known as free convection

for heat transfer in fluid. This is due to the significant role of thermal radiation

in the surface heat transfer when convection heat transfer is small particularly in

free convection problems. Again the boundary layer flow on continuous surfaces

is an important type of flow which occurs in a number of technical processes.

Examples are paper production, crystal growing and glass blowing, aerodynam-

ics extrusion of plastic sheets and fibers. Thus, the study of heat transfer has be-

come important industrially for determining the quality of the final product.

Laminar natural convection flow and heat transfer of fluid with and without heat

source in channels with constant wall temperature have been extensively studied

How to cite this paper:

Hasan, M., Karim

. and Samad, A. (2017) MHD Free Con-

vection Flow past an Inclined Stretching

Sheet with Considering Viscous Dissipation

and Radiation

Open Journal of Fluid D

namics

, 152-168.

https://doi.org/10.4236/ojfd.2017.72010

Received:

April 5, 2017

Accepted:

May 8, 2017

Published:

May 11, 2017

7 by authors and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

DOI: 10.4236/ojfd.2017.72010 May 11, 2017

M. Hasan et al.

by Ostrach [1]. Hossain and Takhar [2] analyzed the effect of radiation effects on

free convection flow of a gas past a semi infinite flat plate using the Cog-

ley-Vincnitine-Giles equilibrium model. Ali

et al.

[3] studied the radiation effect

on natural convection flow over a vertical surface in a gray gas. Following Ali

al.

, Mansour [4] studied the interaction of mixed convection with thermal radia-

tion in laminar boundary layer flow over a horizontal, continuous moving sheet

with suction/injection. Alabraba

et al.

[5] studied the same problem considering

magnetic effect taking into account the chemical reaction and soret-doufour ef-

fects. Sattar and Kalim [6] made a study of the combined unsteady free convec-

tion dynamic boundary layer and thermal radiation boundary layer on a

semi-infinite vertical plate by using the Rosseland diffusion approximation.

Chen [7] investigated natural convection flow over a permeable surface with va-

riable wall temperature and concentration.

The viscous dissipation effect plays an important role in natural convection.

Natural convection flow is often encountered in the cooling of nuclear reactors.

Viscous dissipation effects on non-linear MHD flow in a porous medium over a

stretching porous surface have been studied by S.P. Anjali Devi and B. Ganga

[8]. Jha and Ajibade [9] studied the effect of viscous dissipation on natural con-

vection flow between vertical parallel plates with time-periodic boundary condi-

tions. Ferdows

et al.

[10] described that in the presence of uniform magnetic

field with viscous dissipation at the wall, the thermophoretic parameter is one of

the most useful parameters to control the boundary layer of the fluid.

The analysis and discussion of natural convection flow, the viscous dissipation

effect is generally ignored but here considered the combined effect of viscous

dissipation and radiation on free convection flow an inclined stretching sheet.

2. Mathematical Analysis

A steady-state two-dimensional heat and mass transfer flow of an electrically

conducting viscous incompressible fluid along an isothermal stretching permea-

ble inclined sheet with an angle

to the vertical embedded in a porous me-

dium with heat generation/absorption is considered. A strong magnetic field is

applied in the

axis direction. Here the effect of the induced magnetic field is

neglected in comparison to the applied magnetic field. The electrical current

flowing in the fluid gives rise to an induced magnetic field if the fluid were an

electrical insulator, but here we have taken the fluid to be electrically conducting.

Hence, only the applied magnetic field of strength

plays a role which gives rise

to magnetic forces

-direction,

where is the electrical con-

ductivity assumed to be directly proportional to the

- translational velocity (

)

of the fluid found by Helmy [11] and

is the density of the fluid. Two equal

and opposite forces are introduced along the

- axis so that the sheet is stretched

keeping the origin fixed as shown in

Figure 1.

The fluid is considered to be gray, absorbing emitting radiation but non-scat-

tering medium and the Rosseland approximation is used to describe the radia-

153

M. Hasan et al.

perature within the boundary layer and in the free-stream respectively, while

is the concentration of the fluid within the boundary layer,

is the electric

conductivity,

is the uniform magnetic field strength (magnetic induction),

is the density of the fluid,

is the thermal conductivity of the fluid,

the specific heat at constant pressure,

is the volumetric rate of heat genera-

tion/absorption and

is the chemical molecular diffusivity.

The corresponding boundary conditions are

( )

, , , at 0

0, , at

w ww

u Dx v v x T T C C y

u T TC C y

∞∞

= = = = =

= = = →∞











(5)

where

( )

0D >

is a constant,

( )

is a velocity component at the wall having

positive value to indicate suction,

is the uniform sheet temperature and

is the concentration of the fluid at the sheet.

By using Rosseland approximation,

takes the form

∂

= −

∂

(6)

where

is the Stefan-Boltzmann constant and

is the mean absorption

coefficient. It is assumed that the temperature difference within the flow are suf-

ficiently small such that

may be expressed as a linear function of tempera-

ture.

Thus

43 3

T TT T

∞∞

≅−

(7)

Using the Equations (6) and (7) in Equation (3), we get

( )

pp p p

TT T u T

u v TT

x y c c cy c

κυ

ρ ρ ρκ

∞



∂∂ ∂ ∂ ∂

+ = + −+ +



∂∂ ∂

∂∂



(8)

2.1. Similarity Analysis

In order to obtain similarity solution for the problem under consideration, we

may take the following suitable similarity variables

( ) ( ) ( )

,,,

TT CC

y D xf

TT CC

η ψ υ η θη φη

∞∞

−−

= = = =

−−

(9)

where

is the stream function,

is the dimensionless distance normal to

the sheet,

is the dimensionless stream function,

is the dimensionless

fluid temperature and

is the dimensionless concentration.

Since

∂

and

∂

= −

∂

we have the velocity components from Equa-

tion (6) given by

( ) ( )

,u Dxf v D f

η υη

′

= = −

(10)

where prime denotes the derivative with respect to

Now introducing the similarity variables from Equation (9) and using Equa-

tion (10), Equations (2), (8) and (4) are reduced to the dimensionless equations

given by

155

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