﻿Similarity Solution of the Influence of the Thermal Radiation and Heat Transfer on Steady Compressible Boundary Layer Flow

Open Journal of Fluid Dynamics
Vol.3 No.2(2013), Article ID:33643,4 pages DOI:10.4236/ojfd.2013.32011

Similarity Solution of the Influence of the Thermal Radiation and Heat Transfer on Steady Compressible Boundary Layer Flow

Felix Ilesanmi Alao1, Samson Babatunde Folarin2

1Mathematical Science Department, Federal University of Technology, Akure, Nigeria

2King Fahd University of Petroleum and Mineral, Dhahran, Saudi Arabia

Email: sanmialao@yahoo.com, samson.bfolarin@yahoo.com

Copyright © 2013 Felix Ilesanmi Alao, Samson Babatunde Folarin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received January 25, 2013; revised February 28, 2013; accepted March 8, 2013

Keywords: Radiation; Heat Transfer; Stewartson Transformation; Roseland Approximation

ABSTRACT

This paper determines the influence of the radiation and heat transfer on the compressible boundary layer flow using similarity solutions approach. The Roseland approximation is used to describe the radiative heat flux in the energy equation and the compressible boundary layer equations are transformed using Stewartson transformation. Similarity (invariant) solutions for the governing partial differential equations system are constructed. The shooting method is employed to transform the resulting non-linear boundary value problem into initial value problem, which is solved numerically. The effects of various parameters on the velocity and temperature profiles as well as the Falkner skan exponent and Prandtl number are shown graphically.

1. Introduction

2. Governing Equations

We consider a steady, two-dimensional, laminar boundarylayer flow of viscous compressible fluid, given as,

(2.1)

(2.2)

(2.3)

Boundary conditions are:

at          (2.4)

at            (2.5)

where, is the constant wall temperature, are the Cartesian coordinates with x and y axes along and normal to the surface of the cylinder respectively, are the velocity components along x and y axes, p is the pressure, is the density, k is the thermal conductivity, is the specific heat at constant pressure, R is the gas constant and the suffix o, refers to some standard state, say.

3. Method of Solution

Stewartson transformation variables of the reduced Equations (2.1)-(2.3) is given as,

(3.1)

Then,

(3.2)

(3.3)

(3.4)

see Stewartson [5], with

(3.5)

By applying the stewartson transformation variables on Equations (2.1)-(2.6), we obtained

(3.6)

(3.7)

where is the stream function, a1 and a0 are velocities of sounding main stream.

The roseland approximation for radiation is given by

(see Mukhopadhyay [4]) is the absorption coefficient and is the Stefan-Boltzman constant. Assuming the temperature within the flow is such that, may be expanded in Taylor series about  (free stream temperature) and neglecting the higher orders terms, we have,

(3.8)

replacing by unity and using roseland approximation on Equations (3.6) and (3.7), gives

(3.9)

(3.10)

The non dimensionalized form of Equation (3.10) with the transformed boundary conditions usingbecomes

(3.11)

With the boundary conditions

at       (3.12)

at   (3.13)

The similarity variables defined below, are used to transformed Equations (3.9)-(3.13)

(3.14)

The transformations gives the below coupled non linear ordinary differential equations

(3.15)

(3.16)

because of the presence of temperature T, we define the function S relating to the absolute temperature, with Mach number as

(3.17)

where, (subsonic flow), simplifying Equation (3.17) and substituting it into Equations (3.15) and (3.16), we obtained

(3.18)

(3.19)

With the boundary conditions,

at    (3.20)

at          (3.21)

where and

finally we have Equations (3.18)-(3.21) as

(3.22)

(3.23)

With boundary conditions,

at  (3.24)

at          (3.25)

4. Numerical Solution

Shooting method was employed to transform Equations (3.25)-(3.28) into coupled initial value problems. The approximate solution is constructed using Runge-Kutta fourth order technique. Furthermore, the resulting higher order non-linear coupled differentials are decomposed into systems of first order differential equations given below

(4.1)

(4.2)

(4.3)

Also

(4.4)

(4.5)

5. Results

Discussion of Results

In order to illustrate the results, numerical values were plotted in Figures 1-3). In all cases, we considered the parameters, and. Others are and.

Figure 1 demonstrates the effect of radiative parameter on velocity field, with fixed Falkner skan exponent m, and Prandtl number Pr on both region of the boundary layer flow. It is discovered that, increase in radiation has insignificant effect on the velocity field of the boundary layer. Also, Figure 2 shows the effect of radiation on the velocity field, Radiation has insignificant effect on the fluid flow on the boundary layer.

Furthermore, Figure 3 shows the effects of the radia-

Figure 1. Variation of the velocity profile f(η) with η for several values of Q’s with fixed other parameters.

Figure 2. Variation of the velocity profile f’(η) with η for several values of Q’s with fixed other parameter.

Figure 3. Variation of the temperature profile G(η) with η for several values of Q’s with the other parameter fixed.

tive parameters Q on the temperature field in the presence of some fixed parameter. The temperature decreases as the thermal radiation Q increases. This is in agreement with the physical fact that the thermal boundary layer decreases with increasing Q.

6. Conclusion

The present study gives the similarity solution with the approximate solution using the shooting method to determine radiative the heat transfer on the boundary layer flow. Our results show that due to radiation, the rate of heat transfer increases. It is found that the effect of thermal radiation is insignificant in the fluid velocity. The flow separation can be controlled in the presence of lower radiation. The temperature of the boundary layer decreases with increasing thermal radiation.

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