Article Information

Authors:
Philip O. Olanrewaju^{1}
Jacob A. Gbadeyan^{1}
Tasawar Hayat^{2}
Awatif A. Hendi^{3}
Affiliations:
^{1}Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria
^{2}Department of Mathematics, QuaidIAzam University, Islamabad, Pakistan
^{3}Department of Physics, Faculty of Science, King Saud University, Riyadh,
Saudi Arabia
Correspondence to:
Philip Olanrewaju
Email:
oladapo_anu@yahoo.ie
Postal address:
Department of Mathematics, Covenant University, Ota, Ogun State, PMB 1023, Nigeria
Dates:
Received: 14 Oct. 2010
Accepted: 03 May 2011
Published: 07 Sept. 2011
How to cite this article:
Olanrewaju PO, Gbadeyan JA, Hayat T, Hendi AA. Effects of internal heat generation, thermal radiation and buoyancy force on a boundary
layer over a vertical plate with a convective surface boundary condition. S Afr J Sci. 2011;107(9/10), Art. #476, 6 pages.
doi:10.4102/sajs.v107i9/10.476
Copyright Notice:
© 2011. The Authors. Licensee: AOSIS OpenJournals. This work is licensed under the Creative Commons Attribution License.
ISSN: 00382353 (print)
ISSN: 19967489 (online)



Effects of internal heat generation, thermal radiation and buoyancy force on a boundary layer over a vertical plate with a convective surface boundary condition

In This Research Letter...

Open Access

• Abstract
• Introduction
• Mathematical formulation
• Results and discussion
• Conclusions
• Acknowledgement
• References

In this paper we analyse the effects of internal heat generation, thermal radiation and buoyancy force on the laminar boundary layer about a vertical
plate in a uniform stream of fluid under a convective surface boundary condition. In the analysis, we assumed that the left surface of the plate is in
contact with a hot fluid whilst a stream of cold fluid flows steadily over the right surface; the heat source decays exponentially outwards from the
surface of the plate. The similarity variable method was applied to the steady state governing nonlinear partial differential equations, which were
transformed into a set of coupled nonlinear ordinary differential equations and were solved numerically by applying a shooting iteration technique
together with a sixthorder Runge–Kutta integration scheme for better accuracy. The effects of the Prandtl number, the local Biot number, the
internal heat generation parameter, thermal radiation and the local Grashof number on the velocity and temperature profiles are illustrated and interpreted
in physical terms. A comparison with previously published results on similar special cases showed excellent agreement.
Boundarylayer flows over a moving or stretching plate are of great importance in view of their relevance to a wide variety of technical applications,
particularly in the manufacture of fibres in glass and polymer industries. The first and foremost work regarding boundarylayer behaviour in moving surfaces
in a quiescent fluid was performed by Sakiadis.^{1}
Subsequently, many researchers^{2,3,4,5,6,7,8,9 }worked on the problem of moving or stretching plates under different situations. In the boundarylayer
theory, similarity solutions were found to be useful in the interpretation of certain fluid motions at large Reynolds numbers. Similarity solutions often exist
for the flow over semiinfinite plates and stagnation point flow for twodimensional, axisymmetrical and threedimensional bodies. In special cases, when there
is no similarity solution, one has to solve a system of nonlinear partial differential equations. For similarity boundarylayer flows, velocity profiles are
similar. But this kind of similarity is lost for nonsimilarity flows.^{10,11,12,13,14} Obviously, the nonsimilarity boundarylayer flows are more general
in nature and are more important, not only in theory but also in application.
The heattransfer analysis of boundarylayer flows with radiation is also important in electrical power generation, astrophysical flows, solar power
technology, space vehicle reentry and other industrial areas. Extensive literature that deals with flows in the presence of radiation effects is now
available. Raptis et al.^{15} studied the effect of thermal radiation on the magnetohydrodynamic flow of a viscous fluid past a semiinfinite
stationary plate. Hayat et al.^{16} extended the analysis of reference^{15}
for a secondgrade fluid.
Convective heat transfer studies are very important in processes involving high temperatures, such as gas turbines, nuclear plants and thermal energy storage.
Recently, Ishak^{17} examined the similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition.
Moreover, Aziz^{18,19} studied a similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary
condition and also studied hydrodynamic and thermal slip flow boundary layers over a flat plate with a constant heat flux boundary condition. Very recently,
Makinde and Olanrewaju^{20} investigated the buoyancy effects on a thermal boundary layer over a vertical plate with a convective surface boundary condition.
In this study, the recent work of Ishak^{17}, Aziz^{18} and Makinde and Olanrewaju^{20} was extended to include the effect of thermal
radiation and internal heat generation. The numerical solutions of the resulting momentum and the thermal similarity equations are reported for representative
values of the thermophysical parameters embedded in the fluidconvection process. The objective of this paper was to explore the effects of thermal radiation
and internal heat generation on the fluid under a convective surface boundary condition. The nonlinear equations governing the flow were solved numerically
using a shooting technique together with a sixthorder Runge–Kutta integration scheme, which gives better accuracy than a fourthorder Runge–Kutta
method. Graphical results are first reported for emerging parameters and then discussed.
We considered a twodimensional steady incompressible fluid flow coupled with heat transfer by convection over a vertical plate. A stream of cold
fluid at temperature T_{∞} moved over the right surface of the plate with a uniform velocity U_{∞ }whilst
the left surface of the plate was heated by convection from a hot fluid at temperature T_{f} , which provided a heat transfer
coefficient h_{f }. The density variation as a result of buoyancy force effects was taken into account in the momentum equation
and the thermal radiation and the internal heat generation effects were taken into account in the energy equation (the Boussinesq approximation). The
continuity, momentum and energy equations describing the flow are, respectively:
where u and v are the x (along the plate) and the y (normal to the plate) components of the velocities, respectively.
T is the temperature, υ is the kinematics viscosity of the fluid, α is the thermal diffusivity of the fluid, β
is the thermal expansion coefficient, Q is the heat released per unit per mass, g is the gravitational acceleration, q_{r}
is the radiative heat flux and k is the thermal conductivity. The velocity boundary conditions can be expressed as
and
The boundary conditions at the plate surface and far into the cold fluid may be written as
and
The radiative heat flux q_{r} is described by the Rosseland approximation such that
where σ* and K are the StefanBoltzmann constant and the mean absorption coefficient, respectively. Following Chamkha^{21},
we assume that the temperature differences within the flow are sufficiently small such that T ^{4} can be expressed as a linear function
after using the Taylor series to expand T ^{4} about the free stream temperature T_{∞} and neglecting higherorder terms.
This result is the following approximation:
Using [Eqn 8] and [Eqn 9] in [Eqn 3], we obtain
We then introduce a similarity variable η and a dimensionless stream function f(η) and temperature θ(η) as
where the prime symbol denotes differentiation with respect to η and Re_{x} = U_{∞}x/
υ is the local Reynolds number. [Eqn 1] to [Eqn 7] reduce to:
and
where
Bi_{x} is the local Biot number, Pr is the Prandtl number, Gr_{x} is the local Grashof number, Ra
is the radiation parameter and λ_{x} is the internal heat generation parameter. For the momentum and energy equations to have a
similarity solution, the parameters Gr_{x}, λ_{x} and Bi_{x}_{ }must
be constants and not functions of x as in [Eqn 16]. This condition can be met if the heattransfer coefficient h_{f} is
proportional to x^{–½} , the thermal expansion coefficient β is proportional to x^{–1} and the
heatrelease coefficient Q is proportional to x^{–1}. We therefore assume
where c, d and m are constants. Substituting [Eqn 17] into [Eqn 16], we get
Bi, λ and Gr defined by [Eqn 18] are the Biot number, the internal heat generation parameter and the Grashof number,
respectively. The solutions of [Eqn 12] to [Eqn 15] yield the similarity solutions. However, the solutions generated are the local similarity solutions
whenever Bi_{x}, λ_{x} and Gr_{x} are defined as in [Eqn 13].
The ordinary differential equations ([Eqn 9] and [Eqn 10]) subject to the boundary conditions ([Eqn 11] and [Eqn 12]) were solved numerically using the
symbolic algebra software Maple.^{22} Table 1 presents a comparison of the values of –θ' (0) and θ(0) with those
reported by Aziz^{18}, Ishak^{17} and Makinde and Olanrewaju^{20}, which show an excellent agreement for Pr = 0.72. Table 2
shows the values of the skinfriction coefficient f ''(0) and the local Nusselt number –θ'(0), for various values of embedded
parameters. From Table 2, it can be seen that the skin friction and the rate of heat transfer at the plate surface increased with an increase in the local
Grashof number, the convective surface heat transfer parameter, the internal heat generation parameter and the radiation absorption parameter. However, an
increase in the fluid Prandtl number decreased the skin friction but increased the rate of heat transfer at the plate surface. Figures 1–6 depict the
fluid velocity profiles. Generally, the fluid velocity is zero at the plate surface and increases gradually away from the plate towards the free stream value
satisfying the boundary conditions. It is clearly seen from Figure 1 that the Grashof number had profuse effects on the velocity boundary layer thickness.
It is interesting to note that an increase in the intensity of the convective surface heat transfer (Bi_{x}) produced a slight increase
in the fluid velocity within the boundary layer (Figure 2). The local internal heat generation parameter, the Prandtl number and the local Biot number had
little or no influence on the velocity profiles (Figures 3–4 and Figure 6), which could be justified by [Eqn 12], which had a small value for the Grashof
number. When Ra = 0.1, the Prandtl number effected the velocity profile (Figure 5) and the negative part could have been caused by the value of the
Grashof number used (see [Eqn 12]) and the value of the radiation parameter used (see [Eqn 13]). Figures 7–12 illustrate the fluid temperature profiles
within the boundary layer. The fluid temperature was at a maximum at the plate surface and decreased exponentially to zero away from the plate, thus satisfying
the boundary conditions. From these figures, it is noteworthy that the thermal boundary layer thickness increased with an increase in Bi_{x},
λ_{x} and Ra and decreased with increasing values of Gr_{x} and Pr. Hence, the convective surface
heat transfer, the internal heat generation parameter and the radiation parameter enhanced thermal diffusion whilst an increase in the Prandtl number and the intensity
of buoyancy force slowed down the rate of thermal diffusion within the boundary layer. Figure 13 shows the influence of Prandtl numbers on the thermal boundary
layer, as obtained by Aziz^{18}.
TABLE 1: A comparison of the values obtained for the Nusselt number
[θ'(0)] and surface temperature [θ(0)] with an increase in the local Biot number (Bi_{x})
in this study and by Aziz^{18}, Ishak^{17} and Makinde and Olanrewaju^{20} in previous studies.

TABLE 2: A comparison of the values of the skinfriction coefficient
[ f "(0)], temperature at the wall surface [θ(0)] and the Nusselt number
[θ'(0)] for different parameter values embedded in the flow model.


FIGURE 1: The fluid velocity profile with increasing distance from the plate
surface (where f ’(η) = 0) and increasing Grashof number (Gr_{x}).



FIGURE 2: The fluid velocity profile with increasing distance from the plate
surface (where f ’(η) = 0) and increasing Biot number (Bi_{x}).



FIGURE 3: The fluid velocity profile with increasing distance from the plate
surface (where f ’(η) = 0) and increasing internal heat generation (λ_{x}).



FIGURE 4: The fluid velocity profile with increasing distance from the plate
surface (where f ’(η) = 0) and increasing Prandtl number (Pr) when the radiation parameter, Ra = 0.5.



FIGURE 5: The fluid velocity profile with increasing distance from the plate surface
(where f ’(η) = 0) and increasing Prandtl number (Pr) when the radiation parameter, Ra = 0.1.



FIGURE 6: The fluid velocity profile with increasing distance from the plate
surface (where f ’(η) = 0) and increasing radiation (Ra).



FIGURE 7: The fluid temperature profile with increasing distance from the plate
surface (where θ(η) = maximum) and increasing Grashof number (Gr_{x}).



FIGURE 8: The fluid temperature profile with increasing distance from the plate surface
(where θ(η) = maximum) and increasing Biot number (Bi_{x}).



FIGURE 9: The fluid temperature profile with increasing distance from the plate surface
(where θ(η) = maximum) and increasing internal heat generation (λ_{x}).



FIGURE 10: The fluid temperature profile with increasing distance from the plate
surface (where θ(η) = maximum) and increasing Prandtl number (Pr) when the radiation parameter, Ra = 0.5.



FIGURE 11: The fluid temperature profile with increasing distance from the plate
surface (where θ(η) = maximum) and increasing Prandtl number (Pr) when the radiation parameter, Ra = 0.1.



FIGURE 12: The fluid temperature profile with increasing distance from the plate
surface (where θ(η) = maximum) and increasing radiation (Ra).



FIGURE 13: The fluid temperature profile obtained by Aziz^{18}
with increasing Prandtl number (Pr).


We analysed the effects of internal heat generation, thermal radiation and buoyancy force on the laminar boundary layer about a vertical plate in a uniform
stream of fluid under a convective surface boundary. A similarity solution for the momentum and the thermal boundary layer equations is possible if the
convective heat transfer of the fluid heating the plate on its left surface is proportional to x^{–½} and if the thermal expansion
coefficient β and the heat released per unit per mass Q are proportional to x^{–1}. Numerical solutions of the
similarity equations were reported for the various parameters embedded in the problem. The combined effect of increasing the Prandtl number and the Grashof
number tended to reduce the thermal boundary layer thickness along the plate whilst the effects of increasing the Biot number, the internal heat generation
parameter and the radiation absorption parameter enhanced thermal diffusion.
O.P.O. wishes to thank Covenant University, Ogun State, Nigeria for its generous financial support. A.A.H., as a VP, appreciates the support of
King Saud University, Saudi Arabia (KSUVPP117).
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