# Thermal Radiation and Buoyancy Effects on Heat and Mass Transfer over a Semi-Infinite Stretching Surface with Suction and Blowing

Authors: S. Shateyi
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

This study sought to investigate thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching surface with suction and blowing. Appropriate transformations were employed to transform the governing differential equations to nonsimilar form. The transformed equations were solved numerically by an efficient implicit, iterative finite-difference scheme. A parametric study illustrating the influence of wall suction or
injection, radiation, Schmidt number and Grashof number on the fluid velocity, temperature and concentration is conducted. We conclude from the study that the flow is appreciably influenced by thermal radiation, Schmidt number, as well as fluid injection or suction.


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## Body

## 1. Introduction

There are many transport processes which occur
naturally and artificially in which flow is modified or driven by density
differences caused by temperature, chemical composition differences and
gradients, and material or phase constitution. Boundary layer flow and heat
transfer over a continuously stretched surface has received considerable
attention in recent years. This is because of the various possible engineering
and metallurgical applications such as hot rolling, wire drawing, metal and
plastic extrusion, continuous casting, glass fibre production, crystal growing,
and paper production.Gebhart and Pera [1]
investigated flows resulting from buoyancy forces which arise from a
combination of temperature and species concentration effects of comparable
magnitude. This circumstance arises often, especially in the natural
environment. Pera and Gebhart [2] analyzed the
flow induced by the combined buoyancy effects due to thermal and chemical
species diffusion adjacent to horizontal surfaces having uniform surface
conditions with buoyancy effect primarily away from the surface.Bestman [3] studied
free convection boundary layer flow with simultaneous heat and mass transfer in
a porous medium when the boundary wall moves in its own plane with suction.
Hossain [4] investigated the effect of uniform
transpiration rate on the heat and mass transfer characteristics in mixed
convection flow of a viscous incompressible fluid along a vertical permeable
plate. Acharya et al. [5] analyzed heat and mass
transfer occurring in laminar boundary layer on a linearly accelerating surface
with temperature-dependent heat source subject to suction or blowing.Hossain et al. [6] determined the effect of radiation on natural convection flow of
an optically thick viscous incompressible flow past a heated vertical porous
plate with a uniform surface temperature and a uniform rate of suction where
radiation is included by assuming the Rosseland diffusion approximation. Rahman
and Mulolani [7] examined natural convection
flow over a semi-infinite vertical plate at constant species concentration.Hussain and Hossain [8]
considered the problem of natural  convection boundary layer flow, induced by
the combined buoyancy forces from mass and thermal diffusion from a permeable
vertical flat surface with non uniform surface temperature and concentration
but a uniform rate of suction of fluid through the permeable surface.Chamkha [9]
considered
the problem of steady, hydromagnetic boundary layer flow over an accelerating
semi infinite porous surface in the presence of thermal radiation, buoyancy and
heat generation or absorption. Hossain et al.
[10] numerically investigated the effect of
thermal radiation on natural convection flow along a uniformly heated vertical
porous plate with variable viscosity and uniform suction velocity.Abel et al. [11] investigated numerically natural convective flows, heat and mass
transfer due to the combined effect of thermal and species diffusion in viscoelastic
fluid. Devi and Kandasamy [12] analyzed the
effects of a chemical reaction, heat and mass transfer on an accelerating
surface with a heat source and thermal stratification in the presence of
suction and injection.Chamkha and Quadri [13]
considered simultaneous heat and mass transfer by natural convection from a
vertical semi-infinite plate embedded in a fluid saturated porous medium in the
presence of wall suction or injection, heat generation or absorption effects,
porous medium inertial and thermal dispersion effects. In general, the porous
medium thermal dispersion effects increase the temperature of the fluid causing
higher flow rates along the surface. However, this seems not to be the case in
their study, as the peak values of the temperature and velocity profiles were
lowered as porous medium thermal dispersion parameter increases.Saha and Hossain [14] numerically studied the
problem of laminar doubly diffusive free convection flows adjacent to a
vertical surface in a stable thermally stratified medium. Abel et al. [15]
analyzed the effect of the buoyancy force and thermal radiation in MHD boundary
layer viscoelastic fluid flow over a continuously moving stretching surface.Azizi et al. [16] investigated numerically the
effects of thermal and buoyancy forces on both upward flow and downward flow of
air in a vertical parallel-plates channel. Shateyi et al. [17] studied
magnetohydrodynamic flow past a vertical plate with radiative heat transfer.Motivated by the above referenced work and the vast
possible industrial applications, it is of paramount interest in this study to
consider effects of thermal radiation, buoyancy and suction/blowing on natural
convection heat and mass transfer over a semi-infinite stretching surface. The
essential difference between the current work and Chamkha [9] and other
related work arises from the absence of an electrically conducting fluid. The
inclusion of mass transfer, as well as suction and blowing and the exclusion of
an electrically conducting fluid in the current work makes it different from
Shateyi et al. [17]. The inclusion of thermal radiation differentiates this
current work from other similar free convection heat and mass transfer studies.


## 2. Mathematical Formulation

We consider a
steady two-dimensional laminar boundary layer flow of an incompressible viscous
fluid over a semi-infinite porous stretching surface. Heat and the
concentration are supplied from the plate to the fluid at uniform rates. The
chemical species diffuses into the nearby fluid inducing a buoyancy force. A
change in the temperature of the fluid near the plate surface also results in
additional buoyancy.Introducing a Cartesian coordinate system,x-axis is chosen along the plate in the
direction of flow and y-axis normal to it. The plate is maintained at
a constant temperature Tw and the concentration is maintained at a
constant value Cw. The ambient temperature of the flow is T∞ and the concentration of uniform flow is C∞. The concentration of diffusing species is very
small in comparison to other chemical species and hence the thermal diffusion
and diffusing thermal energy effects are neglected. Viscous dissipation in the
energy is negligible. Variations in fluid properties are limited only to those
density variations which affect the buoyancy terms and the radiative heat flux
in the x-direction is considered negligible in
comparison with that in the y-direction. The concentration is assumed to be
nonreactive.Under the usual Boussinesq approximation, the
conservation equations for the steady, laminar, two-dimensional boundary layer
flow problem under consideration can be written as(2.1)∂u∂x+∂v∂y=0,(2.2)u∂u∂x+v∂u∂y=ν∂2u∂y2+gβt(T−T∞)+gβc(C−C∞),(2.3)u∂T∂x+v∂T∂y=αt∂2T∂y2−1ρcp∂qr∂y,(2.4)u∂C∂x+v∂C∂y=D∂2C∂y2. The boundary conditions
are(2.5)u(x,0)=ax,v(x,0)=Vw,T(x,0)=Tw,C(x,0)=Cw,u(x,∞)=0,T(x,∞)=T∞,C(x,∞)=C∞, where u,v are velocity components along x-axis and y-axis, respectively, g is the acceleration due
to gravity, T is temperature, Tw is the wall temperature, T∞ is the temperature of the uniform flow, αt is thermal conductivity, cp is the specific heat at constant pressure, ρ is density of the ambient fluid, and qr is the component of radiative heat flux. C is the concentration of species, Cw is the wall concentration, C∞ is the concentration of the uniform flow, D is the molecular diffusivity, Vw is suction/injection velocity, ν is the kinematic viscosity, βt is the volumetric coefficient of thermal
expansion, βc is the volumetric coefficient of thermal
expansion with concentration, and a is a stretching constant.The radiative heat fluxqr is described by the Rosseland approximation
such that(2.6)qr=−4σ*3K∂T4∂y,where σ* and K are the Stefan-Boltzman constant and the
Roseland mean absorption coefficient, respectively. Following Chamkha [18]
and others, we assume that the
temperature differences within the flow are sufficiently small so that the T4 can be expressed as a linear function after
using Taylor series to expand T4 about the free stream temperature T∞ and neglecting higher-order terms. This
results in the following approximation:(2.7)T4≈4T∞3T−3T∞4.Using (2.6) and (2.7) in the
last term of (2.3), we obtain(2.8)∂qr∂y=−16σ*T∞33K∂2T∂y2.We then nondimensionalize
(2.1)–(2.4) using the following transformations:(2.9)x=Lξ,y=LGr−1/4η,u=νLGr1/2F,v=νLGr1/4H,θ=T−T∞Tw−T∞,ϕ=C−C∞Cw−C∞,with L being the characteristic length, and the two
Grashof numbers are given by Gr=gβt(Tw−T∞)L3/ν2 and Grc=gβc(Cw−C∞)L3/ν2.Using these transformations, the governing equations
become(2.10)∂F∂ξ+∂H∂η=0,F∂F∂ξ+H∂F∂η=θ+Nϕ+∂2F∂η2,F∂θ∂ξ+H∂θ∂η=1Pr(1+R)∂2θ∂η2,F∂ϕ∂ξ+H∂ϕ∂η=1Sc∂2ϕ∂η2, where N=Grc/Gr is the buoyancy ratio, Pr is the Prandtl number, Sc is the Schmidt number, and R=16σT∞3/3Kλc is the dimensionless thermal radiation
coefficient.The boundary conditions become(2.11)F(ξ,0)=1,H(ξ,0)=V0,θ(ξ,0)=ϕ(ξ,0)=1,F(ξ,∞)=H(ξ,∞)=0,θ(ξ,∞)=ϕ(ξ,∞)=0,where V0=LVwGr−1/4/ν is the dimensionless wall normal velocity such
that V0>0 indicates injection and V0<0 indicates suction at the surface.


## 3. Method of Solution

Equations (2.10) are coupled
with nonlinear partial differential ones which possess no
similarity or closed-form solutions. Therefore a numerical solution of the
problem under consideration is needed (see, e.g., Abel et al.
[11], Abel et al. [15],
Chamkha and Khaled [19], among others). In general,
closed-form or similarity solutions are very useful in validating numerical methods.
Since 1970, when Blottner first discussed it, the implicit finite-difference
method has proven to be adequate and accurate for equations similar to
(2.10). For this reason, the implicit finite-difference method
discussed by Blottner [20] is employed in the present work. Also since the
finite-difference method is more accurate and more flexible in setting the
limiting condition far from the surface than most numerical methods such as
Runge-kutta methods, it is one of the reasons why it is adopted in the present
work. Finite-difference procedure leads to a system which is triadiagonal and
therefore speedy to solve and also economical of memory space to store the
coefficients. Although the shooting methods can be used for solving problems
presented in this study, they often present problems of instability.
Finite-difference methods have better stability characteristics, though they
generally require more work to obtain a specified accuracy.Owing to the nonlinear nature of the equations, we
employed an iterative procedure with10−6 as the maximum absolute error between two
successive iterations. The computational domain consisted of more than 300
nodal points of nonuniform distribution employed to accommodate steep changes
in the velocity, temperature, and chemical species in the immediate vicinity of
the wall. After many numerical experiments were performed to assess grid
independence and accuracy of the results, the choice of an initial step size of ∇η1 of 0.001 and a growth factor k of 1.02, such that ∇ηi+1=k∇ηi, was made. We chose the relative difference
between the current and the previous iterations to be the convergence
criterion. When this difference reached 10−6, the solution was assumed converged and the
iteration process was terminated.


## 4. Results and Discussion

A graphical
representation of the numerical results is illustrated in Figure1 through
Figure 7 to show the influence of the wall suction or blowing, Schmidt number,
radiation parameter, and Grashof number. Figure 1 depicts the influence of the
suction/injection parameter V0 on the flow velocity in the boundary layer. It
is now known that imposition of wall fluid injection increases the hydrodynamic
boundary layer which indicates an increase in the fluid velocity. However, the
exact opposite behaviour is produced by imposition of wall fluid suction. These
behaviours are clear from Figures 1(a) and 1(b). As is clearly depicted in
Figure 1, the velocity profiles rise from an initial velocity 1 up to
respective maximum values before asymptotically tend to zero as we move away
from the moving wall. In Figure 1(b), we can see that as the suction parameter
increases, the maximum fluid velocity decreases. This can be physically
interpreted by the fact that suction is to take away the warm solute on the
vertical plate thereby decreasing the velocity with a reduction in the
intensity of the natural convection rate.The variation of the velocity distribution
with increasing (a) injection parameter and (b) suction parameter withξ=5,Gr=Gc=1,Pr=0.72,Sc=1,R=1.
(a)(b)Figure2 shows the effect of the injection parameter
on the temperature and concentration profiles. As injection rate increases,
more warm fluid is added and thus the thermal and concentration boundary layer
thicknesses increase. In Figure 3, it is shown that as suction parameter value
increases, both the temperature and concentration profiles decrease. This is
because as the suction rate is increased, more warm fluid is taken away from
the boundary layer.The variation of (a) the temperature
distribution and (b) the concentration profiles with increasing injection
parameter numbers withξ=5,Gr=Grc=1,Pr=0.72,Sc=1,R=1.
(a)(b)The variation of (a) the temperature
distribution and (b) the concentration profiles with increasing injection
parameter numbers withξ=5,Gr=Grc=1,Pr=0.72,Sc=1,R=1.
(a)(b)Diffusing
chemical species of most interest in air has Schmidt numbers in the range from
0.1 to 10 [8]. In the present investigation, we consider
hydrogen (Sc=0.22), water vapour (Sc=0.66), and carbon dioxide (Sc=0.94). The effect of these chemical species on the
velocity and concentration distribution is shown in Figure 4. It can be seen
that the presence of a heavier species (lager Schmidt number) is to decrease
both the fluid velocity and the concentration in the boundary layer. This is
due to the thinning of the momentum and concentration boundary layer with the
introduction of a heavier species diffusion.The variation of (a) the velocity and (b) the
concentration profiles with different Schmidt numbers withξ=5,Gr=Grc=1,Pr=0.72,V0=0.1,R=1.
(a)(b)The effects of
thermal radiation parameterR on the velocity and temperature profiles in
the boundary layer are illustrated in Figures 5(a) and 5(b), respectively.
Increasing the thermal radiation parameter R produces an increase in the thermal condition
of the fluid and its thermal boundary layer. More flow is induced in the
boundary layer by the increase in the fluid temperature thereby causing the
velocity of the fluid to increase as well.The variation of (a) velocity and (b) the
temperature profiles with increasing radiation parameter numbers withξ=5,Gr=Grc=1,Pr=0.72,V0=0.1,Sc=1.
(a)(b)Figure6(a)
shows that the velocity rises steeply near the vertical wall as the Grashof
number is increased. Moving away from the wall, a cross flow in the velocity is
induced as the velocity profiles turn to zero at slower rates for small Grashof
numbers. The thermal boundary layer and the concentration boundary layer reduce
as the Grashof number increases causing the fluid temperature to reduce at
every point other than the wall. It is observed that the effect of the Grashof
number is to reduce the concentration distribution as concentration species is
dispersed away. This is clearly depicted in Figures 6(b) and 7(a). In
Figure 7(b) we see that radiation has no significant effect on the
concentration composition of the flow. However, it can be seen that increasing
radiation slightly reduces the concentration boundary layer.The variation of (a) velocity and (b) the
temperature distribution with increasing Grashof number values withξ=5,Pr=0.72,Sc=1,V0=0.1,R=1.
(a)(b)The variation of concentration distribution
(a) with increasing buoyancy numbers withξ=5,Pr=0.72,Sc=1,V0=0.1,R=1 and (b) with increasing radiation parameter
numbers with ξ=5,Gr=Grc=1,Pr=0.72,Sc=1,V0=0.1.
(a)(b)


## 5. Conclusion

In this paper, investigations
were made on the effects of thermal radiation, combined buoyancy and
suction/blowing on natural convection heat and mass transfer over a
semi-infinite stretching surface. Implicit finite difference method was
employed and graphical results were obtained to illustrate the details of flow
characteristics and their dependence on some of the physical parameters. It was
found that when the Grashof number increased, the fluid velocity increased.
However, this same effect was found to decrease both thermal and concentration
boundary layers. The present analysis has shown that the flow is appreciably
influenced by thermal radiation. It was observed that increasing the thermal
radiation parameter produces significant increases in the thermal conditions of
the fluid temperature which consequently induces more fluid in the boundary layer through buoyancy
effect, causing the velocity in the fluid there to increase. The hydrodynamic
boundary layer and thermal boundary layer thicknesses were observed to increase
as a result of increasing radiation. However, the concentration boundary layer
thickness was reduced as a result of increases in the thermal radiation
parameter. It was also observed that increasing the Schmidt number caused
reduction in the concentration distribution in the boundary layer. The study
noted that velocity, temperature, and concentration profiles decrease with
increases in the suction effect and that injection has opposite effects on
these profiles. It is hoped that the present work will serve as a tool for
understanding more complex problems involving various physical effects
investigated in this study.



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