# On Nonperturbative Techniques for Thermal  Radiation Effect on Natural Convection past  a Vertical Plate Embedded in a Saturated  Porous Medium

Authors: O. D. Makinde; R. J. Moitsheki
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

In this article, the heat transfer characteristics of natural convection about a vertical permeable flat surface embedded in a saturated porous medium are studied by taking into account the thermal radiation effect. The plate is assumed to have a power-law temperature distribution. Similarity variables are employed in order to transform the governing partial differential equations into a nonlinear ordinary differential equation. Both Adomian decomposition method (ADM) and He's variational iteration method (VIM) coupled with Padé approximation technique are implemented to solve the reduced system. Comparisons with previously published works are performed, and excellent agreement between the results is obtained.


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## Body

## 1. Introduction

Heat transfer from different
geometrics embedded in porous media has many engineering and geophysical
applications such as geothermal reservoirs, drying of porous solids, thermal
insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling of
nuclear reactors, and underground energy transport [1]. Nakayama and Koyama [2] studied free convection over a vertical flat
plate embedded in a thermally stratified porous medium by exploiting the
similarity transformation procedure. Cheng and Minkowycz [3] studied the steady
free convection about a vertical plate embedded in a porous media using the
boundary layer assumptions and Darcy model by the similarity method. Cheng [4]
extended the work by studying the effect of lateral mass flux with prescribed
temperature and velocity as power law on the vertical surface. Other
investigators [5–8] studied some similar porous medium cases using Darcy and
Boussinesq approximations with different power-law velocity and temperature variations
at the boundaries.Meanwhile, the boundary
layer equations for free-convective flow through a porous medium constitute a nonlinear
problem. The theory of nonlinear differential equations is quite elaborate and
their solutions are of practical relevance in the engineering sciences. Several
numerical approaches have been developed in the last few decades (e.g., finite
differences, spectral method, shooting method, etc.) to tackle this problem.
More recently, the ideas of classical analytical methods have experienced a
revival in connection with the proposition of novel hybrid numerical-analytical
schemes for nonlinear differential equations. Among such trends are Adomian
decomposition method (ADM) [9–12] and He’s variational iteration method
(VIM) [13–16] coupled with Padé approximation method [17] especially when
dealing with boundary value problems [18]. These techniques, over the last few
years, have proved themselves as a powerful tool and a potential alternative to
traditional numerical techniques in various applications in science and
engineering. This seminumerical approach is also extremely useful in the
validation of purely numerical schemes.The aim
of the present work is to construct a nonperturbative solution for natural convection
boundary layer flow through a porous medium on an unbound domain in the
presence of radiation using both ADM and VIM coupled with Padé approximation
technique. The chief merit of the methods is
that they are capable of greatly reducing the size of computation work
while still maintaining accuracy of the numerical solution. However, VIM gives successive approximations of high
accuracy of the solution and VIM does not require specific treatments as in ADM
for nonlinear terms. Both numerical and graphical results are presented and
discussed quantitatively with respect to various parameters embedded in the
problem.


## 2. Mathematical Formulation

We
consider the steady two-dimensional flow of an incompressible viscous fluid induced
by a heated vertical plate embedded in a homogeneous porous medium of uniform
ambient temperatureT∞. The fluid is assumed to be Newtonian,
and a constant fluid suction or blowing is imposed at the plate surface. Under
Darcy and Boussinesq approximations, the governing boundary layer equations for
this problem can be written as [5, 6] (2.1)∂u∂x+∂v∂y=0,(2.2)∂u∂y=gKβυ∂T∂y,(2.3)ρcp(u∂T∂x+v∂T∂y)=k∂2T∂y2−∂qr∂y. Here, u, v are the velocity components along x, y coordinates, ρ the
fluid density, T the temperature of
the fluid, cp specific
heat at constant pressure, k the
thermal conductivity, υkinematic viscosity, g the gravitational acceleration, K permeability of the porous medium, and β thermal expansion coefficient. It is also assumed
that the temperature distribution of the plate is governed by the power law Tw(x)=T∞+Axλ, where A is a constant >
0 for heated plate. Using Roseland approximation [7, 19], we takethe radiative heat flux as (2.4)qr=−4σ3γ∂T4∂y, where σis the
Stefan-Boltzmann constant and γthe mean
absorption coefficient. Assume that the temperature differences
within the flow are sufficiently small such that T4 may expressed as a linear function of temperature (2.5)T4≈4T∞3T−3T∞4. The
boundary conditions are given by (2.6)T(x,0)=Tw(x),v(x,0)=V(x),T(x,∞)=T∞,u(x,∞)=0. We
introduced the following similarity variables and parameters [5, 8]: (2.7)Ψ=αRaxF(η),Rax=gKβ(Tw−T∞)xυα,T=T∞+Axλθ(η),N=16σT∞33γk,η=(yx)Rax1/2,α=kρcp,θ=T−T∞Tw−T∞, where Rax is the modified local Rayleigh number. The continuity equation (2.1) is satisfied by the stream function Ψ(x,y) defined by (2.8)u=∂Ψ∂y=(αx)RaxF′(η),v=−∂Ψ∂x=−(α2x)Rax1/2[(λ+1)F+(λ−1)F′], and (2.2) and (2.3) become (2.9)F′′=θ′,θ′′+λ+12(N+1)Fθ′−λN+1F′θ=0, where the primes denote
differentiation with respect to η, N is the Radiation parameter, and λ is
the temperature exponent. In view of (2.7), the boundary conditions
(2.6) transform
into (2.10)θ(0)=1,θ(∞)=0,F(0)=m,F′(∞)=0. The suction
or injection speed at the plate surface becomes (2.11)v(x,0)=−(α2x)Rax1/2(λ+1)F(0), where m=F(0) is the suction or
injection parameter according to m>0 or m<0, respectively. The entrainment
velocity of the fluid is given by (2.12)v(x,∞)=−(α2x)Rax1/2(λ+1)F(∞). Equation (2.9) together with the boundary conditions (2.10)
can be easily reduced to give (2.13)F′′′+λ+12(N+1)F′′F−λN+1F′2=0, with (2.14)F′(0)=1,F(0)=m,F′(∞)=0, since it is very obvious
from (2.9) and (2.10) that F′=θ (i.e., the vertical velocity and the temperature
profiles are identical). The local surface heat flux can be expressed as a
function of the local Rayleigh and Nusselt numbers as (2.15)NuxRax−1/2=−θ′(0).


## 3. Adomian Decomposition Method

In order to explicitly construct approximate nonperturbative solutions
of the problem described by (2.13) and (2.14), Adomian decomposition method well
addressed in [9–11] is employed and implemented in Maple (a symbolic algebra
package). We rewrite (2.13) in the form (3.1)LηF=λ(N+1)(Fη)2−(λ+1)2(N+1)FFηη, where the subscript η represents
differentiation with respect to η and
the differential operator employs the first three derivatives in the form Lη=d3/dη3. The
inverse operator Lη−1 is considered a threefold integral operator
defined by (3.2)Lη−1=∫0η∫0η∫0η(⋅)dηdηdη. Applying Lη−1 to both sides of
(3.1), using the boundary conditions in (2.14), we obtain (3.3)F(η)=m+η+bη22+a1Lη−1(Fη2)−a2Lη−1(FFηη), where a1=λ/(N+1),a2=(λ+1)/2(N+1),
and b=F′′(0) is to be determined from the boundary condition
at infinity in (2.14). As usual in Adomian decomposition method, the solution of (3.3)
is approximated as an infinite series (3.4)F(η)=∑j=0∞Fj, and the nonlinear terms are
decomposed as (3.5)Fη2=∑j=0∞Hj,FFηη=∑j=0∞Gj, where Hj, Gj, are polynomials (called Adomian polynomials)
given by (3.6)Hj=1j!djdSj[(∑i=0∞FηiSi)2]S=0,Gj=1j!djdSj[(∑i=0∞FiSi)(∑i=0∞FηηiSi)]S=0. Thus, we can identify (3.7)F0=m+η,F1=bη22+a1Lη−1(H0)−a2Lη−1(G0),Fj+1=a1Lη−1(Hj)−a2Lη−1(Gj),forj≥1. Using Maple, we obtained a few terms
approximation to the solution as (3.8)F1=12bη2+(1/6)λη3N+1,F2=(160λ2(N+1)2−160(1+λ)λ(2N+2)(N+1))η5+(112λbN+1−(1+λ)((1/24)(mλ/(N+1))+(1/24)b)2N+2)η4−16(1+λ)mbη32N+2  and so on. Substituting (3.7) into
(3.4), we obtain(3.9)F(η)=m+η+12bη2−112(bm+bmλ−2λ)η3N+1+1961(N+1)2((−2bN−2b+6bλN+6bλ+bm2+2bm2λ−2mλ+bm2λ2−2mλ2)η4)+14801(N+1)2((−2b2N−2b2+6b2λN+6b2λ−2bmλ−5bmλ2+4λ2+3bm−4λ)η5)+O(η6),where other terms up to O(η13) were derived. Let WL=∑j=0LFj represent the decomposition series partial sum
obtained, then F(η)=limL→∞(Wl).


## 4. He’s Variational Iteration Method

In 1978, Inokuti et al. [20] proposed a
general Lagrange multiplier method to solve nonlinear problems, which was first
proposed to solve problems in quantum mechanics. The modified method, or
variational iteration method (VIM) proposed by He [13–16], has been shown to
solve effectively, easily, and accurately a large class of nonlinear problems
with approximations converging rapidly to accurate solutions. To illustrate the basic idea of the method, we consider the
general nonlinear system (4.1)L[F(η)]+N[F(η)]=g(η), where L is a linear operator, N is a nonlinear operator, and g(η) is
a given continuous function. The basic character of the method is to construct
a correction functional for the system, which reads (4.2)Fn+1(η)=Fn(η)+∫0ηB(s)[LFn(s)+NF˜n(s)−g(s)]ds, where B is a Lagrange multiplier which can be identified optimally via variational
theory, Fn is the nth
approximate solution, and F˜n denotes a restricted variation (i.e., δF˜n=0). This technique provides a
sequence of functions which converges to the exact solution of the problem. The
initial values F(0),F′(0), and F′′(0)are usually
used for selecting the zeroth approximation F0.
Consequently, the exact solution may be obtained by usingF(η)=limn→∞Fn.In what
follows, we will apply the VIM for the problem in (2.13) to illustrate the
strength of the method. The correction functional for (2.13) reads (4.3)Fn+1(η)=Fn(η)+∫0ηB(s)[Fn′′′+λ+12(N+1)F˜n′′F˜n−λN+1F˜n′2]ds. Making the above correction
functional stationary with respect to Fn yields the stationary conditions (Euler equations) (4.4)∂3B∂s3=0,1+∂2B∂s2|s=η=0,∂B∂s|s=η=0,B(s)|s=η=0. Solving the above equations
results in B=−(1/2)(s−η)2, and (4.3) then becomes (4.5)Fn+1(η)=Fn(η)−12∫0η(s−η)2[Fn′′′(s)+λ+12(N+1)Fn′′(s)Fn(s)−λN+1Fn′2(s)]ds. We select the initial value F0(η)=m+η+(1/2)bη2by using the conditions in (2.14), where b=F′′(0) is to
be determined from the boundary condition at infinity in (2.14). Using (4.5), we obtain the next
successive approximation as(4.6)F1(η)=(−1120(1+λ)b22N+2+160λb2N+1)η5+(−124(1+λ)b2N+2+112λbN+1)η4+(−16(1+λ)bm2N+2+16λN+1)η3+12bη2+η+mand after few
iterations, we obtain(4.7)F(η)=m+η+12bη2−112(bm+bmλ−2λ)η3N+1+1961(N+1)2((−2bN−2b+6bλN+6bλ+bm2+2bm2λ−2mλ+bm2λ2−2mλ2)η4)+14801(N+1)2((−2b2N−2b2+6b2λN+6b2λ−2bmλ−5bmλ2+4λ2+3bm−4λ)η5)+O(η6)andF(η)=limn→∞Fn.


## 5. Padé Approximation Technique

It
is now well known that Padé approximants [17] have
the advantage of manipulating the polynomial approximation into rational
functions of polynomials. By this manipulation we gain more information about
the mathematical behavior of the solution. In addition, power series is not
useful for large values of η. Boyd [18] and others have
formally shown that power series in isolation are not useful to handle boundary
value problems. This can be attributed to the possibility that the radius of
convergence may not be sufficiently large to contain the boundaries of the
domain. It is therefore essential to combine the series solution, obtained by
the ADM and VIM or any series solution methods, with the Padé approximants to
provide an effective tool to handle boundary value problems on an infinite or
semi-infinite domain. Recall that the Padé approximants can be easily evaluated
by using built-in function in a symbolic computational package such as Maple.
The essential behavior of the solution will be addressed by using several
diagonal Padé approximants of different degrees. Furthermore, the undetermined value
of b=F′′(0)is calculated from the boundary condition at
infinity in (2.14). The difficulty at infinity is overcome by employing the
diagonal Padé approximants [10, 11, 18] that approximate F′(η) using WL′(η). For instance, the series is transformed into
diagonal Padé approximants as follows: (5.1)WL[M,M]′(η)=∑i=0Mhiηi∑i=0Mgiηi, where P=2(M+1) is the order
of the series required for each approximant. In the Maple
environment, the simultaneous evaluation of limη→∞WL[M/M]′(η)=0for M=2,3,4,… in (3.9) gives the numerical results for b=F′′(0) as shown in Tables 1 and 2.Table 1
Comparison of the values ofF′′(0)with previous results for N=0,λ=1.
mPostelnicu et al. [6]Ali [5]Present results (Padé [3/3])Present results (Padé [6/6])−1.0−0.6180−0.61803−0.61728−0.61803−0.8−0.6770−0.67703−0.67516−0.67703−0.6−0.7440—−0.74412−0.74404−0.4−0.8198−0.81980−0.82101−0.81982−0.2−0.9050—−0.90513−0.904990.0−1.0000−1.00000−0.99998−1.000000.2−1.1049—−1.10524−1.104970.4−1.2198—−1.22781−1.219760.6−1.3440—−1.34462−1.343900.8−1.4770—−1.47789−1.477011.0−1.6180−1.61803−1.62351−1.61803Table 2
Numerical values ofF′′(0)for N>0 using Padé approximants [6/6].
NλmF′′(0)1.01.01.0−0.9998222.01.01.0−0.7675303.01.01.0−0.6403544.01.01.0−0.5930695.01.01.0−0.5286241.01.00.0−0.7071001.01.00.5−0.8430481.01.0−0.5−0.5930841.01.0−1.0−0.500004


## 6. Numerical Results and Discussion

The
governing equation (2.13) subject to the boundary conditions (2.14) is solved using
both ADM and VIM together with Padé technique as described in Sections 3–5. Solutions are
obtained for the plate temperature with uniform lateral mass flux (λ=1) controlled by the
suction/injection parameter m and radiation parameter N as shown in Tables 1 and 2 and Figures 1 and 2.Figure 1
Vertical velocity or temperature profiles forλ=1;m=1;N=1
(solid line), N=2 (circles), N=3 (plus signs).Figure 2
Vertical velocity or
temperature profiles forN=1;λ=1;m=1 (solid line), m=0 (circles), m=−1 (plus signs).The results presented in Table1 are in
good agreement with those given by Postelnicu et al. [6] and Ali [5] who solved
numerically the case of permeable surface without considering the thermal
radiation effect. In Table 2, we observed that the local surface
heat flux rate decreases with increasing values of
radiation parameter. Figures
1 and 2 confirm the exponential decay velocity F′(η) or
temperature θ(η) profiles across the boundary layers [5–8]. As mentioned
earlier, suction corresponds to m>0, injection to m<0, and m=0 to impermeable plate. Therefore, it is clear from Figure 1 that suction
reduces the boundary layer thickness sharply as seen for m=1 while injection
increases it as for m=−1; however, the surface heat flow is always
positive regardless of the sign of m where the heat is directed from the
plate to the porous medium. Figure 2 shows that the fluid velocity and temperature increase as
the radiation parameter N increases.
This can be explained by the fact that the effect of radiation N is to increase the rate of energy
transport to the fluid and accordingly to increase the fluid temperature.


## 7. Conclusions

We employed
both ADM and VIM to compute a nonperturbative solution for thermal radiation
effect on natural convection boundary layer flow past a vertical plate embedded
in a saturated porous medium. The results demonstrate the reliability and the
efficiency of both methods in an unbounded domain. The two methods are powerful and efficient in obtaining approximations of higher accuracy and closed-form solutions if
existing. However, He’s variational iteration method gives several successive
approximations through using the iteration of the correction functional and Adomian
decomposition method provides the components of the exact solution that will be
added to get the series solution. Moreover, the VIM requires the evaluation of
the Lagrangian multiplier whereas ADM requires the evaluation of the Adomian polynomials
that mostly require tedious algebraic calculations.



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