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Research ArticleChemical Entropy Generation and MHD Effects on
theUnsteady Heat and Fluid Flow through a Porous Medium
Gamal M. Abdel-Rahman Rashed
Department of Mathematics, Faculty of Science, Benha University,
Benha 13518, Egypt
Correspondence should be addressed to Gamal M. Abdel-Rahman
Rashed; [email protected]
Received 25 November 2015; Accepted 27 December 2015
Academic Editor: Oluwole D. Makinde
Copyright © 2016 Gamal M. Abdel-Rahman Rashed. This is an open
access article distributed under the Creative CommonsAttribution
License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work isproperly
cited.
Chemical entropy generation and magnetohydrodynamic effects on
the unsteady heat and fluid flow through a porous mediumhave been
numerically investigated. The entropy generation due to the use of
a magnetic field and porous medium effects onheat transfer, fluid
friction, and mass transfer have been analyzed numerically. Using a
similarity transformation, the governingequations of continuity,
momentum, and energy and concentration equations, of nonlinear
system, were reduced to a set ofordinary differential equations and
solved numerically. The effects of unsteadiness parameter, magnetic
field parameter, porosityparameter, heat generation/absorption
parameter, Lewis number, chemical reaction parameter, and Brinkman
number parameteron the velocity, the temperature, the
concentration, and the entropy generation rates profiles were
investigated and the results werepresented graphically.
1. Introduction
Industries developing technology related to heat transfer
aremore concerned with the design of new thermal systems;thus
research is in progress to investigate the hydrodynamicand heat
transfer behavior of new forms of heat transferfluid. Recently,
entropy generation (or production) has beenused to gauge the
significance of irreversibility related toheat transfer, friction,
and other nonideal processes withinthermal system by Bejan [1].
Entropy generation and itsminimization have been considered as an
effective toolto improve the performance of any heat transfer
process.Entropy generation minimization of diabetic distillation
col-umn with trays has been investigated using a new approachby
Spasojević et al. [2] in which the exchanged heat has
beenconsidered as a control variable instead of temperature.
Andersson et al. [3] analyzed the momentum and heattransfer in a
laminar liquid film on a horizontal stretchingsheet governed by
time-dependent boundary layer equations.Tsai et al. [4] studied the
nonuniform heat source/sink effecton the flow and heat transfer
from an unsteady stretching
sheet through a quiescent fluidmedium extending to
infinity.Elbashbeshy and Bazid [5] presented similarity solutions
ofthe boundary layer equations, which describe the unsteadyflow and
heat transfer over an unsteady stretching sheet.Shateyi and Motsa
[6] investigated thermal radiation effectson heat andmass transfer
over an unsteady stretching surface.Bouabid et al. [7] studied
analysis of the magnetic fieldeffect on entropy generation at
thermosolutal convectionin a square cavity. Oliveski et al. [8]
proposed an entropygeneration and natural convection in rectangular
cavities.Achintya [9] Analyzed the entropy generation due to
naturalconvection in square enclosures with multiple discrete
heatsources. Magherbi et al. [10] investigated second law
analysisin convective heat and mass transfer. El Jery et al. [11]
studiedeffect of external oriented magnetic field on entropy
gen-eration in natural convection and Abd El-Aziz [12]
studiedradiation effect on the flow and heat transfer over an
unsteadystretching sheet.
This study is a complementary study to the work of[6] to include
the thermal radiation effects on heat andmass transfer over an
unsteady stretching surface. This also
Hindawi Publishing CorporationJournal of Applied
MathematicsVolume 2016, Article ID 1748312, 9
pageshttp://dx.doi.org/10.1155/2016/1748312
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2 Journal of Applied Mathematics
presents chemical entropy generation and magnetohydrody-namic
effects on the unsteady heat and fluid flow througha porous medium
which are numerically investigated. Theentropy generation due to
the use of a magnetic fieldand porous medium effects on heat
transfer, fluid friction,and mass transfer have been analyzed
numerically. Theeffects of unsteadiness parameter, magnetic field
parameter,porosity parameter, heat generation/absorption
parameter,Lewis number, chemical reaction parameter, and
Brinkmannumber parameter on the velocity, the temperature,
theconcentration, and the entropy generation rates profiles
wereinvestigated and the results were presented graphically.
2. Problems Development
The flow, assumed to be unsteady, laminar, and incompress-ible
fluid on a horizontal sheet with chemical entropy gen-eration and
magnetohydrodynamic effects through a porousmedium, has been
considered. The viscous dissipation effectis also taken into
consideration. The radiative heat flux inthe 𝑥-direction is
negligible in comparison with that in the𝑦-direction. The fluid
motion arises due to the stretching ofthe elastic sheet. The
continuous sheet aligned with the 𝑥-axis at 𝑦 = 0 moves in its own
plane with a surface velocity𝑈𝑤(𝑥, 𝑡), the surface temperature
𝑇
𝑤(𝑥, 𝑡), and the surface
concentration𝐶𝑤(𝑥, 𝑡) varying both along the sheet and with
time. The magnetic Reynolds number of the flow is taken tobe
small enough that the inducedmagnetic field is assumed tobe
negligible in comparison with the applied magnetic field,that is, 𝐵
= (0, 𝐵
0, 0), where 𝐵
0is the uniform magnetic
field acting normal to the plate. 𝑢 and V are the velocity of𝑥
and 𝑦 component, and 𝑇 and 𝐶 are the temperature andconcentration,
respectively. The governing boundary layerequations of continuity,
momentum, energy, and concentra-tion equations under Boussinesq
approximations could bewritten as follows.
The continuity equation:
𝜕𝑢
𝜕𝑥
+
𝜕V𝜕𝑦
= 0. (1)
The momentum equations:
𝜕𝑢
𝜕𝑡
+ 𝑢
𝜕𝑢
𝜕𝑥
+ V𝜕𝑢
𝜕𝑦
= ]𝜕2
𝑢
𝜕𝑦2−
𝜎𝐵2
(𝑥, 𝑡)
𝜌
𝑢 − 𝑆1(𝑥, 𝑡) 𝑢. (2)
The energy equation:
𝜕𝑇
𝜕𝑡
+ 𝑢
𝜕𝑇
𝜕𝑥
+ V𝜕𝑇
𝜕𝑦
= 𝛼
𝜕2
𝑇
𝜕𝑦2−
1
𝜌𝐶𝑝
𝜕𝑞𝑟
𝜕𝑦
+
𝑄1(𝑥, 𝑡)
𝜌𝐶𝑝
(𝑇 − 𝑇∞) .
(3)
The concentration equations:
𝜕𝐶
𝜕𝑡
+ 𝑢
𝜕𝐶
𝜕𝑥
+ V𝜕𝐶
𝜕𝑦
= 𝐷
𝜕2
𝐶
𝜕𝑦2− 𝑘1(𝑥, 𝑡) (𝐶 − 𝐶
∞) . (4)
The boundary conditions are
𝑢 (𝑥, 0) = 𝑈𝑤(𝑥, 𝑡) ,
V (𝑥, 0) = 0,
𝑇 (𝑥, 0) = 𝑇𝜔(𝑥, 𝑡) ,
𝐶 (𝑥, 0) = 𝐶𝑤(𝑥, 𝑡) ,
𝑢 (𝑥,∞) → 0,
𝑇 (𝑥,∞) → 𝑇∞,
𝐶 (𝑥,∞) → 𝐶∞,
(5)
where 𝜎 is the electrical conductivity, 𝜌 is the density ofthe
fluid, 𝜇 is the viscosity of the fluid, 𝛼 = 𝑘/𝜌𝐶
𝑝is
the thermal diffusivity of the fluid, 𝐶𝑝is the heat capacity
at constant pressure, 𝑘1is the thermal conductivity, 𝑄
1is
heat generation/absorption rate (is positive in the case of
thesheet’s generation of heat and is negative in the case of
thesheet’s absorption of heat from the fluid flow), 𝐷 is the
massdiffusivity, 𝑇
∞is temperature of the fluid at infinity, and 𝑞
𝑟is
the radiative heat flux in the 𝑦-direction. Using the
Rosselandapproximation (Sparrow andCess [13] andMoradi et al.
[14]),the radiative heat flux 𝑞
𝑟is given by
𝑞𝑟= −
4𝜎∗
3𝑘∗
𝜕𝑇4
𝜕𝑦
, (6)
where 𝜎∗ is the Stefan-Boltzmann constant and 𝑘∗ is themean
absorption coefficient. Assuming that the temperaturedifference
within the flow is sufficiently small such that 𝑇4could be
approached as the linear function of temperature:
𝑇4
≅ 4𝑇3
∞𝑇 − 3𝑇
4
∞. (7)
Following Andersson et al. [15], the surface velocity 𝑈𝑤(𝑥,
𝑡)
is assumed to be 𝑈𝑤(𝑥, 𝑡) = 𝑏𝑥/(1 − 𝑎𝑡), where both 𝑎 and
𝑏 are positive constants with dimension reciprocal time. Wehave
𝑏 as the initial stretching rate and 𝑏/(1−𝑎𝑡) is increasingwith
time. In the context of polymer extrusion, the materialproperties
particularly the elasticity of the extruded sheetmay vary with time
even though the sheet is being pulledby a constant force. With
unsteady stretching, however,𝑎−1 becomes the representative time
scale of the resulting
unsteady boundary layer problem. We assume that all ofthe
surface temperature 𝑇
𝑤(𝑥, 𝑡), the surface concentration
𝐶𝑤(𝑥, 𝑡), the applied transversemagnetic field𝐵(𝑥, 𝑡), the
vol-
umetric heat generation/absorption rate𝑄1(𝑥, 𝑡), the thermal
conductivity 𝑘1(𝑥, 𝑡), and the porous medium 𝑆
1(𝑥, 𝑡) are on
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Journal of Applied Mathematics 3
a stretching sheet to vary with the distance 𝑥 along the
sheetand time in the following forms:
𝐵 (𝑥, 𝑡) = 𝐵0(1 − 𝑎𝑡)
−1/2
,
𝑄1(𝑥, 𝑡) = 𝑄
0(1 − 𝑎𝑡)
−1
,
𝑇𝑤(𝑥, 𝑡) = 𝑇
∞+ 𝑇0(
𝑏𝑥2
2]) (1 − 𝑎𝑡)
−3/2
,
𝐶𝑤(𝑥, 𝑡) = 𝐶
∞+ 𝐶0(
𝑏𝑥2
2]) (1 − 𝑎𝑡)
−3/2
,
𝑆1(𝑥, 𝑡) =
]𝜅
= 𝑆0(1 − 𝑎𝑡)
−1
,
𝑘1(𝑥, 𝑡) = 𝑘
0(1 − 𝑎𝑡)
−1
.
(8)
We introduce the similarity transformations
𝜓 = √]𝑏 (1 − 𝑎𝑡)−1/2 𝑥𝑓 (𝜂) ,
𝑇 = 𝑇∞+ 𝑇0(
𝑏𝑥2
2]) (1 − 𝑎𝑡)
−3/2
𝜃 (𝜂) ,
𝐶 = 𝐶∞+ 𝐶0(
𝑏𝑥2
2]) (1 − 𝑎𝑡)
−3/2
𝜑 (𝜂) ,
𝜂 = √𝑏
](1 − 𝑎𝑡)
−1/2
𝑦.
(9)
Equation (1) is satisfied automatically and so are
governingequations (2)–(5); we have
𝑓
− 𝐴(
𝜂𝑓
2
+ 𝑓
) + 𝑓𝑓
− 𝑓2
− (𝑀 + 𝑆) 𝑓
= 0,
(3𝑅 + 4) 𝜃
+ 3𝑅𝑃𝑟(𝑓𝜃
− 2𝑓
𝜃 − (
𝐴
2
) (3𝜃 + 𝜂𝜃
) + 𝐻𝜃)
= 0,
𝜑
+ 𝑃𝑟𝐿𝑒(𝑓𝜑
− 2𝑓
𝜑 − (
𝐴
2
) (3𝜑 + 𝜂𝜑
) − 𝛾𝜑)
= 0,
(10)
with the boundary conditions
𝑓 (0) = 0,
𝑓
(0) = 1,
𝜃 (0) = 1,
𝜑 (0) = 1,
𝑓
(∞) = 0,
𝜃 (∞) = 0,
𝜑 (∞) = 0.
(11)
Here the prime denotes a partial differentiation with respectto
𝜂, 𝜅 is the permeability of the porous medium, 𝐴 = 𝑎/𝑏 isthe
unsteadiness parameter,𝑀 = 𝜎𝐵2
0/𝑏𝜌 is themagnetic field
parameter, 𝑆 = 𝑆0/𝑏 is the porosity parameter, 𝑃
𝑟= 𝜇𝐶𝑝]/𝑘 is
the Prandtl number,𝑅 = 𝑘∗𝑘/4𝜎∗𝑇3∞is the thermal radiation
parameter,𝐻 = 𝑄0/𝑏𝜌𝐶𝑝is the heat generation/absorption
parameter, 𝐿𝑒= 𝑘/𝐷𝜌𝐶
𝑝is the Lewis number, and 𝛾 = 𝑘
0/𝑏
is the chemical reaction parameter.
3. Entropy Generation
In the present problem, the volumetric entropy generationis
therefore the sum of irreversibilities due to heat transfer,fluid
friction, mass transfer by pure concentration gradients,and mass
transfer by mixed product of concentration andthermal gradients
with magnetic field and porous mediumeffects which is given by
𝑆𝐺=
𝑘
𝑇2
∞
(
𝜕𝑇
𝜕𝑦
)
2
+
𝜇
𝑇∞
(
𝜕𝑢
𝜕𝑦
)
2
+
𝜇
𝐶∞
(
𝜕𝐶
𝜕𝑦
)
2
+
𝜇
𝐶∞
(
𝜕𝑇
𝜕𝑦
)(
𝜕𝐶
𝜕𝑦
) +
𝜎𝐵2
0
𝑇∞
𝑢2
+
𝜇
𝜅𝑇∞
𝑢2
.
(12)
Using the nondimensional quantities, we obtain the localentropy
generation rates in nondimensional form:
𝑁𝑠= 𝜆1
𝑘
𝑇2
∞
(
𝜕𝑇
𝜕𝑦
)
2
+ 𝜆2
𝜇
𝑇∞
(
𝜕𝑢
𝜕𝑦
)
2
+ 𝜆3
𝜇
𝐶∞
(
𝜕𝐶
𝜕𝑦
)
2
+ 𝜆4
𝜇
𝐶∞
(
𝜕𝑇
𝜕𝑦
)(
𝜕𝐶
𝜕𝑦
)
+ 𝜆5
𝜎𝐵2
0
𝑇∞
𝑢2
+ 𝜆6
𝜇
𝜅𝑇∞
𝑢2
,
(13)
where 𝜆1= 𝑥2
𝑇2
∞(1 − 𝑎𝑡)
4
/𝑘(Δ𝑇)2, 𝜆2= 𝑥2
𝑇2
∞(1 − 𝑎𝑡)
3
/
𝑘(Δ𝑇)2, 𝜆3= 𝑇∞(Δ𝐶/Δ𝑇)
2
(1−𝑎𝑡)4
/𝑘, 𝜆4= 𝑇∞(Δ𝐶/Δ𝑇)(1−
𝑎𝑡)4
/𝑘, 𝜆5= 𝑥2
𝑇∞(1 − 𝑎𝑡)
2
/𝑘(Δ𝑇)2, and 𝜆
6= 𝑥2
𝑇∞(1 −
𝑎𝑡)3
/𝑘(Δ𝑇)2 are the characteristic entropy generation rate.
The total dimensionless entropy generation is obtained by
𝑁𝑠= 𝐵𝑟𝑅𝑒[𝜃
/2+ 𝑓
//2+
Ω1
Ω
𝜑/2+ Ω1𝜃/𝜑/
+
1
Ω
(𝑀 + 𝑆) 𝑓/2] ,
(14)
where 𝐵𝑟= 𝜇𝑇
∞𝑏2
𝑥2
/𝑘(Δ𝑇) is the Brinkman number,𝑅𝑒= 𝑏𝑥2
/] is local Reynolds number, Ω = Δ𝑇/𝑇∞
is thedimensionless temperature ratio, and Ω
1= Δ𝐶/𝐶
∞is the
dimensionless concentration ratio.
4. Result and Discussion
The system of nonlinear ordinary differential equations
(10)together with the boundary conditions (11) is locally
similarand solved numerically by using the Control Volume
Finite-Element Method. Numerical values of the velocity, the
tem-perature, and the concentration profiles have been used to
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4 Journal of Applied Mathematics
A = 0.5, 1.0, 2.0
1 2 3 4 50𝜂
0.0
0.2
0.4
0.6
0.8
1.0f (𝜂)
S = 0.1, M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
(a)
A = 0.5, 1.0, 2.0
0.0
0.2
0.4
𝜃(𝜂)
0.6
0.8
1.0
1 2 3 4 50𝜂
S = 0.1,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
(b)
A = 0.5, 1.0, 2.0
1 2 3 4 50𝜂
0.0
0.2
0.4
𝜙(𝜂)
0.6
0.8
1.0
S = 0.1,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
(c)
A = 0.5, 1.0, 2.0
0.5 1.0 1.5 2.0 2.50.0𝜂
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Ns(𝜂)
S = 0.1, M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
(d)
Figure 1: Influence of unsteadiness parameter on (a) the
velocity profile, (b) the temperature profile, (c) the
concentration profile, and (d)the entropy generation rates
profile.
compute the entropy generation rates, found for differentvalues
of the various parameters occurring in the prob-lem with
unsteadiness parameter, magnetic field parameter,porosity
parameter, heat generation/absorption parameter,Lewis number,
chemical reaction parameter, and Brinkmannumber parameter; the
results are displayed in Figures 1–7, for the velocity, the
temperature, the concentration, andthe entropy generation rates
profiles. In order to verify theaccuracy of our present method, we
have compared ourresults with those of Shateyi and Motsa [6] and
Abd El-Aziz[12]. Table 1 shows the values of −𝜃(0) for several of𝐴
and𝑃
𝑟.
The comparisons in all the above cases are found to
agreewitheach other excellently. Also the results are found to be
similarto those by Shateyi and Motsa [6] and Abd El-Aziz [12]. So
itis good.
In Figures 1(a), 1(b), 1(c), and 1(d), respectively,
theinfluence of the unsteadiness parameter on the velocityprofile,
the temperature profile, the concentration profile, andthe entropy
generation rates profile is shown. It is observedthat the velocity
profile, the temperature profile, and theconcentration profile
decrease while the entropy generationrates profile increases with
the increase of the unsteadiness
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Journal of Applied Mathematics 5
A = 1.0, S = 0.1, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
M = 1.0, 3.0, 5.0
0.0
0.2
0.4
0.6
0.8
1.0f (𝜂)
1 2 3 4 50𝜂
(a)
A = 1.0, S = 0.1, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
M = 1.0, 3.0, 5.0
1 2 3 4 50𝜂
0.0
0.2
0.4
𝜃(𝜂)
0.6
0.8
1.0
(b)
A = 1.0, S = 0.1, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
M = 1.0, 3.0, 5.0
1 2 3 4 50𝜂
0.0
0.2
0.4
𝜙(𝜂)
0.6
0.8
1.0
(c)
A = 1.0, S = 0.1, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
M = 1.0, 3.0, 5.0
0
1
2
3
4
Ns(𝜂)
0.5 1.0 1.5 2.0 2.50.0𝜂
(d)
Figure 2: Influence of the magnetic field parameter on (a) the
velocity profile, (b) the temperature profile, (c) the
concentration profile, and(d) the entropy generation rates
profile.
parameter. When 𝐴 = 0, we have a steady state flow and for𝐴 >
0, we have an unsteady flow.
In absence of both magnetic field parameter and
porosityparameter effects on the velocity profile, the
temperatureprofile, the concentration profile, and the entropy
generationrates profile are illustrated in Figures 2(a), 2(b),
2(c), 2(d),3(a), 3(b), 3(c), and 3(d), respectively; also, we have
found thatthe velocity profile decreases while the temperature
profile,the concentration profile, and the entropy generation
ratesprofile increase with the increase of each of themagnetic
field
parameter and porosity parameter, and this is due to the
factthat the thermal boundary layer increases withmagnetic
fieldparameter and porosity parameter. The presence of each ofthe
magnetic field and porosity creates additional entropy.
The effects of heat generation/absorption parameter onthe
temperature and the entropy generation rates profilesare presented
in Figures 4(a) and 4(b), respectively. It isobserved that the
temperature profile increases while theentropy generation rates
profile decreases with increase ofheat generation/absorption
parameter. This is due to the fact
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6 Journal of Applied Mathematics
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
S = 0.1, 1.0, 4.0
𝜂0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0f (𝜂)
(a)
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
S = 0.1, 1.0, 4.0
𝜃(𝜂)
1 2 3 4 50𝜂
0.0
0.2
0.4
0.6
0.8
1.0
(b)
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
S = 0.1, 1.0, 4.0
0.0
0.2
0.4
0.6
𝜙(𝜂)
0.8
1.0
1 2 3 4 50𝜂
(c)
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
S = 0.1, 1.0, 4.0
0
1
2
3
4
Ns(𝜂)
0.5 1.0 1.5 2.0 2.50.0𝜂
(d)
Figure 3: Influence of the porosity parameter on (a) the
velocity profile, (b) the temperature profile, (c) the
concentration profile, and (d) theentropy generation rates
profile.
that the increase of the heat source/sink parameter meansan
increase of the heat generated inside the boundary layerleading to
higher temperature profile.
The effect of the Lewis number parameter on the concen-tration
and the entropy generation rates profiles is shown onFigures 5(a)
and 5(b), respectively, and we have found thatthe concentration
profile decreases while entropy generationrates profile increases
with increase of the values of the Lewisnumber parameter.
The concentration profile for different values of thechemical
reaction parameter is plotted in Figure 6;we observe
that the concentration profile decreases with the increaseof the
chemical reaction parameter. The influence of theBrinkman number
parameter on the entropy generation ratesprofile is shown in Figure
7; we have found that the entropygeneration rates profile increases
with the increase of theBrinkman number parameter.
5. Conclusions
The entropy generation due to the use of a magnetic fieldand
porous medium effects on heat transfer, fluid friction,
-
Journal of Applied Mathematics 7
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7, S = 0.1,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
H = −0.5, 0.0, 0.5
0.0
0.2
0.4
0.6
𝜃(𝜂)
0.8
1.0
1 2 3 4 50𝜂
(a)
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7, S = 0.1,
Le = 1.0, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
H = −0.5, 0.0, 0.5
0.0
0.5
1.0
1.5
2.0
2.5
Ns(𝜂)
0.5 1.0 1.5 2.0 2.50.0𝜂
(b)
Figure 4: Influence of the heat generation/absorption parameter
on (a) the temperature profile and (b) the entropy generation rates
profile.
𝜙(𝜂)
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
S = 0.1, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
Le = 0.5, 1.0, 1.5
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4 50𝜂
(a)
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
S = 0.1, 𝛾 = 0.2, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
Le = 0.5, 1.0, 1.5
0.0
0.5
1.0
1.5
2.0
2.5
Ns(𝜂)
0.5 1.0 1.5 2.0 2.50.0𝜂
(b)
Figure 5: Influence of the Lewis number parameter on (a) the
concentration profile and (b) the entropy generation rates
profile.
andmass transfer have been analyzed numerically.Numericalvalues
of the velocity, the temperature, and the concentrationprofiles
have been used to compute the entropy generationrates, found for
different values of the various parame-ters occurring in the
problem. The effects of unsteadinessparameter,magnetic field
parameter, porosity parameter, heatgeneration/absorption parameter,
Lewis number, chemicalreaction parameter, and Brinkman number
parameter on the
velocity, the temperature, the concentration, and the
entropygeneration rates profiles are discussed graphically. The
mainconclusions derived from this study are given below:
(1) The increasing values of porosity parameter andmag-netic
field parameter lead to increase in the entropygeneration rates
profile, and we also observed thatthe temperature profile and the
concentration profileincrease, while the velocity profile
decreases.
-
8 Journal of Applied Mathematics
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, S = 0.1, Br = 0.5,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
𝛾 = 0.2, 0.4, 0.8
𝜙(𝜂)
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4 50𝜂
Figure 6: Influence of the chemical reaction parameter on
theconcentration profile.
A = 1.0,M = 1.0, R = 1.0, Pr = 0.7,H = 0.0,
Le = 1.0, 𝛾 = 0.2, S = 0.1,
Re = 1.0, Ω = 10.0, Ω1 = 1.0
Br = 0.5, 1.0, 5.0
0
5
10
15
20
25
Ns(𝜂)
0.5 1.0 1.5 2.0 2.50.0𝜂
Figure 7: Influence of the Brinkman number parameter on
theentropy generation rates profile.
(2) The entropy generation rates profile decreases withthe
increase of the heat generation/absorption param-eter. And also,
the temperature profile increases.
(3) As the Lewis number parameter increases, we foundthat the
entropy generation rates profile increases,while the concentration
profile decreases.
Table 1: Comparison of the value −𝜃(0) for several of𝐴 and
𝑃𝑟with
𝑀 = 0.0, 𝑆 = 0.0,𝐻 = 0.0, 𝐿𝑒= 0.0, 𝛾 = 0.0, and 𝐵
𝑟= 0.0.
𝐴 𝑃𝑟
Abd El-Aziz [12] Shateyi and Motsa [6] Present study
0.80.1 0.4517 0.45149 0.45181.0 1.6728 1.67285 1.6727510.0
5.70503 5.70598 5.70573
1.20.1 0.5087 0.50850 0.508411.0 1.818 1.81801 1.8179310.0
6.12067 6.12102 6.12012
2.00.1 0.606013 0.60352 0.603411.0 2.07841 2.07841 2.0783010.0
6.88506 6.88615 6.88586
(4) The entropy generation rates profile decreases withthe
increase of the Brinkman number parameter.
(5) The unsteadiness parameter increases with theincrease of the
entropy generation rates profile, andalso, the velocity profile,
the temperature profile, andthe concentration profile decrease.
Nomenclature
𝐴: Unsteadiness parameter (= 𝑎/𝑏)𝑎: Positive constant𝐵: Magnetic
field𝐵0: Uniform transverse magnetic field
𝐵𝑟: Brinkman number (= 𝜇𝑇
∞𝑏2
𝑥2
/𝑘(Δ𝑇))𝑏: Positive constant𝐶: Concentration profile𝐶𝑝: Heat
capacity at constant pressure
𝐶𝑤: Surface concentration
𝐷: Mass diffusivity𝐻: Heat generation/absorption parameter (=
𝑄
0/𝑏𝜌𝐶𝑝)
𝑘: Thermal conductivity𝑘∗: Mean absorption coefficient𝐿𝑒: Lewis
number (= 𝑘/𝐷𝜌𝐶
𝑝)
𝑀: Magnetic field parameter (= 𝜎𝐵20/𝑏𝜌)
𝑁𝑠: Entropy generation rates
𝑄1: Volumetric heat generation/absorption rate
𝑄0: Heating source
𝑞𝑟: Radiative heat flux
𝑇: Temperature profile𝑇𝑤: Surface temperature
𝑇∞: Temperature of the fluid at infinity
𝑆: Porosity parameter (= 𝑆0/𝑏)
𝑆𝐺: Volumetric entropy generation
𝑃𝑟: Prandtl number (= 𝜇𝐶
𝑝]/𝑘)
𝑅: Thermal radiation parameter (= 𝑘∗𝑘/4𝜎∗𝑇3∞)
𝑅𝑒: Local Reynolds number (= 𝑏𝑥2/])
𝑢: Velocity in the 𝑥-direction𝑈𝑤: Surface velocity
V: Velocity in the 𝑦-direction𝑥: Horizontal distance𝑦: Vertical
distance.
-
Journal of Applied Mathematics 9
Greek Symbols𝜆1, 𝜆2, 𝜆3, 𝜆4, 𝜆5: Characteristic entropy
generation rate
𝛼: Thermal diffusivity of the fluid(= 𝑘/𝜌𝐶
𝑝)
𝛾: Chemical reaction parameter (= 𝑘0/𝑏)
Ω: Dimensionless temperature ratio(= Δ𝑇/𝑇
∞)
Ω1: Dimensionless concentration ratio
(= Δ𝐶/𝐶∞)
𝜂: Similarity variable𝜃: Dimensionless temperature
distribution𝜓: Stream function𝜇: Viscosity of the fluid𝜌: Density
of the fluid]: Kinematic viscosity of the fluid𝜅: Permeability of
the porous medium𝜎: Electrical conductivity𝜎∗: Stephan-Boltzmann
constant.
Subscripts𝑤,∞: Conditions at the surface and in the free
stream.
Conflict of Interests
The author declares that there is no conflict of
interestsregarding the publication of this paper.
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