Research Article Chemical Entropy Generation and MHD Effects … · 2019. 7. 30. · Research Article Chemical Entropy Generation and MHD Effects on the Unsteady Heat and Fluid Flow

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

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

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


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


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


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,


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.


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.


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.

References

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