IJAAMMInt. J. Adv. Appl. Math. and Mech. 7(3) (2020) 31 – 42 (ISSN: 2347-2529)
Journal homepage: www.ijaamm.com
International Journal of Advances in Applied Mathematics and Mechanics
Analysis of Entropy Generation of MHD Micropolar Fluid through aRectangular Duct with Effect of Induced Magnetic Field and SlipBoundary Conditions
Research Article
A. A. El Desoukya,∗, Hassan Nasr Ahamed Ismaila, Aly Maher Abourabiab, D. A. Hammada, Nasrelden A. Ahmeda
a Basic Engineering Sciences Department, Benha Faculty of Engineering, Benha University,13512, Benha, Egyptb Mathematics Department, Faculty of Science, Menoufia University, 32511, Shebin Elkom, Egypt
Received 29 January 2020; accepted (in revised version) 18 February 2020
Abstract: The entropy generation and Bejan numbers (Be,Bm and B f ) for steady, incompressible and laminar micropolar fluidthrough a rectangular duct with the effects of slip flow and slip convective boundary conditions are calculated. Theflow is induced by a constant pressure gradient under an external magnetic field applied in a perpendicular plane tothe flow direction. The governing nonlinear partial differential equations of momentum, induction, microrotations,and the energy is used to evaluate the entropy generation and Bejan number numerically using the finite differencemethod. The effect of various parameters and numbers such as Hartman, Brinkman, Reynolds, magnetic Reynolds,coupling numbers, slip flow and convective parameters is represented graphically.
In the last years, several of fluid flow and heat transfer have been studied because the fluid flow and heat transferare very important for many industrial applications which the energy consumption can be reduced. Eringen [1–3]studied the theory micropolar fluids and prove that the Navier–Stokes theory does not inadequate to describe thecorrect behavior of all types of fluids with microstructure like muddy water, animal blood, colloidal fluids, chemi-cal suspensions and lubricants. According to Bejan [4, 5] the entropy generation was analyzed in the fluid flow, heattransfer and processes. After this study, the entropy generation has been conducted from many researches under thedifferent flow configuration. Kumar et al. [6] studied analytically of free convective flow of micropolar and viscousfluids through a vertical channel, the separate solutions for each fluid are obtained and these solutions are comparedwith each other. Kuang Chen et al. [7] investigated numerically the effect of thermal radiation over a wavy surfaceon the micropolar fluid. H. N. Ismail et al. [8] investigated numerically mass and heat transfer due to the laminar,steady and incompressible (MHD) micropolar fluid passing through a rectangular duct under the effects of the slip
32 Analysis of Entropy Generation of MHD Micropolar Fluid through a Rectangular Duct ...
flow and slip convective boundary conditions. Kiyasatfar et al. [9] studied numerically entropy production of the twodimensional, laminar and fully developed MHD fluid flows in the rectangular microchannels by ADI finite differencemethod. Murthy and Srinivas [10] studied entropy generation and Bejan number analytical of steady, incompressiblePoiseuille flow of two immiscible micropolar fluids through a channel with two horizontal parallel plates and con-stant temperature for each wall. Ibáñez [11] studied the optimization for applied the entropy generation minimiza-tion method for steady, incompressible and electrically conducting MHD fluid through a porous horizontal channelwith the effect of slip flow and convective boundary conditions. Srinivasacharya et al. [12] used Chebyshev spec-tral collocation method to study the entropy generation rate of micropolar fluid flow in an inclined channel with slipflow and convective boundary conditions. Srinivasacharya et al. [13] investigated numerically the entropy genera-tion due to heat transfer, fluid friction and magnetic field for steady, incompressible micropolar fluid flow through arectangular duct with constant wall temperatures. The fluid motion depends on the constant pressure gradient andan external uniform magnetic field is applied to the plane perpendicular to the flow direction. Srinivasacharya et al.[14] investigated numerically the entropy generation due to heat transfer, fluid friction and magnetic field for steady,incompressible micropolar fluid flow through a rectangular duct with slip and convective boundary conditions. Thefluid motion depends on the constant pressure gradient and an external uniform magnetic field is applied to the planeperpendicular to the flow direction. Jangili and Murthy [15] studied analytically the heat transfer for two immisciblemicropolar fluids through a horizontal channel under an external magnetic field applied by the first and second lawsof thermodynamics. Ashmawy [16] investigated the fully developed natural convective micropolar fluid flow througha vertical channel with flow slip boundary conditions. Srinivas and Murthy [17] investigated the effect of entropy gen-eration rate with the motion of two immiscible micropolar fluids through a horizontal channel which bounded fromthe top and bottom by two porous beds. Srinivas et al. [18] they examined the flow, heat transfer, entropy generationrate and Bejan number characteristics through an inclined channel of two immiscible micropolar fluids by Homotopyanalysis method (HAM). Nezhad and Shahri [19] studied the entropy generation rate of two immiscible fluids throughan inclined channel under apply a uniform magnetic field, the channel consists of two regions where a viscous fluidoccupies the first region and an electrically conducting fluid occupies the second region. Srinivasacharya and Bindu[20] studied the velocity, microrotation, temperature, entropy generation rate and Bejan number of micropolar fluidthrough an inclined channel which consists of the two parallel plates. Srinivasacharya and Bindu [21] investigated nu-merically for velocity, microrotation, temperature, entropy generation and the Bejan number of micropolar fluid flowin an annulus with porous walls under the external magnetic field is applied with slip flow and convective boundaryconditions. Srinivasacharya and Bindu [22] used Chebyshev spectral collocation method to study the velocity, micro-rotation, temperature entropy generation and the Bejan number of micropolar fluid flow inside an inclined porouspipe with the effect of convective boundary conditions. Jangili et al. [23] the heat transfer and entropy generationrate of micropolar fluid through infinite channel saturated with porous material has been investigated analytically,the walls of the channel are maintained at different constant temperatures. Ayano et al. [24] used the numericalmethod to solve mixed convection micropolar fluid flow through a rectangular duct under the transverse magneticfield applied. Srinivasacharya and Himabindu [25] studied the effect of slip flow and convective heating on entropygeneration and Bejan number of incompressible micropolar fluid inside a porous channel, the flow is moved by a con-stant pressure gradient in the flow direction. Jangili and Bég [26] investigated analytically the buoyancy force effectson the entropy generation of MHD non-Newtonian micropolar fluid through infinite vertical parallel plate channelat different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27, 28] studied numericallythe heat and mass transfer of the laminar, incompressible, steady and viscous fluid inside the T-shaped cavity using afinite difference approximation. [29–31] studied the effect of MHD in the different shapes of channels.
This paper is organized as follows: In Section 2, the physical and mathematical Modeling of the problem ofsteady, laminar, incompressible and convection micropolar fluid through a rectangular duct. The flow is subject to theapplied magnetic field and the effect of the induced magnetic field and slip conditions are presented. The governingequations for velocity, microrotations, induction and temperature due to micropolar fluid are simplified and used tocalculate the entropy generation and Bejan numbers by the finite difference method. Section 3 contains the resultsand discussion of the entropy generation and the Bejan number in both x and y directions. Finally, a conclusion isgiven in Section 4.
2. The physical problem and mathematical Modeling
Consider the steady, laminar and incompressible of an electrically conducting micropolar fluid moving in a rect-angular duct under transverse external magnetic flux density B0 applied in X -direction. The coordinate system ischooses as Z -axis to be the length of the duct and (X &Y ) axis to be the cross-section area of the duct and the fluidmotion is occurring due to constant pressure gradient ∂P/∂Z along the Z -direction as shown in Fig. 1. Each wall of theduct is kept at a uniform temperature (Tw ). Assume that the slip flow condition for the velocity at the horizontal wallsand the slip convective conditions are applied to the vertical walls. Assume that the velocity vector of micropolar fluid
is*V =Vz (X ,Y )
*k and the microrotation vector
*ω=ωx (X ,Y )
*i +ωy (X ,Y )
*j . The induced magnetic field is significantly
A. A. El Desouky et al. / Int. J. Adv. Appl. Math. and Mech. 7(3) (2020) 31 – 42 33
Fig. 1. Geometrical model of the investigated problem.
produced inside the fluid so that the fluid has a large magnetic Reynolds number (Rm).
2.1. Governing Equations
The governing equations of incompressible and electrically conducting micropolar fluid with the induced magneticfield are [8]
Continuity equation:
∂ρ
∂t+∇.(ρ
*V ) = 0 (1)
Momentum equation:
ρD*V
Dt=−∇P − (µ+k)∇×∇×*V +k∇×*ω+*J ×*B (2)
Microrotation equation:
ρ j∗D*ω
Dt= k∇×*V −2k
*ω−γ∇×∇×*ω+ (α+β+γ)∇(∇.
*ω) (3)
Energy equation:
ρCpDT
Dt= K f ∇2T +λ(∇.
*V )2 +2µ(D : D)+4k(
1
2∇× *V −*ω)2 + α(∇.
*ω)2 +γ(∇*ω) : (∇*ω)+β(∇*ω) : (∇*ω)T +
*
J 2
σ(4)
Where ρ, T and P are the fluid density, temperature and pressure respectively. σ, Cp , K f and j∗ are the electricalconductivity, the specific heat at constant pressure, the thermal conductivity and microgyration parameter respec-
tively. D is the deformation tensor i.e. D = 0.5(*
Vi j +*
V j i ), J is the electric current density,*B is the total magnetic field
vector*B = (B0
*i +Bz(X ,Y )
*k ) and Bz is the induced magnetic field. µ,k, α,β and γ are the material constants (viscosity
coefficients) which are satisfying the following inequalities;
k ≥ 0, 2µ+k ≥ 0, 3α+β+γ≥ 0, γ≥ |β| (5)
Ohm’s law, the induction equation and Maxwell’s equations related to the electric current density*J , the magnetic
field*B , and electric field density
*E are given by
*J =σ(
*E +*V ×*B ),
∂*B
∂t=∇× (
*V ×*B )+ 1
σµm∇2*B ,∇×*E = ∂
*B
∂t,∇×*B =µm
*J ,∇.
*E = 0,∇.
*B = 0 (6)
Where µm is the magnetic permeability. The electric field density*E is neglected during the system does not ap-
ply polarization voltage (each wall is an electrical insulator). Under these above assumptions, the MHD equationsbecome;
Induction equation:
B0
[∂Vz
∂X
]+ 1
σµm
[∂2Bz
∂X 2 + ∂2Bz
∂Y 2
]= 0 (7)
34 Analysis of Entropy Generation of MHD Micropolar Fluid through a Rectangular Duct ...
Momentum equation:
−[∂P
∂Z
]+ (µ+k)
[∂2Vz
∂X 2 + ∂2Vz
∂Y 2
]+k
[∂ωy
∂X− ∂ωx
∂Y
]+ B0
µm
[∂Bz
∂X
]= 0 (8)
Microrotation equation in X -direction;
−2kωx +k
[∂Vz
∂Y
]−γ ∂
∂Y
[∂ωy
∂X− ∂ωx
∂Y
]+ (α+β+γ)
∂
∂X
[∂ωx
∂X+ ∂ωy
∂Y
]= 0 (9)
Microrotation equation in Y -direction;
−2kωy +k
[∂Vz
∂X
]−γ ∂
∂X
[∂ωy
∂X− ∂ωx
∂Y
]+ (α+β+γ)
∂
∂Y
[∂ωx
∂X+ ∂ωy
∂Y
]= 0 (10)
Energy equation:
k f
[∂2T
∂X 2 + ∂2T
∂Y 2
]+ (µ+k)
[(∂Vz
∂X
)2
+(∂Vz
∂Y
)2]+ α
[∂ωx
∂X+ ∂ωy
∂Y
]2
+2k
[ω2
x +ω2y −ωx
∂Vz
∂Y+ωy
∂Vz
∂X
]+γ
[(∂ωx
∂X
)2
+(∂ωx
∂Y
)2
+(∂ωy
∂X
)2
+(∂ωy
∂Y
)2]
+β[(∂ωx
∂X
)2
+(∂ωy
∂Y
)2
+2
(∂ωx
∂Y
∂ωy
∂X
)]+ 1
σµ2m
(∂2Bz
∂X 2 + ∂2Bz
∂Y 2
)= 0
(11)
The boundary conditions are classified as following [8, 14].
*B = 0, T = Tw , ωx = 0, ωy = 0, atY = 0andY = b; (12c)
Vz =α∗ dVz
dYatY = 0;andVz =−α∗ dVz
dYatY = b. (12d)
Where α∗and h are the slip flow and the slip convection coefficients respectively, T f is the initial temperature offluid. The dimensionless variables are introduced as following;
x = X
a, y = Y
a, v = Vz
V0, ω1 = ωx a
V0, ω2 =
ωy a
V0, θ = T −Tw
T f −Tw, B = Bz
B0,
∂P
∂Z=−ρV 2
0
aPl (13)
Substituting the equation (13) into equations (7) to (11), we get;[∂v
∂x
]+ 1
Rm
[∂2B
∂x2 + ∂2B
∂y2
]= 0 (14)
Re Pl +(
1
1−N
)[∂2v
∂x2 + ∂2v
∂y2
]+
(N
1−N
)[∂ω2
∂x− ∂ω1
∂y
]+ H a2
Rm
[∂B
∂x
]= 0 (15)
−ω1 +(
1
2
)[∂v
∂y
]+
(2−N
2m2
)[∂2ω1
∂y2
]+
(1
l 2
)[∂2ω1
∂x2
]+
(2m2 − l 2(2−N )
2m2l 2
)[∂2ω2
∂x∂y
]= 0 (16)
−ω2 +(
1
2
)[∂v
∂x
]+
(2−N
2m2
)[∂2ω1
∂x2
]+
(1
l 2
)[∂2ω1
∂y2
]+
(2m2 − l 2(2−N )
2m2l 2
)[∂2ω1
∂x∂y
]= 0 (17)
[∂2θ
∂x2 + ∂2θ
∂y2
]+Br
[(1
1−N
)[(∂v
∂x
)2
+(∂v
∂y
)2]+
(2N
1−N
)[ω2
1 +ω22 −ω1
∂v
∂y+ω2
∂v
∂x
]+
A
[∂ω1
∂x+ ∂ω2
∂y
]2
+(
N (2−N )
m2(1−N )
)[(∂ω1
∂x
)2
+(∂ω1
∂y
)2
+(∂ω2
∂x
)2
+(∂ω2
∂y
)2]+
C
[(∂ω1
∂x
)2
+(∂ω2
∂y
)2
+2
(∂ω1
∂y
∂ω2
∂x
)]]+ H a2Br
R2m
(∂2B
∂x2 + ∂2B
∂y2
)= 0
(18)
A. A. El Desouky et al. / Int. J. Adv. Appl. Math. and Mech. 7(3) (2020) 31 – 42 35
Where m2 = a2k(2µ+k)γ(µ+k) , A = α
µa2 and C = β
µa2 are the micropolar parameters, N = kµ+k is the coupling number, l 2 =
2a2kα+β+γ is a dimensionless parameter, Re = ρV0a/µ is the Reynolds number, Rm = σµmV0a is the magnetic Reynolds
number, H a = B0a√σ/µ =
√Re Rm/Al 2 is the Hartman number, Al = V0/
√B 2
0 /µmρ) is the Alfven number, Br =µV 2
0k f (T f −Tw ) is the Brinkman number and Pl = 1/
(∂P∂Z
)is inverted the pressure gradient.
The boundary conditions in dimensionless form are:
B = 0, v = 0, ω1 = 0, ω2 = 0 at x = 0 and x = 1; (19a)
dθ
d X−Bi (θ−1) = 0 at x = 0 and
dθ
d X−Biθ = 0 at x = 1; (19b)
B = 0, θ = 0, ω1 = 0, ω2 = 0 at y = 0 and y = yo; (19c)
v =αd v
d yat y = 0 and v =−αd v
d yat y = yo. (19d)
Where yo = (b/a) is the aspect ratio, α = (α∗/a) is the slip flow parameter and Bi = (a h/k f ) is the slip convectionparameter.
2.2. Entropy Generation
The velocity, microrotations, temperature and induced magnetic fields already obtained will be used for the calcu-lation of the volumetric rate of entropy generation within a rectangular duct. The mechanism of entropy generationare produced by fluid friction, heat transfer, and magnetic effects. The volumetric rate of entropy generation of in-compressible micropolar fluid in dimension form can be written as follows;
Sg = k f
T 2w
[(∂T
∂X
)2
+(∂T
∂Y
)2]+ µ+k
Tw
[(∂Vz
∂X
)2
+(∂Vz
∂Y
)2]+ 2k
Tw
[ω2
x +ω2y −ωx
∂Vz
∂Y+ωy
∂Vz
∂X
]+
γ
[(∂ωx
∂X
)2
+(∂ωx
∂Y
)2
+(∂ωy
∂X
)2
+(∂ωy
∂Y
)2]+ β
Tw
[(∂ωx
∂X
)2
+(∂ωy
∂Y
)2
+2
(∂ωx
∂Y
∂ωy
∂X
)]+
α
Tw
[∂ωx
∂X+ ∂ωy
∂Y
]2
+ 1
Tw
[(*J −Q
*V ).(
*E +*V ×*B )
].
(20)
Where Q electric charge density and equal zero, the dimensionless entropy generation rate production becomes:
N S =[(∂θ
∂x
)2
+(∂θ
∂y
)2]+ Br
τ
[1
1+N
[(∂v
∂x
)2
+(∂v
∂y
)2]+ 2N
1−N
[ω2
1 +ω22 −ω1
∂v
∂y+ω2
∂v
∂x
]+
N (2−N )
m2(1−N )
[(∂ω1
∂x
)2
+(∂ω1
∂y
)2
+(∂ω2
∂x
)2
+(∂ω2
∂y
)2]+ A
[∂ω1
∂x+ ∂ω2
∂y
]2
+
C
((∂ω1
∂x
)2
+(∂ω2
∂y
)2
+2
(∂ω1
∂y
∂ω2
∂x
))]+
[H a2Br
τR2m
(∂2B
∂x2 + ∂2B
∂y2
)]= 0
(21)
Where N s = (Sg a2/k f τ2) is the characteristic entropy generation rate and τ= ((T −Tw )/Tw ) is the dimensionless
temperature difference. The equation (21) can be expressed alternatively as follows.
N s = N h +N f +N m (22)
The first term on the right-hand side of the equation (22) denotes the entropy generation rate due to heat transferirreversibility, the second term refers to the entropy generation rate due to fluid friction irreversibility and the thirdterm refers to the entropy generation rate due to the magnetic field. To evaluate the irreversibility distribution, Bejannumber (Be) is used, which defined as the ratio of heat transfer entropy generation to the overall entropy generationand can write as follows;
Be = N h
N h +N f +N m(23)
If the Bejan number (Be) equals 0, this case refer to the irreversibility is dominated by the sum of fluid friction andmagnetic field. If the Bejan number equals 1, this case refers to the irreversibility is dominated by the heat transfer.The Bejan number equals 0.5 when the heat transfer equal to the sum of fluid friction and magnetic field.
Because this research is interested to study the effect of the induced magnetic field, we need to find the irreversibil-ity values due to the magnetic field on its own. We can add another definition for Bejan number and defined as the
36 Analysis of Entropy Generation of MHD Micropolar Fluid through a Rectangular Duct ...
ratio of magnetic field entropy generation to the overall entropy generation and refer to it by the symbol (Bm) andwritten as follows;
Bm = N m
N h +N f +N m(24)
By the same way, the Bejan number due to the fluid friction irreversibility is defined as the ratio of fluid frictionentropy generation to the overall entropy generation and refer to it by the symbol (B f ) and written as follows;
B f = N f
N h +N f +N m(25)
3. Results and Discussion
In this study, the Magnetohydrodynamic flow and heat transfer for the micropolar fluid through a rectangular ducthave been solved numerically by the finite difference method with 41x41 mesh points in both directions. The dimen-sionless momentum, induction, microrotation and temperature equations (14) – (18) are solved together with theboundary conditions (19a)-(19d) then, used that equations to investigate the entropy generation and Bejan numberthat have been described in equations (21), (23), (24) and (25). The effect of various parameters Rm ,Re , H a, N ,α,Biand Br are presented and discussed graphically. Fig. 2 shows the profiles of entropy generation (N s) and Bejan num-bers (Be , Bm and B f ) in x and y directions at Rm = 10,Re = 8, H a = 5, N = 0.4,α = 0.1,Bi = 1,Br = 1,Pl = 1,m =1, l = 0.5, A = 1 and C = 0.1. Fig. 2(a) it is observed that the maximum magnitude of the entropy generation near to thewalls and decreases at the center of the duct, also it is reaching to the maximum values near to x = 0 and x = 1. It’sclear from Fig. 2(b) the maximum value of Bejan number Be is occurring at the vertical walls of the rectangular ductand equal one, this phenomenon was caused because the boundary conditions for the velocity, microrotation and themagnetic field on these walls equal zero and the heat only is effecting on these walls. It’s clear also, the minimum valueof Bejan number Be is occurring at the horizontal walls of the duct and equal zero because the boundary conditionsfor the heat transfer is equal zero. Fig. 2(c) presents the maximum value of Bejan number Bm is occurring near to thecenter of the duct and equal zero in each wall due to the magnetic field at each wall is equal zero. Fig. 2(d) presents themaximum value of Bejan number B f is occurring at each horizontal wall of the duct, this phenomenon was causedbecause the entropy generations due to the magnetic field and heat transfer at the horizontal walls are equal zero dueto the equation (25).
Fig. 3 presents the effect of magnetic Reynolds number on the entropy generation and Bejan at the centerlines ofthe duct. Fig. 3(a and b) show the change of entropy generation rate in x and y directions and presents the entropygeneration decreases with increasing the magnetic Reynolds number. Fig. 3(c and d) present the change of Bejannumber Be in x and y directions. Fig. 3(e and f) present the change of Bejan number Bm in x and y directions, it’spresent from the figures the Bejan number Bm decreases with increase the magnetic Reynolds number. Fig. 3(g andh) present the change of Bejan number B f in x and y directions, it’s clear from the figures the Bejan number B fdecreases with increase the magnetic Reynolds number. Fig. 4 presents the effect of Reynolds number on the entropygeneration and Bejan at the centerlines of the duct. The entropy generation values in Fig. 4(a and b) increase with theincrease Reynolds number [14]. Fig. 4(c and d) show that the Bejan number Be increase with increasing the valuesof the Reynolds number in both x and y direction. Fig. 4(e and f) show that the Bejan number Bm increase withincreasing the values of the Reynolds number in x and y directions.
Fig. 5(a and b) present that the entropy generation rate increase with the decrease in the Hartmann number inx and y directions. Fig. 5(c and d) present that the Bejan number Be increase with the decrease in the values ofHartmann number inside the duct in x and y directions. Fig. 5(e and f) show that the Bejan number Bm increase withthe decrease in the values of Hartmann number in x and y directions [13, 14]. Fig. 6(a and b) show that the entropygeneration rate increase with decreasing the coupling number in x and y directions. Fig. 6(c and d) present the Bejannumber Be in x and y directions with various values of coupling number. Fig. 6(e and f) show that the Bejan numberBm increase with decrease the values of coupling number in x and y directions [13, 14]. Fig. 7(a and b) present thatthe entropy generation rate increase with increase the values of Brinkman number in x and y directions. Fig. 7(c andd) show that the Bejan number Be increase with the increase in the values of Brinkman number in x and y directions.Fig. 7(e and f) show Bejan number Bm in x and y directions are no noticeable changes with change with Brinkmannumber [13, 14].
Fig. 8(a, c and e) present that the entropy generation and Bejan number ( Be and Bm) does not change with changeslip flow parameter in the x-direction while, Fig. 8(b and d) present that the entropy generation and Bejan number Beincreases with decrease slip flow parameter in the y-direction. Fig. 8(f) shows that the Bejan number Bm increase withincrease the values of the slip flow parameter in the y-direction. Fig. 8(f) show that the Bejan number Bm increasewith increase the slip flow parameter in y-direction [14]. Fig. 9(a and b) present the entropy generation rate in x andy directions with various values of slip convection parameters, the entropy generation rate increased with increaseslip convection parameter in x-direction while, it remains constant with y-direction. Fig. 9(c and d) present the Bejan
A. A. El Desouky et al. / Int. J. Adv. Appl. Math. and Mech. 7(3) (2020) 31 – 42 37
Fig. 2. 3-Dimensional of (a) entropy generation profile, (b) Bejan number Be profile, (c) Bejan number Bm profile and (d) Bejannumber B f profile.
Table 1. Percentages of Bejan numbers that indicate to the irreversibility domination at the main central axis of the duct
Rm = 10 Re = 4 H a = 5 N = 0.1 Br = 1 α= 0.1 Bi = 1Be (%) 0.41 1.03 1.03 0.34 0.34 1.03 1.03Bm (%) 94.18 95.08 95.08 99.3 95.08 95.08 95.08Bf (%) 5.41 3.89 3.89 0.36 4.58 3.89 3.89
Rm = 20 Re = 6 H a = 10 N = 0.2 Br = 2 α= 0.2 Bi = 2Be(%) 0.06 0.43 0.03 0.46 0.52 1.03 2.69Bm(%) 86.58 99.25 88.49 99.00 95.50 95.20 93.40B f (%) 13.36 0.32 11.48 0.54 3.98 3.77 3.91
Rm = 30 Re = 8 H a = 15 N = 0.3 Br = 3 α= 0.3 Bi = 3Be(%) 0.016 0.26 0.001 0.65 0.36 1.02 4.26Bm(%) 86.58 99.54 77.36 98.30 95.70 95.23 91.90B f (%) 13.36 0.20 22.639 1.05 3.94 3.75 3.84
Rm = 40 Re = 10 H a = 20 N = 0.4 Br = 4 α= 0.4 Bi = 4Be(%) 0.005 0.18 0. 00006 1.03 0.27 1.02 5.61Bm(%) 82.95 99.6 65.9 95.08 95.8 95.3 90.68B f (%) 17.045 0.22 34.099 3.89 3.93 3.68 3.71
number Be with various values of slip convection parameter in x and y directions. Fig. 9(e and f) show the Bejannumber Bm with various values of slip convection parameters in x and y directions [14].
Table 1 presents the percentage of Bejan numbers (Be, Bm and B f ) that refer to the irreversibility dominationat the main central axis of the duct (x = 0.5, y = 0.5). It is clear from the table that the Bejan number Bm has themaximum percentage values than Bejan numbers Bm and B f .
38 Analysis of Entropy Generation of MHD Micropolar Fluid through a Rectangular Duct ...
Fig. 3. The effect of magnetic Reynolds number on (a-b) entropy generation, (c-d) Bejan number Be, (e-f ) Bejan number Bm and(g-h) Bejan number B f profiles at the centerlines of the duct.
4. Conclusion
In this article, we presented an analysis of entropy generation and Bejan numbers (Be,Bm and B f ) of the steady,incompressible micropolar fluid in a rectangular duct with the effects of the induced magnetic field, slip flow and con-vection conditions. The analysis of the problem has been conducted by the various of Reynolds, magnetic Reynolds,Hartmann, coupling and Brinkman numbers, and with slip flow and convection parameters. Due to the obtainedresults, we can conclude that:
• The entropy generation increases with increase Reynolds and Brinkman numbers in x and y directions and byincrease slip convective parameter in the x-direction.
• The entropy generation decreases with increase magnetic Reynolds, Hartmann, coupling numbers in x and ydirections and by increase the slip flow parameter in the y-direction.
• The Bejan number Be increases with increase slip convective parameter, Reynolds and Brinkman numbers in xand y directions while, it decreases with increase magnetic Reynolds, Hartmann, coupling numbers in x and ydirections and by increasing the slip flow parameter in the y-direction.
• The Bejan number Bm increases with increase slip convective parameter in the y-direction and Reynolds num-ber in x and y directions while, it decreases with increase magnetic Reynolds, Hartmann, coupling numbersand the slip convection parameter in x and y directions.
To optimizing of the entropy generation rate, increasing in magnetic Reynolds, Hartmann, coupling numbers anddecreasing in Reynolds and Brinkman numbers. It is clear that from the Bejan number graphs and table, the irre-versibility is dominated by the magnetic field in the main central axis of the duct.
A. A. El Desouky et al. / Int. J. Adv. Appl. Math. and Mech. 7(3) (2020) 31 – 42 39
Fig. 4. Effect of Reynolds number on (a-b) entropy generation, (c-d) Bejan number Be and (e-f ) Bejan number Bm profiles at thecenterlines of the duct.
Fig. 5. Effect of Hartmann number on (a-b) entropy generation, (c-d) Bejan number Be and (e-f ) Bejan number Bm profiles at thecenterlines of the duct.
References
[1] A. C. Eringen, Simple microfluids, International Journal of Engineering Science, 2 (1964) 205-217.[2] A. C. Eringen, Theory of micropolar fluids, Journal of Mathematics, 16 (1966) 1-18.[3] A. C. Eringen, Theory of thermomicrofluids, Journal of Mathematical analysis, 38 (1972) 480-496.[4] A. Bejan, "A Study of entropy generation in fundamental convective heat transfer," Journal of Heat Transfer,
101(1979) 718-725.
40 Analysis of Entropy Generation of MHD Micropolar Fluid through a Rectangular Duct ...
Fig. 6. Effect of coupling number on (a-b) entropy generation, (c-d) Bejan number Be and (e-f ) Bejan number Bm profiles at thecenterlines of the duct.
Fig. 7. Effect of Brinkman number on (a-b) entropy generation, (c-d) Bejan number Be and (e-f) Bejan number Bm profiles at thecenterlines of the duct.
[5] A. Bejan, Second-law analysis in heat transfer and thermal design, Advances in Heat Transfer, 15 (1982) 1-58.[6] J. P. Kumar, J. C. Umavathi, A. J. Chamkha, and I. Pop, Fully-developed free-convective flow of micropolar and
viscous fluids in a vertical channel, Applied Mathematical Modelling, 34 (2010) 1175-1186.[7] C. o.-K. Chen, Y.-T. Yang, and K.-H. Chang, The effect of thermal radiation on entropy generation due to micro-
polar fluid flow along a wavy surface, entropy, 13 (2011) 1595-1610.[8] H. N. A. Ismail, A. M. Abourabia, D. A. Hammad, N. A. Ahmed, and A. A. El Desouky, On the MHD flow and
heat transfer of a micropolar fluid in a rectangular duct under the effects of the induced magnetic field and slipboundary conditions, SN Applied Sciences, 2 (1) (2019) 25.
[9] M. Kiyasatfar, N. Pourmahmoud, M. M. Golzan, and I. Mirzaee, Thermal behavior and entropy generation rateanalysis of a viscous flow in MHD micropumps, Journal of mechanical science, 26 (2012) 1949-1955.
A. A. El Desouky et al. / Int. J. Adv. Appl. Math. and Mech. 7(3) (2020) 31 – 42 41
Fig. 8. Effect of slip flow parameter on (a-b) entropy generation, (c-d) Bejan number Be and (e-f ) Bejan number Bm profiles at thecenterlines of the duct.
Fig. 9. Effect of slip convection parameter on (a-b) entropy generation, (c-d) Bejan number Be and (e-f ) Bejan number Bmprofiles at the centerlines of the duct.
[10] J. R. Murthy and J. Srinivas, Second law analysis for Poiseuille flow of immiscible micropolar fluids in a channel,International Journal of Heat and Mass Transfer, 65 (2013) 254-264.
[11] G. Ibáñez, Entropy generation in MHD porous channel with hydrodynamic slip and convective boundary condi-tions, International Journal of Heat and Mass Transfer, 80 (2015) 274-280.
[12] D. Srinivasacharya and K. H. Bindu, Entropy generation in a micropolar fluid flow through an inclined channelwith slip and convective boundary conditions, Energy, 91 (2015) 72-83.
[13] D. Srinivasacharya and K. H. Bindu, Effect of magnetic field on entropy generation due to micropolar fluid flowin a rectangular duct, Procedia Engineering, 127 (2015) 1150-1157.
[14] D. Srinivasacharya and K. Himabindu, Analysis of entropy generation due to micropolar fluid flow in a rectangularduct subjected to slip and convective boundary conditions, Journal of Heat Transfer, 139 (2017) p. 072003.
42 Analysis of Entropy Generation of MHD Micropolar Fluid through a Rectangular Duct ...
[15] S. Jangili and J. Murthy, Thermodynamic analysis for the mhd flow of two immiscible micropolar fluids betweentwo parallel plates, Frontiers in Heat, 6(4) (2015) 1-11.
[16] E. Ashmawy, Fully developed natural convective micropolar fluid flow in a vertical channel with slip, Journal ofthe Egyptian Mathematical Society, 23 (2015) 563-567.
[17] J. Srinivas and J. R. Murthy, Second law analysis of the flow of two immiscible micropolar fluids between twoporous beds, Journal of Engineering thermophysics, 25 (2016) 126-142.
[18] J. Srinivas, J. R. Murthy, and A. J. Chamkha, Analysis of entropy generation in an inclined channel flow containingtwo immiscible micropolar fluids using HAM, International Journal of Numerical Methods for Heat and FluidFlow, 26 (2016), 1027-1049.
[19] A. Nezhad and M. Shahri, Entropy generation case studies of two-immiscible fluids under the influence of auniform magnetic field in an inclind channel, Journal of Mechanics, 32 (2016), 749-757.
[20] D. Srinivasacharya and K. H. Bindu, Entropy generation in a micropolar fluid flow through an inclined channel,Alexandria Engineering Journal, 55 (2016) 973-982.
[21] D. Srinivasacharya and K. H. Bindu, Entropy generation in a porous annulus due to micropolar fluid flow withslip and convective boundary conditions, Energy, 111 (2016) 165-177.
[22] D. Srinivasacharya and K. H. Bindu, Entropy generation of micropolar fluid flow in an inclined porous pipe withconvective boundary conditions, Sadhana, 42 (2017), 729-740.
[23] S. Jangili, S. Adesanya, J. Falade, and N. Gajjela, Entropy generation analysis for a radiative micropolar fluid flowthrough a vertical channel saturated with non-Darcian porous medium, International Journal of Applied Com-putational Mathematics, 3 (2017), 3759-3782.
[24] M. S. Ayano, S. T. Sikwila, and S. Shateyi, MHD mixed convection micropolar fluid flow through a rectangularduct, Mathematical Problems in Engineering, 2018 (2018) 9840862.
[25] D. Srinivasacharya and K. Himabindu, Effect of slip and convective boundary conditions on entropy generationin a porous channel due to micropolar fluid flow, International Journal of Nonlinear Sciences and NumericalSimulation, 19 (2018), 11-24.
[26] S. Jangili and O. A. Bég, Homotopy study of entropy generation in magnetized micropolar flow in a vertical parallelplate channel with buoyancy effect, Heat Transfer Research, 49 (2019) 529-553.
[27] H. N. A. Ismail, A. M. Abourabia, A. A. Saad, and A. A. El Desouky, Numerical simulation for steady incompress-ible laminar fluid flow and heat transfer inside T-Shaped cavity in the parallel and anti-parallel wall motions,International Journal of Innovative Science, Engineering & Technology, 2 (2015) 271-280.
[28] H. N. A. Ismail, A. M. Abourabia, A. A. Saad, and A. A. El Desouky, Numerical simulation for steady incompressiblelaminar fluid flow and heat transfer inside T-Shaped cavity using stream function and vorticity, InternationalJournal of Innovative Science, Engineering & Technology, 2 (2015) 40-48.
[29] C. Deepa, Impacts of hall and heat transfer on peristaltic blood flow of a MHD Jeffrey fluid in a vertical asymmetricporous channel, International Journal of Advances in Applied Mathematics and Mechanics, 6(4) (2019) 55 – 63.
[30] P. G. Moakher, M. Abbasi, and M. Khaki, New analytical solution of MHD fluid flow of fourth grade fluid throughthe channel with slip condition via collocation method, International Journal of Advances in Applied Mathemat-ics and Mechanics, 2(3) (2015) 87 - 94.
[31] David Chepkonga, Roy Kiogora, and K. e. Giterere, Fluid flow and heat transfer through a vertical cylindrical col-lapsible tube in presence of magnetic field and an obstacle, International Journal of Advances in Applied Mathe-matics and Mechanics, 6(4) (2019) 41 – 49.
Submit your manuscript to IJAAMM and benefit from:
I Rigorous peer reviewI Immediate publication on acceptanceI Open access: Articles freely available onlineI High visibility within the fieldI Retaining the copyright to your article—————————————————————————————Submit your next manuscript at I [email protected]
![Page 1: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/1.jpg)
![Page 2: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/2.jpg)
![Page 3: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/3.jpg)
![Page 4: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/4.jpg)
![Page 5: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/5.jpg)
![Page 6: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/6.jpg)
![Page 7: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/7.jpg)
![Page 8: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/8.jpg)
![Page 9: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/9.jpg)
![Page 10: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/10.jpg)
![Page 11: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/11.jpg)
![Page 12: Analysis of Entropy Generation of MHD Micropolar Fluid ... · at different temperatures, (HAM) has been used to solve the problem. H. Ismail et al. [27,28] studied numerically the](https://reader030.documents.pub/reader030/viewer/2022041107/5f0b0cb27e708231d42e99e3/html5/thumbnails/12.jpg)
