DOI 10.1140/epjp/i2013-13051-y Regular Article Eur. Phys. J. Plus (2013) 128: 51 T HE EUROPEAN P HYSICAL JOURNAL PLUS Entropy effects in hydromagnetic free convection flow past a vertical plate embedded in a porous medium in the presence of thermal radiation Adnan Saeed Butt a and Asif Ali Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Received: 1 January 2013 / Revised: 4 April 2013 Published online: 22 May 2013 – c Societ` a Italiana di Fisica / Springer-Verlag 2013 Abstract. This article investigates the irreversibility effects in hydromagnetic free convective flow past an infinite vertical plate embedded in a porous medium. Thermal radiation effects are also assumed to be present and the Rosseland approximation is used to describe the heat flux in the energy equation. The governing equations are non-dimensionalised by suitable transformations and the Laplace transformation technique is used to obtain exact solutions. The expressions for velocity and temperature are utilized to compute the local entropy generation number and the Bejan number. The effects of magnetic field parameter, permeability parameter, Grashof number and radiation parameter on velocity, temperature, entropy generation rate and Bejan number are analyzed and presented through graphs. 1 Introduction In the last 40 years, there has been an increased interest in hydromagnetic, buoyancy driven convective flows over flat surfaces. This is mainly because of its wide applications in natural and industrial processes such as the atmospheric and ocean circulation, filtration processes, heat exchangers, cooling or heated systems, fibre and granular insulations, nuclear reactors, aerospace engineering etc. Also the interaction of thermal radiation effects with natural convection is important in processes where the temperature is high. Much literature relating to these topics is available. Soundal- gekar [1] was the first who studied the free convection effects in viscous incompressible fluid flow past an infinite vertical plate. He used the Laplace transformation technique and gave the exact solution of the problem. Revankar [2] examined the natural convection in a MHD flow past an impulsive permeable plate. Na and Pop [3] numerically inves- tigated a free convective flow past past a vertical semi-infinte vertical flat plate embedded in a highly saturated porous medium. Raptis and Singh [4] used the finite difference method to examine the free convective flow past an impulsive vertical plate immersed in a porous medium. Soundalgekar and Takhar [5,6] investigated the magnetrohydrodynamic and radiation effects on free convective flow past a vertical plate. Raptis [7] discussed the effects of radiation in viscous fluid flow past a permeable plate through a porous medium. He found that there is a decrease in the velocity with an increase in the radiation parameter. The effects of free convection on the unsteady viscous flow on an accelerated porous plate were studied by Makinde et al. [8]. Sankar et al. [9] analysed the natural convection flow in a vertical annulus filled with a fluid saturated porous medium with the inner wall subjected to discrete heating. Gorder et al. [10] examined a non-linear hydromagnetic convective flow along a vertical permeable cylinder in a porous medium in the presence of heat source/sink. Hamad et al. [11] studied the effects of the magnetic field on a free convection flow of a nanofluid past a vertical semi-infinite flat plate. The free convective boundary layer flow of a viscous fluid past a vertical surface embedded in a porous medium with temperature-dependent fluid properties were studied by Vajravelu et al. [12]. Khanafer [13] examined the non-darcian effects on natural convection flow in a square enclosure with a porous medium using the finite element formulation. Recently, Fan et al. [14] used the local radius basis function (RBF) collocation method to analyze the double-diffusive natural convection in a porous medium filled with fluid. It is well known that the complex systems in which flow and heat transfer processes occur are subjected to changes which are irreversible. Due to these changes, it is impossible to bring the considered system back to its initial position. Sometimes these changes can cause a lot of energy losses which increase the entropy of the system. Bejan [15] proposed that entropy generation can be used as a tool to study the irreversibility effects in the system. He proposed that energy a e-mail: [email protected]
Transcript
DOI 10.1140/epjp/i2013-13051-y
Regular Article
Eur. Phys. J. Plus (2013) 128: 51 THE EUROPEANPHYSICAL JOURNAL PLUS
Entropy effects in hydromagnetic free convection flow past avertical plate embedded in a porous medium in the presence ofthermal radiation
Adnan Saeed Butta and Asif Ali
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
Abstract. This article investigates the irreversibility effects in hydromagnetic free convective flow past aninfinite vertical plate embedded in a porous medium. Thermal radiation effects are also assumed to bepresent and the Rosseland approximation is used to describe the heat flux in the energy equation. Thegoverning equations are non-dimensionalised by suitable transformations and the Laplace transformationtechnique is used to obtain exact solutions. The expressions for velocity and temperature are utilizedto compute the local entropy generation number and the Bejan number. The effects of magnetic fieldparameter, permeability parameter, Grashof number and radiation parameter on velocity, temperature,entropy generation rate and Bejan number are analyzed and presented through graphs.
1 Introduction
In the last 40 years, there has been an increased interest in hydromagnetic, buoyancy driven convective flows over flatsurfaces. This is mainly because of its wide applications in natural and industrial processes such as the atmosphericand ocean circulation, filtration processes, heat exchangers, cooling or heated systems, fibre and granular insulations,nuclear reactors, aerospace engineering etc. Also the interaction of thermal radiation effects with natural convectionis important in processes where the temperature is high. Much literature relating to these topics is available. Soundal-gekar [1] was the first who studied the free convection effects in viscous incompressible fluid flow past an infinitevertical plate. He used the Laplace transformation technique and gave the exact solution of the problem. Revankar [2]examined the natural convection in a MHD flow past an impulsive permeable plate. Na and Pop [3] numerically inves-tigated a free convective flow past past a vertical semi-infinte vertical flat plate embedded in a highly saturated porousmedium. Raptis and Singh [4] used the finite difference method to examine the free convective flow past an impulsivevertical plate immersed in a porous medium. Soundalgekar and Takhar [5,6] investigated the magnetrohydrodynamicand radiation effects on free convective flow past a vertical plate. Raptis [7] discussed the effects of radiation in viscousfluid flow past a permeable plate through a porous medium. He found that there is a decrease in the velocity withan increase in the radiation parameter. The effects of free convection on the unsteady viscous flow on an acceleratedporous plate were studied by Makinde et al. [8]. Sankar et al. [9] analysed the natural convection flow in a verticalannulus filled with a fluid saturated porous medium with the inner wall subjected to discrete heating. Gorder et al. [10]examined a non-linear hydromagnetic convective flow along a vertical permeable cylinder in a porous medium in thepresence of heat source/sink. Hamad et al. [11] studied the effects of the magnetic field on a free convection flow ofa nanofluid past a vertical semi-infinite flat plate. The free convective boundary layer flow of a viscous fluid past avertical surface embedded in a porous medium with temperature-dependent fluid properties were studied by Vajraveluet al. [12]. Khanafer [13] examined the non-darcian effects on natural convection flow in a square enclosure with aporous medium using the finite element formulation. Recently, Fan et al. [14] used the local radius basis function(RBF) collocation method to analyze the double-diffusive natural convection in a porous medium filled with fluid.
It is well known that the complex systems in which flow and heat transfer processes occur are subjected to changeswhich are irreversible. Due to these changes, it is impossible to bring the considered system back to its initial position.Sometimes these changes can cause a lot of energy losses which increase the entropy of the system. Bejan [15] proposedthat entropy generation can be used as a tool to study the irreversibility effects in the system. He proposed that energy
losses can be reduced by knowing about the factors that are the source of entropy production. Later, several researchersstudied entropy generation effects in flow and heat transfer along flat surfaces. Odat et al. [16] examined the magneticfield effects on entropy generation in laminar flow past a flat plate. The entropy analysis of a gravity-driven film alonga permeable inclined surface was made by Makinde and Osalus [17]. Makinde [18] studied the entropy generationeffects in a non-Newtonian liquid film falling under the effects of gravity along an inclined plate. Makinde [19] madethe thermodynamical analysis of a variable viscosity liquid film flow along an inclined plate. He also considered theconvective cooling effects and found that for an isoflux heated inclined surface, the entropy generation rate decreaseswith an increase in the convective cooling. However, the effects are opposite in the case of an isothermally heatedinclined plate. Butt et al. [20] investigated the radiation effects on entropy generation during viscous flow over ahorizontal flat surface. Recently, entropy generation in a hydromagnetic, free convective viscous flow over a verticalplate with convective boundary and slip velocity were analyzed by Butt et al. [21].
The aim of the present article is to examine the entropy effects in a free convective, MHD flow past an infinite verticalplate embedded in a porous medium in the presence of thermal radiation. The Laplace transformation technique is usedto obtain the exact solutions and a comprehensive parametric study is done through graphs for velocity, temperature,entropy generation rate and Bejan number.
2 Mathematical formulation of the problem
The unsteady free-convective flow of an electrically conducting, viscous incompressible fluid past an impulsively startedinfinite vertical plate through a porous medium is considered. The x∗-axis is coincident with the plate and the positivedirection is taken vertically upwards, whereas the y∗-axis is taken normal to the plate. A magnetic field of uniformstrength Bo is applied in the direction perpendicular to the plate, i.e., along the y∗-axis. The induced magnetic fieldis assumed to be negligible as compared to the applied magnetic field. Moreover, there is no external electrical fieldapplied. Initially, both plate and fluid are at the same temperature T∞
∗. At time t∗ > 0, the plate starts movingimpulsively in the upward direction in its own plane with a velocity Uo. At the same time, the temperature of theplate is raised to Tw
∗ which is of an adequately high magnitude such that radiation effects are imperative. It is assumedthat the radiation flux qr is unidirectional and ∂qr
∂y∗ � ∂qr
∂x∗ . Since the plate is infinite in the x∗-axis, all physical variablesare functions of y∗ and t∗ only and do not depend upon x∗. All fluid properties are assumed to be constant exceptfor the density variation in the buoyancy term. Using the Boussinesq approximation, the equations governing the flowcan be written as
∂u∗
∂t∗= ν
∂2u∗
∂y∗2 − σB2ou∗
ρ− νu∗
K∗ + βg (T ∗ − T∞∗) , (1)
∂T ∗
∂t∗=
k
ρcp
∂2T ∗
∂y∗2 − 1ρcp
∂q∗r∂y∗ . (2)
The corresponding initial and boundary conditions are
t∗ ≤ 0 : u∗ = 0, T ∗ = T∞∗ for all y∗,
t∗ > 0 : u∗ = Uo, T ∗ = Tw∗ at y∗ = 0,
t∗ > 0 : u∗ → 0, T ∗ → T∞∗ at y∗ → ∞. (3)
The radiative heat flux can be simplified by using the Rosseland approximation,
qr = −4σ1
3k1
∂T ∗4
∂y∗ . (4)
The temperature differences within the flow are assumed to be sufficiently small such that the term T ∗4 can beexpressed as a linear function of the temperature. By expanding T ∗4 in a Taylor series about free stream temperatureT∞
∗ and neglecting higher-order terms, T ∗4 has a form
T ∗4 = 4T∞∗3T ∗ − 3T∞
∗4. (5)
Substituting (4) and (5) in the last term of eq. (2), the equation takes the form
∂T ∗
∂t∗=
(k
ρcp+
16σ1T∞∗3
3ρcpk1
)∂2T ∗
∂y∗2 . (6)
Eur. Phys. J. Plus (2013) 128: 51 Page 3 of 15
Let us introduce the following non-dimensional variables:
u =u∗
Uo, t =
t∗U2o
ν, y =
y∗Uo
ν, K =
U2o K∗
ν2,
M =σB2
oν
ρU2o
, Pr =μcp
k, Nr =
kk1
4σ1T 3∞
,
Gr =gβν (Tw
∗ − T∞∗)
U3o
, θ =T ∗ − T∞
∗
Tw∗ − T∞
∗ . (7)
Using (7) in eqs. (1) and (6), the equations in non-dimensional form are
∂u
∂t=
∂2u
∂y2−
(M +
1K
)u + Grθ, (8)
∂θ
∂t=
1Pr
(4 + 3Nr
3Nr
)∂2θ
∂y2. (9)
The initial and boundary conditions become
t ≤ 0 : u = 0, θ = 0 for all y,
t > 0 : u = 1, θ = 1 at y = 0,
t > 0 : u → 0, θ → 0 at y → ∞. (10)
All the physical variables involving in the considered problem are defined in nomenclature.
3 Entropy generation
According to Bejan [15], the local volumetric rate of entropy generation SG for a viscous fluid flow in the presence ofa magnetic field is defined by
SG =k
T∞∗2
[(∂T ∗
∂y∗
)2]
︸ ︷︷ ︸entropy due toheat transfer
+μ
T∞∗
(∂u∗
∂y∗
)2
+μ
K∗T∞∗ u∗2
︸ ︷︷ ︸entropy due tofluid friction
+σB2
0
T∞∗ u2.
︸ ︷︷ ︸entropy due tomagnetic field
(11)
In dimensionless variables, the entropy generation is expressed as
Ns =SG
So=
(∂θ
∂y
)2
+Br
Ω
(∂u
∂y
)2
+Br
Ω
1K
u2 +Br
ΩMu2, (12)
where So = k(Tw∗−T∞
∗)2U2o
T∞∗2ν2
is the characteristic entropy generation rate, Ω−1 = T∗∞
Tw∗−T∞∗ is the dimensionless tem-
perature difference and Br = μU2o
k(Tw∗−T∞∗) is the Brinkman number. Here Ns is known as entropy generation number
which is obtained by dividing the local volumetric entropy generation rate SG to a characteristic entropy generationrate So.
Another imperative parameter known as the Bejan number gives an idea whether the entropy effects due to fluidfriction and magnetic field dominates over the heat transfer entropy effects or vice versa. It is defined as the ratio ofthe entropy generation due to heat transfer to the total entropy generation,
Be =Entropy generation due to heat transfer
Total entropy generation. (13)
It is quite evident from (13) that the range of the Bejan number is between 0 and 1. When the value of Be is greaterthan 0.5, the entropy due to heat transfer dominates over entropy due to fluid friction and magnetic field, and Be < 0.5refers to the dominance of fluid friction and magnetic field entropy. When Be = 0.5, the contribution of the entropygeneration due to heat transfer and the entropy generation due to fluid friction and magnetic field are equal.
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4 Solution of the problem
The non-dimensional equations (8) and (9), subject to initial and boundary conditions (10) are solved with the helpof the Laplace transform technique and the solutions are obtained as follows:
θ(y, t) = erfc(
y
2
√a
t
), (14)
u(y, t) =12
[e√
Hy erfc(
y
2√
t+√
Ht
)+ e−
√Hy erfc
(y
2√
t−√
Ht
)]
−Gr
2H
[e√
Hy erfc(
y
2√
t+√
Ht
)+ e−
√Hy erfc
(y
2√
t−√
Ht
)]
+Gr
Herfc
(y
2
√a
t
)+
Gr
H
[12ebt
{ey
√ab erfc
(y
2√
t+√
abt
)+ e−y
√ab erfc
(y
2√
t−√
abt
)}]
−Gr
H
[12ebt
{ey
√ab erfc
(y√
a
2√
t+√
bt
)+ e−y
√ab erfc
(y√
a
2√
t−√
bt
)}]. (15)
where H = M + 1K , a = (3 Pr Nr
4+3Nr ), b = Ha−1 .
The Skin friction and the Nusselt number, which are measures of shear stress and heat transfer rate at the surfaceof the plate, are given in non-dimensional form as
τ = −(
∂u
∂y
)y=0
=Gr
H
√a
πt+
Gr
2Hebt
[√ab erfc
(√bt
)−√
ab erfc(−√
bt)− 2
√a
πte−bt
]
−Gr
2Hebt
[√ab erfc
(√abt
)−√
ab ercf(−√
bt)− 2√
πte−abt
]
+Gr
2H
[√H erfc
(√Ht
)−√
H erfc(−√
Ht)− 2√
πte−Ht
]
−12
[√H erfc
(√Ht
)−√
H erfc(−√
Ht)− 2√
πte−Ht
], (16)
Nu = −(
1 +4
3Nr
)(∂θ
∂y
)y=0
=(
1 +4
3Nr
)√a
πt. (17)
In the same manner, the entropy generation number (12) and the Bejan number (13) can be calculated by using (14)and (15). Due to very lengthy expressions, they are not shown here.
5 Results and discussion
In this section, the influence of various physical parameters involved in the considered problem on velocity, temperature,local entropy generation number and Bejan number are represented through graphs. Moreover, skin friction coefficientand Nusselt number are computed for different parameters and are shown in tables. The whole analysis has beencarried out by choosing the values of the Prandtl number, Pr, as 0.71 (air) and 7.0 (water).
The velocity profiles are plotted against y for different parameters in figs. 1–5. It is observed that in all cases,the magnitude of velocity of air (Pr = 0.71) is greater than for water (Pr = 7.0). Physically, this is possible becausefluids having high Prandtl number are more viscous and their motion is slow as compared to fluids having low valuesof Prandtl number. Figure 1 depicts that with an increase in magnetic field parameter M , the velocity decreases forboth air and water. This is because the presence of the transverse magnetic field creates a drag known as the Lorentzforce which retards the fluid motion. The effects of variation in the permeabilitiy parameter K on velocity is presentedin fig. 2. As the permeability parameter K increases, a decrease in the resistance of porous medium is noticed whichaccelerates the fluid flow. The velocity profiles are shown in fig. 3 for different values of the Grashof number Gr. Itis noteworthy that velocity increases for both water and air with an increase in Gr due to the enhancement in thebuoyancy force. However, the rise in velocity profiles is more prominent in the case of air (Pr = 0.71). A decrease inthe velocity is observed in fig. 4 for both types of fluids as the radiation parameter Nr increases. In fig. 5, the velocityprofiles due to variation in time t are displayed. It is observed that there is an increase in the velocity with time.
Eur. Phys. J. Plus (2013) 128: 51 Page 5 of 15
Fig. 1. Effects of the magnetic field parameter M on the velocity profile.
Fig. 2. Effects of the permeability parameter K on the velocity profile.
Fig. 3. Effects of the Grashof number Gr on the velocity profile.
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Fig. 4. Effects of the radiation parameter Nr on the velocity profile.
Fig. 5. Effects of the non-dimensional time t on the velocity profile.
The temperature profiles of air (Pr = 0.71) and water (Pr = 7.0) due to variations in the radiation parameter,Nr, and time, t, are displayed in figs. 6 and 7. Figure 6 depicts that the effects of Nr on the temperature aredecreasing for both fluids. This can be explained physically by the fact that the increase in the radiation parameterNr = kk1/4σ1T
3∞ for fixed values of k and T∞ increases the Rosseland radiation absorptivity k1 which significantly
reduces the temperature. The temperature increases with increase in time t for both air and water as shown in fig. 7.The numerical values of skin friction τ and Nusselt number Nu are calculated for different values of M , K, Nr,
Gr, t by choosing values of Prandtl number Pr as 0.71 (air) and 7.0 (water). The effects of these parameters on skinfriction are tabulated in tables 1 and 2. It is observed that the skin friction increases with the magnetic field parameterM and the radiation parameter Nr and it decreases with the permeability parameter K, the Grashof number Gr andtime t for both values of the Prandtl number. The effects of the radiation parameter Nr and time t on the Nusseltnumber Nu are decreasing for both air and water as illustrated in tables 3 and 4.
The variations due to different parameters involved in the problem on the local entropy generation number Ns arepresented in figs. 8–12. Figure 8 shows the influence of the magnetic field parameter M on Ns. It is noteworthy thatthe entropy generation number Ns is maximum at the plate surface moving impulsively and decreases asymptoticallywith increase in distance. An increase in the value of M causes an increase in entropy production rate. Thus, themagnetic field is a major source of entropy production in the fluid. Also, it is observed that the entropy generationrate near the impulsive plate is higher in the case of water as compared to air because fluids with high Prandtl numbershave more viscosity, which leads to more entropy effects due to fluid friction.
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Fig. 6. Effects of the radiation parameter Nr on the temperature profile.
Fig. 7. Effects of the non-dimensional time t on the temperature profile.
Table 1. Effects of various parameters on skin friction when Pr = 0.71.
Table 3. Effects of various parameters on the Nusselt number Nu when Pr = 0.71.
t Nr Nu0.5 0.5 1.287370
1.0 1.0269701.5 0.9240012.0 0.867948
0.2 1.0 1.6237800.5 1.0269700.6 0.937491
Table 4. Effects of various parameters on the Nusselt number Nu when Pr = 7.0.
t Nr Nu0.5 0.5 4.04227
1.0 3.224611.5 2.901302.0 2.72529
0.2 1.0 5.098560.5 3.224610.6 2.94365
Fig. 8. Effects of the magnetic field parameter M on the local entropy generation number Ns.
Eur. Phys. J. Plus (2013) 128: 51 Page 9 of 15
Fig. 9. Effects of the permeability parameter K on the local entropy generation number Ns.
Fig. 10. Effects of the Grashof number Gr on the local entropy generation number Ns.
In fig. 9, the effects of the permeability parameter K on the entropy generation number are shown. Entropy effectsrecede with increase in the value of K for both air and water. However, entropy effects last long in the flow regime forair as compared to water. Figure 10 shows that in the surrounding of impulsively moving plate, the entropy generationnumber Ns decreases with an increase in the Grashof number Gr for both values of the Prandtl number. However, theeffects are reversed away from the plate and an increase in entropy is seen. The effects of radiation parameter Nr onNs are displayed in fig. 11. The entropy generation rate is much higher in the case of water (Pr = 7.0) as compared toair (Pr = 0.71) in the surroundings of impulsively moving flat surface. However, entropy effects diminish very quicklyin the case of water. An increase in the radiation parameter Nr enhances the entropy effects. The influence of thegroup parameter Br/Ω on the entropy generation number Ns is presented in fig. 12. An increase in Ns is observed forboth water and air with increase in Br/Ω. In figs. 13 and 14, the local entropy generation number Ns is presented asa function of y and t for both air (Pr = 0.71) and water (Pr = 7.0) by keeping all other parameters fixed. It is noticedthat Ns decreases with increase in the distance y. In the case of air, entropy effects become strong with an increasein time t. An opposite behavior is observed in the case of water with respect to t.
Figures 15–19 are drawn to see the effects of pertinent parameters on the Bejan number Be. The Bejan number isuseful as it gives information about dominancy of fluid friction and magnetic field entropy over heat transfer entropy orvice versa. Figure 15 shows the effects of magnetic field parameter M on the Bejan number Be for both air and water.It is noticed that entropy effects due to fluid friction and magnetic field become dominant with increase in M at thesurface of impulsively moving plate for both air and water. However, these effects are more prominent in case of air.
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Fig. 11. Effects of the radiation parameter Nr on the local entropy generation number Ns.
Fig. 12. Effects of the group parameter Br/Ω on the local entropy generation number Ns.
Fig. 13. Effects of the distance, y, and time, t, on the local entropy generation number, Ns, in the case of air (Pr = 0.71) whenM = 1.0, K = 0.5, Gr = 5.0, Nr = 5.0, Br/Ω = 1.0.
Eur. Phys. J. Plus (2013) 128: 51 Page 11 of 15
Fig. 14. Effects of the distance, y, and time, t, on the local entropy generation number, Ns, in the case of air (Pr = 0.71) whenM = 1.0, K = 0.5, Gr = 5.0, Nr = 5.0, Br/Ω = 1.0.
Fig. 15. Effects of the magnetic field parameter M on the Bejan number Be.
Fig. 16. Effects of the permeability parameter K on the Bejan number Be.
Page 12 of 15 Eur. Phys. J. Plus (2013) 128: 51
Fig. 17. Effects of the Grashof number Gr on the Bejan number Be.
Fig. 18. Effects of the radiation parameter Nr on the Bejan number Be.
As the distance from the plate increases, heat transfer entropy effects start to appear and become fully dominant infar away region in case of air (Pr = 0.71). For water (Pr = 7.0), entropy effects due to the heat transfer attain a peakvalue a small distance away from the surface. However, the entropy due to fluid friction and magnetic field is againfully dominant in a far away region due to more viscous effects. Figure 16 depicts that entropy production due to fluidfriction and magnetic field become lower with increase in the permeability parameter K in the neighbourhood of theplate. In the far away region, same effects are observed as in the case of the magnetic field parameter. In fig. 17, itis seen that the Grashof number Gr causes slight increase in the fluid friction and magnetic field entropy adjacent tothe impulsively moving plate. This situation immediately becomes different as the distance increases. For air, entropyeffects due to heat transfer are prominent and in the case of water, fluid friction and magnetic field entropy effectsdominate in the far region. The effects of the radiation parameter, Nr, on Be are illustrated in fig. 18. It is quite clearthat the Bejan number increases with an increase in the value of Nr for Pr = 0.71. This shows that heat transferentropy effects become strong with increase in Nr in the case of air. For Pr = 7.0, heat transfer entropy effectsare prominent in the surrounding of impulsively moving plate and entropy due to fluid friction and magnetic fielddominates in the far region. Figure 19 shows that the fluid friction and the magnetic field entropy strengthen with anincrease in the group parameter Br/Ω.
The Bejan number Be is plotted against y and t for both values of Prandtl number in figs. 20 and 21 by fixingvalues of other involved parameters. It is detected that fluid friction and magnetic field entropy effects dominate onincreasing y and t in both air and water.
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Fig. 19. Effects of the group parameter Br/Ω on the Bejan number Be.
Fig. 20. Effects of the distance y and time t on the Bejan number Be in the case of air (Pr = 0.71) when M = 1.0, K = 0.5,Gr = 5.0, Nr = 5.0, Br/Ω = 1.0.
Fig. 21. Effects of the distance y and time t on the Bejan number Be, in the case of air (Pr = 7.0) when M = 1.0, K = 0.5,Gr = 5.0, Nr = 5.0, Br/Ω = 1.0.
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6 ConclusionsIn this article, entropy effects are analyzed in unsteady, free convective hydromagnetic flow past a vertical flat surfacethrough a porous medium when thermal radiation is also present. Using the Laplace transformation technique, exactsolutions are obtained for velocity and temperature in terms of exponential and complementary error functions. Theexpressions for skin frictin, the Nusselt number, the local entropy generation number and the Bejan number areevaluated using velocity and temperature solutions. The results are displayed through graphs and tables for two valuesof the Prandtl number which correspond to air and water. The following observations are made from the above analysis:
– The velocity decreases with increasing values of the magnetic field parameter M and the radiation parameter Nrand increases with increase in the Grashof number Gr, the permeability parameter K and time t.
– The velocity of air (Pr = 0.71) is greater than the velocity of water (Pr = 7.0).– The radiation parameter, Nr, has decreasing effect on temperature for both air and water whereas an increase in
the temperature is observed with an increase in time t.– For both values of Pr, the skin friction τ increases with Nr and M and decreases when values of K, Gr and t are
increased.– A decrease in the Nusselt number Nu is detected by increasing time t and radiation parameter Nr for both air
and water.– Entropy effects in the case of water are more prominent in the neighbourhood of impulsive plate due to strong
viscosity effects. However, for air, these effects last for long distance.– An increase in the entropy generation number, Ns, is observed with an increase in M , Nr and Br/Ω and decreasing
effects are seen in case of K.– For the Grashof number Gr, a slight decrease in Ns is noticed in the neighbouring region of impulsive flat surface
for both air and water. The situation becomes reverse as the distance from the surface increases.– Local entropy generation number Ns increases with time t for Pr = 0.71 (air) whereas for Pr = 7.0 (water), Ns
decreases as time t increases.– In the case of air, fluid friction and magnetic field entropy effects are dominant adjacent to impulsive plate and
entropy effects due to heat transfer dominate in the far away region. In the case of water, entropy due to fluidfriction and magnetic field is fully dominant.
– For both air and water, increase in M and Br/Ω cause entropy due to fluid friction and magnetic field to increasein the surrounding of impulsively moving surface whereas heat transfer entropy effects start to appear in thatregion with increase in the values of K and Nr.
– At the surface of impulsively moving plate, Gr causes a slight decrease in entropy effects due to fluid friction andmagnetic field. As the distance increases from the surface, the situation changes and increasing effects are observed.
– With an increase in time t, the entropy due to the fluid friction and the magnetic field becomes dominant.
Nomenclature
Bo Uniform magnetic field normal to the plate t∗ Time
Br Brinkman number t Dimensionless time
Be Bejan number T ∗ Temperature of the fluid
Gr Grashof number Tw∗ Temperature of the fluid at the surface
g Acceleration due to gravity T∞∗ Temperature of the fluid away from the plate
K Permeability parameter u∗ Velocity in the x∗-direction
M Magnetic field parameter u Dimensionless velocity
Nr Radiation parameter Uo Velocity of the plate
Nu Nusselt number x∗ x-coordinate
Pr Prandtl number x Dimensionless coordinate axis parallel to the plate
qr Radiative flux y∗ y-coordinate
SG Volumetric rate of entropy generation y Dimensionless coordinate axis normal to the plate
So Characteristic entropy generation rate
Eur. Phys. J. Plus (2013) 128: 51 Page 15 of 15
Greek symbols
β Volumetric coefficient of thermal expansion
cp Specific heat at constant pressure
θ Dimensionless temperature
μ Coefficient of viscosity
ν Kinematic viscosity
ρ Fluid density
σ Electrical conductivity
σ1 Stefan-Boltzmann constant
κ Thermal conductivity
κ1 Mean absorption coefficient
τ Dimensionless skin friction
Ω Dimensionless temperature difference
References
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