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Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates
Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates
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Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates M. Ijaz Khan, Mujeeb ur Rahman, Sohail A. Khan, T. Hayat, M. Imran Khan PII: DOI: Reference:
S0169-2607(19)31730-4 https://doi.org/10.1016/j.cmpb.2019.105149 COMM 105149
To appear in:
Computer Methods and Programs in Biomedicine
Received date: Revised date: Accepted date:
7 October 2019 17 October 2019 18 October 2019
Please cite this article as: M. Ijaz Khan, Mujeeb ur Rahman, Sohail A. Khan, T. Hayat, M. Imran Khan, Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105149
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Highlights • Here we have discussed MHD squeezed flow between two plates. • The upper plate squeezed towards the lower plates. • Energy equation is modeled subject to dissipation and Joule heating. • Entropy rate is calculated through thermodynamics law.
1
Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates M. Ijaz Khana,1 , Mujeeb ur Rahmanb , Sohail A. Khana , T. Hayata,c and M. Imran Khand,2 a
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
b
Department of Mathematics, Karakoram International University Gilgit 15100, Pakistan c
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University P. O. Box 80207, Jeddah 21589, Saudi Arabia
d
School of Engineering, University of Portsmouth, Winston Churchill Avenue Portsmouth PO1 2UP, United Kingdom Abstract: Background: Nanomaterials have advanced behaviors that make them possibly beneficial in
various applications in mass and heat transports such as engine cooling, pharmaceutical processes, fuel cells, engine cooling and domestic refrigerator etc. Therefore here we deliberated the entropy generation in unsteady magnetohydrodynamic squeezing flow of viscous nanomaterials between two parallel plates. The upper plate is squeezing towards lower plate. The lower plate exhibits porous character. Energy attributes are discussed through heat flux, dissipation and Joule heating. Furthermore the irreversibility analysis with cubic autocatalysis chemical reaction is also accounted.
Methods: Nonlinear differential systems are converted to ordinary differential system by transformations. For convergent series solution the given system are solved by homotopy analysis method (HAM).
Results: Characteristics of various interesting variables on velocity, Bejan number, concentration, entropy optimization and temperature are deliberated through graphs. Gradient of velocity
(Cf x ) and Nusselt number (N ux ) are numerically computed against various physical variables. Entropy generation and Bejan number both quantitatively enhance versus radiation parameter. For larger squeezing parameter the velocity and temperature field are increased.
Conclusions: The obtained results show that for larger squeezing parameter the velocity field boosts up. Velocity have opposite impact For larger magnetic and porosity parameters. Temperature is decreased for higher values of radiation parameter and Prandtl number. Temperature and concentration have same outcome for thermophoresis parameter. Entropy generation and Bejan 1 2
Corresponding author E-mail address: ijazfmg [email protected] (M. I. Khan) Corresponding author E-mail address: [email protected] (M. I. Khan)
2
number both quantitatively enhance versus radiation parameter, while reverse is hold for Brinkman number.
Keywords: Entropy generation; Thermal radiation; Homogenous heterogenous reactions; Viscous dissipation; Squeezing flow and Joule heating.
1
Introduction
Colloidal suspension of nano size solid particles (1-100nm) in traditionally working materials is known as nanoliquids. Most commonly metals are used as nanoparticles like (Cu, Ag, Au, F e), carbon nanotubes, nitride (SiN, SiC, T iC) and metallic oxides (T iO2 , Al2 O3 , CuO). Heat conduction phenomenon of conventionally working materials are improved by enclosure of nanoparticles into it. Nanoparticles have wide range of application in heat exchanger, compressor, combustors, computer processor, hybrid powered engines and air conditioners etc. Khanafer et al. [1] studied heat transfer enhancement through nanoparticles. Sheikholeslami et al. [2] discussed effect of magnetohydrodynamic on flow of copper water nanoliquid with natural convection in an enclosure. Mixed convective flow of Ag and Cu nanoparticles with heat flux is illustrated by Hayat et al. .[3]. Hayat et al. [4] discussed the study of Jeffrey nanomaterials flow with Hall and ion effect. Farooq et al. [5] analyzed viscoelastic nanofluid flow with stagnation point and heat flux. Magnetohydrodynamic effect on CuO-water nanofluid with mix convection is presented by Sheikholeslami et al. [6]. Ellahi et al. [7] examined shape effect of nanoparticles in Cu − H2 O on entropy generation. Thermal behaviors of hydromagnetic Al2 O3 -water nanoliquid in a micro annular cylinder is examined by Malvindi et al. [8]. Hayat et al. [9] examined copper (Cu) and silver (Ag) nanoparticles flow with entropy optimization. Entropy generation is the measurement of energy in a system, which are not used to do significant work. As a significant of thermodynamics second law entropy generates. According to thermodynamics second law entropy optimization of an isolated system does not decays over time. Entropy of a reversible processes never change, while in an irreversible processed entropy always enhances and as a results total entropy increases. These irreversible processes contains radiation, Joule heating, fluid viscosity, chemical reaction, friction between solid surfaces and diffusion in a system. For effectiveness of equipment entropy generation is minimized. Entropy minimization is used in curved pipes, microchannel, reactor, chillers, reg3
ular graded materials, fuel cell and helical coil. Alsaadi et al. [10] discussed the irreversibility effect in Williamson nanoliquids with MHD and activation energy over a stretchable surface. Nouri et al. [11] studied the entropy optimization in nanoliquids flow with heat generation/absorption. Entropy optimization study in magnetohydrodynamic flow of CNTs based nanoliquids with Joule heating and heat flux due to stretchable rotating disks is discussed by Hosseinzadeh et al. [12]. Impact of irreversibility analysis in unsteady MHD squeezing flow of viscous nanomaterial between two parallel plates is highlighted by Ahmad et al. [13]. Qing et al. [14] discussed entropy optimization analysis in magnetohydrodynamic flow of Casson nanoliquids with chemical reaction and radiation over a porous medium. Effect of entropy optimization and heat flux in magnetohydrodynamic flow of second grade nanoliquids is highlighted by Sith et al. [15]. Nouri et al. [16] reported the exploration of entropy optimization in nanoliquids flow inside a channel. Impact of frictional and Joule heating in mixed convective magnetohydrodynamic viscous fluid flow with entropy optimization is explored by Afridi et al. [17]. Physical behaviors of irreversibility in Ree-Eyring nanoliquids with cubic autocatalysis chemical reaction is presented by Khan et al. [18]. Shah et al. [19] studied impact of entropy optimization and heat flux in viscous flow of nanoliquids by a stretchable Sheet. Kumar et al. [20] discussed the entropy generation in CNTs nanoliquids with heat flux and quartic autocatalysis chemical reaction. Aziz et al. [21] reported the irreversibility analysis in magnetohydrodynamic flow of Casson fluids with Hall current. Khan et al. [22] studied the MHD dissipative flow of third grade nanoliquids with chemical reaction and entropy generation. Irreversibility analysis on magnetohydrodynamic flow of viscous liquids with heat generation/absorption and heat flux by a stretching surface is illustrated by Hayat et al. [23]. In this article irreversibility analysis in 3D squeezing flow of nanofluids between two parallel plates is investigated. Energy attributes are discussed through heat flux, dissipation and Joule heating are also considered. Furthermore cubic autocatalysis chemical reactions are also taken into account. Nonlinear system are altered to ordinary one by suitable transformations. For series solution the obtained system are solved HAM [24-29]. Effect of numerous engineering parameters on velocity, Bejan number, concentration, entropy generation and temperature are deliberated through graphs. Gradients of temperature and velocity are numerically computed via various flow parameters.
4
2
Mathematical modeling
Here we discussed incompressible unsteady magnetohydrodynamic squeezing flow of viscous nanoliquid between two parallel plates. The flow is caused due to stretching of sheet. Energy characteristics are taken into account through Joule heating, dissipation and heat flux. Furthermore physical aspects of cubic autocatalysis chemical reactions with entropy optimization are also examined. The upper plate is squeezing towards the lower plate with squeez ing velocity Vh = dh(t) . The fluid is electrically conducting with magnetic field strength dt ax 0 B = √B the stretching velocity with stretching rate . Let us assume that Uw = 1−ct 1−ct q a > 0. h(t) = ν(1−ct) is the distance between the two plates. a Isothermal homogeneous-heterogeneous chemical reaction is given as [25-27]: A∗ + 2B ∗ → 3B ∗ , rate = k1 C1 C22 ,
(1)
On the surface an isothermal chemical reaction is A∗ + B ∗ = 3B ∗ , rate = k2 C2 ,
(2)
in which A∗ and B ∗ represent the chemical reaction species, k1 and k2 the reaction rates and C1 and C2 the concentrations. The boundary layer equations are ∂u ∂u ∂w + + = 0, ∂x ∂y ∂z
∂u ∂u ∂u ∂u Ω0 −1 ∂p +u +v +w +2 w= +ν ∂t ∂x ∂y ∂z 1 − ct ρ ∂x
∂v ∂v ∂v ∂v −1 ∂p +u +v +w = +ν ∂t ∂x ∂y ∂z ρ ∂y
∂w ∂w ∂w ∂w Ω0 +u +v +w −2 u=ν ∂t ∂x ∂y ∂z 1 − ct
(3)
2 σB0 u ∂ 2u ∂ 2u ∂ 2u ν + 2+ 2 − + 0 , 2 ∂x ∂y ∂z ρ k 1 − ct (4)
∂ 2v ∂ 2v ∂ 2v + + ∂x2 ∂y 2 ∂z 2
−
ν v , k 0 1 − ct
(5)
2 ∂ 2w ∂ 2w ∂ 2w σB0 ν w + + − + , (6) ∂x2 ∂y 2 ∂z 2 ρ k 0 1 − ct
5
2 2 2 16σ ∗ T 3 2 kf σB 2 (u +w ) ∂T ∂T ∂ T ∂2T ∂2T + u ∂T + v + w = + 3(ρcp )kh∗ ∂∂yT2 + (ρcp0) (1−ct) + + ∂x ∂y ∂z (ρcp ) ∂x2 ∂y 2 ∂z 2 2 2 2 µnf ∂u 2 ∂v ∂w 2 ∂v ∂u ∂u ∂w 2 ∂v ∂w + (ρcp ) 2 ∂x + 2 ∂y + 2 ∂z + ∂x + ∂y + ∂z + ∂x + ∂z + ∂y 2 DT ∂T ∂C1 ∂T ∂C2 ∂T 2 ∂T ∂T ∂C1 ∂T ∂C1 ∂T ∂C2 ∂T ∂C2 +τ Dc1 ∂x ∂x + ∂y ∂y + ∂z ∂z + τ Dc2 ∂x ∂x + ∂y ∂y + ∂z ∂z + Th + + ∂x ∂y ∂T ∂t
(7)
∂C1 ∂C1 ∂C1 ∂ 2 C1 DT ∂ 2 T ∂C1 +u +v +w = Dc1 + − k1 C1 C22 , ∂t ∂x ∂y ∂z ∂y 2 Th ∂y 2
(8)
∂C2 ∂C2 ∂C2 ∂C2 ∂ 2 C2 DT ∂ 2 T +u +v +w = Dc2 + + k1 C1 C22 , 2 2 ∂t ∂x ∂y ∂x ∂y Th ∂y
(9)
∂T 2 ∂z
with
u = uw (x) =
ax , 1−ct
v=
−v0 , 1−ct
u = 0, v = Vh =
−c 2
∂C2 1 w = 0, T = Tw , Dc1 ∂C = −DC = k C and C = C at y = 0, 2 ∂y 2 1 w ∂y q . ν , w = 0, T → Th , C1 → Ch , C2 → 0 when y → h(t) a(1−ct) (10)
where u, v, w show the velocity component in x−, y−and z−directions, ρ the density, k 0 the porosity of medium, ν the kinematic viscosity, σ ∗ the Stefan Boltzmann constant, µnf
the viscosity, kf the thermal conductivity, k ∗ the mean absorption coefficient, σ nf the electric conductivity, Cp the specific heat, Ω0 angular speed, Dc1 and Dc2 the diffusion coefficients due to concentrations C1 and C2 respectively and DT the thermal diffusion. Considering the transformations u= f (η) , v = − g (η) , (η) , w = y −Th θ (η) = TTw−T , φ (η) = CCh1 , χ (η) = CCh2 , η = h(t) . ax 1−ct
One can write
0
p
0
h
000
f (4) + f f 00 + f f −
aν f 1−ct
ax 1−ct
β 00 000 00 0 3f + ηf − (M + λ) f − 2αg = 0, 2
η 0 0 g + f g − f g − β g + g − (M + λ) g − 2αf = 0, 2 00
0
0
(11)
(12)
(13)
02 02 02 1 β 2 002 00 [1 + T r] θ + f − η θ+Ec 4f + g 2 +Ed g + 2f + M f + g 2 +f θ0 +N tθ0 Pr 2 (14) 6
,
1 00 β 0 1 N t 00 K1 2 φ − ηφ + f φ0 + θ − φχ = 0, Sc 2 Sc N b Sc
(15)
1 N t 00 K1 2 δ 1 00 β 0 χ − ηχ + f χ0 + θ + φχ = 0, Sc 2 Sc N b Sc
(16)
0 f (0) = δ, f 0 (0) = 1, g (0) = 0, θ(0) = 1, φ (0) = K2 φ (0) , δ 1 χ0 (0) = K2 φ (0) , 0 f (1) = β , f (1) = 0, g (1) = 0, θ(1) = 0, φ(1) = 1, χ0 (0) = 0.
(17)
2
σ f B02 where β = ac indicates the squeezing parameter, M = aρ the magnetic paramef ν V0 ter, λ = akf0 the porosity parameter, α = Ωa0 the rotation parameter, δ = ah the suc µ (ρc ) 2 av p tion parameter, Pr = f kf f the Prandtl number, Ec = (cp ) (Twf−Th )h2 the local Eckert f 2 16σ ∗ Th3 Uw number, Ed = (cp ) (T the radiation parameter, the Eckert number, T r = w−Th ) 3kf k∗ f Tw −Th Dc1 Ch DT N t = τ Th the thermophoresis parameter, N b = τ ν f the Brownian moνf ν Dc2 the diffusion coefficients ratio, Sc = Dcf1 the Schmidt number, tion parameter, δ 1 = Dc 1 k C 2 h2 2 h(t) K1 = 1Dch1 the homogeneous reaction parameter and K2 = kDc the heterogeneous 1 reaction parameter.
For equal diffusion coefficients i.e. Dc1 = Dc2 we have
φ (η) + χ (η) = 1,
(18)
1 00 β 0 1 N t 00 K1 φ − ηφ + f φ0 + θ − φ (1 − φ)2 = 0, Sc 2 Sc N b Sc
(19)
From Eqs. (15) and (16) one has
0
φ (0) = K2 φ (0) , φ(1) = 1.
3 3.1
(20)
Physical quantities Skin friction coefficient
Surface drag force (Cf x ) of present flow is defined as
Cf x =
−2τ w , ρnf Uw2 7
(21)
here τ w shear stress is expressed as
τ = µnf
∂v ∂u + ∂x ∂y
,
(22)
Dimensionless form is 1
00
Cf x (Rex ) 2 = −2f (0) .
3.2
(23)
Nusselt number
Gradient of temperature (N ux ) is define as
N ux = −
xqw , knf (Tw − Th )
(24)
in which qw shows the heat flux is given as
qw = − knf
16σ ∗ Th3 + 3k ∗
∂T , ∂y
(25)
One can write
N ux (Rex ) Rex (=
4
xUw ) νf
−1 2
= − (1 + T r) θ0 (0).
(26)
shows the local Reynold number.
Entropy modeling
Mathematically entropy optimization is defined as
EG =
kf Th2
1+ µ
+ Thf
16σ ∗ T3 3knf k∗ h
2
∂u 2 ∂x
1 + RDc C1
∂T 2 ∂x
+ 2
+2 2 ∂C1 ∂y
∂v ∂y
+
∂T ∂y
+2
RDc1 T
2
+
∂w 2 ∂z
∂C1 ∂y
∂T 2 ∂z
+ ∂T ∂y
The final version are
8
+
σ f B02 Th (1−ct)
∂v ∂x
+
∂u ∂y
+
RDc2 C2
2
+
∂C2 ∂y
2
2
(u + w ) + ∂u ∂z
2
+
+
∂w 2 ∂x
RDc2 T
µf Th k0 (1−ct)
+ ∂w ∂y ∂C2 ∂T , ∂y ∂y +
∂v ∂z
{u + v + w } 2 2
2
(27)
2
0 2 02 02 0 0 0 0 φ φ L1 L2 2 NG = (1 + T r) θ + Tc L1 φ + (Tc +θ) θ φ + L2 (1−φ) − (Tc +θ) θ φ 002 02 02 02 02 1 2 2 2 2 Tc Br f + g + 4f + g + M f + g + λ f + Rex f + g
Bejan number is given as
.
(28)
Entropy generation due to heat and mass transfer (29) Total entropy generation Th where Br (= Pr Ec) the Brinkman number, Tc = Tw −Th the temperature difference parameter, L1 = RDckf1 Ch the diffusion parameter with respect to homogeneous reaction, EG Th2 h2 RDc2 Ch EG0 = k (T −T the shows entropy generation rate and L = the diffusion 2 2 2 2 kf ) T h Be =
f
w
h
h
variable with respect to heterogeneous.
5
Solution methodology
Suitable linear operators and initial guesses are f0 (η) = δ + η + (−4 + 3β − 6δ) η + (1 − β + 2δ) η 2 g0 (η) = η − η , θ0 (η) = 1 − η φ0 (η) = η 1 2
2
0000
3
00
00
00
£f = f , £g = g , £θ = θ , £φ = φ ,
(30)
(31)
with h 3 i 2 £f d1 η6 + d2 η2 + d3 η + d4 , £g = d5 η + d6 , £ =d η+d , £ =d η+d , θ
7
8
φ
9
(32)
10
in which di (i = 1 − 10) represents the arbitrary constants.
6
Convergence analysis
Convergence region is precise with the help of auxiliary variables (~f , ~g , ~θ and ~φ ) in HAM. The ~-curves for temperature velocity and concentration expression are sketched in Figs. 1 and 2. The convergence region of auxiliary variables are −1.7 ≤ ~f ≤ −0.2, −1.8 ≤ ~g ≤ 9
−0.2, −0.6 ≤ ~θ ≤ −0.2 and −1.1 ≤ ~φ ≤ −0.1. Table 1 it is scrutinized that our expressions starts convergence at 5th , 10th , 10th and 10th respectively. 00
0
0
0
Table 1: Different order of iterations for f (0), g (0), θ (0) and φ (0).
7
Order of approximations
−f 00 (0)
−g 0 (0)
−θ0 (0)
φ0 (0)
1
5.1969
0.04833
5.1969
0.04833
5
5.1966
0.000916
5.1966
0.000916
10
5.1966
0.000914
5.1966
0.000914
15
5.1966
0.000914
5.1966
0.000914
20
5.1966
0.000914
5.1966
0.000914
25
5.1966
0.000914
5.1966
0.000914
Computational results and discussion
Here features of various engineering variables on entropy optimization, temperature, Bejan number, velocity and concentration are scrutinized in this section.
7.1
Velocity
0 Influences of different variables like (M ), (β) and (λ) on velocity f (η), f (η) are plotted 0 in Figs. (3 − 8). Figs. (3) and (4) show the behaviors of (M ) on velocity f (η), f (η) .
Here velocity components are decreased versus larger (M ). Physically magnetic field creates resistive force, which causes resistance in flow region and thus velocity is diminished. Effect 0
of (β) on f (η) and f (η) is sketched in Figs. (5 and 6). In fact for larger (β) the upper plate moves downward which exerts more force on nanoliquids and therefore velocity components 0 0 f (η), f (η) are enhanced. Figs. (7 and 8) show the effect of (λ) on f (η) and f (η). Clearly for rising values of porosity parameter velocity components is decayed.
7.2
Temperature
Salient behaviors of pertinent variables like (T r), (Ec), (N t), (β) and (Pr) on temperature (θ (η)) is highlighted in Figs. (9 − 13). Fig. 9 illustrates the temperature (θ (η)) versus radiation parameter (T r). Clearly noticed that θ (η) is decaying function of (T r). Influence of Eckert number (Ec) on θ (η) is displayed in Fig. 10. From this Fig it is analyzed that 10
θ (η) boosts up against (Ec). Fig. 11 highlights the result of temperature field through thermophoresis parameter (N t). The augmentation of (N t) boosts up the temperature (θ (η)). Effect of (β) on θ (η) is illustrated in Fig. 12. For larger (β) the upper plate move towards downwards and inter atomic collision of nanoliquids enhances and consequently temperature boosts up. The variation of θ (η) with (Pr) is presented in Fig. 13. In fact for rising values of (Pr) thermal diffusivity decays and therefore θ (η) is diminished.
7.3
Concentration
Salient feature of thermophortic parameter (N t) on concentration (φ (η)) is presented in Fig. 14. In fact for larger (N t) thermophortic force enhances which conveyances nanoparticles from high to low temperature region. Therefore temperature enhances. Fig. 15 indicates the influences of (N b) on concentration (φ (η)). Here one can found that φ (η) decreases versus higher values of (N b). The variation of concentration (φ (η)) via (Sc) is sketched in Fig. 16. Here for larger (Sc) mass diffusivity reduces which leads to reduction in concentration. Behavior of (K2 ) on φ (η) is illustrated in Fig. 17. Here φ (η) decreases for higher values of (K2 ).
7.4
Entropy generation and Bejan number
Figs. (18 and 19) highlight the outcome of radiation parameter (T r) on (NG ) and (Be). From these Figs. one can found that both (NG ) and (Be) quantitatively boost up for larger (T r). Impact of (Br) on (NG ) and (Be) is illustrated in Figs. (20 and 21). For larger (Br) thermal conductivity decays and consequently it boosts up (NG ). While reverse effect is noticed for Be. Figs. (22 and 23) show the effect of (L1 ) on (NG ) and (Be). Clearly for higher estimation of (L1 ) (NG ) has enhancing effect while opposite holds for (Be). Characteristic of (L2 ) on (NG ) and (Be) is displayed in Figs. (24 and 25). Here one can found that both (NG ) and (Be) have increasing impact against (L2 ).
7.5
Physical quantities of interest
Influences of various interesting parameters on gradients of velocity (Cf x ) and temperature (N ux ) are scrutinized in Table 2. From Table 2 clearly observed that for larger estimation of porosity and magnetic parameters the (Cf x ) boosts up. One can found that heat transfer 11
rate increases against larger values of (M ) while opposite is hold for (λ). Table 2: Numerical results for Cf x and N ux . λ
M
Cf x
N ux
0.1
1.0
2.62732
0.865741
0.3
2.98765
0.658967
0.5
3.1467
0.523456
1.0
5.98674
4.65743
2.0
6.09832
4.84968
3.0
6.98364
5.02342
0.2
8
Conclusions
The following observations are observed on the basis of obtained results: • Velocity of fluid particles increases via squeezing variable(β). • Velocity decays via (λ) and (M ). • θ (η) boosts up via rising value of (Ec). • Temperature is increased against (β) and (β). • For larger (N t) and (T r) temperature boosts up. • φ (η) decays versus higher estimation of (N b), (Sc) and (K2 ). • Concentration enhances for rising values of (N t). • NG and Be have opposite impact for (Br) and (L1 ). • NG and Be have similar effect for (T r) and (L2 ). • Cf x increases for larger values of (λ) and (M ). • (λ) and (M ) have opposite impact on N ux .
12
References [1] K. Khanafer, K. Vafai, M. lightstone, Buoyancy-driven heat transfer enhancement in two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 36393653. [2] M. Sheikholeslami, M. G. Bandpy, D. D. Ganji, S. Soheil, Effect of magnetic field on natural convection in an inclined half annulus enclosure filled with Cu-water nanofluid using CVFEM, Adv. Powder technol. 24 (2013) 980-991. [3] T. Hayat, S. Qayyum, M. Imtiaz, A. Alsaedi, Comparative study of silver and copper water nanofluids with mixed convection and nonlinear thermal radiation, Int. J. Heat Mass Transf., 102 (2016) 723-732. [4] T. Hayat, M. Shafique, A. Tanveer, A. Alsaedi, Hall and ion slip effects on peristaltic flow of Jeffery nanofluid with joule heating, J. Magn. Magn. Mater .407 (2016) 51-59. [5] M. Farooq, M. I. Khan, M. Waqas, T. Hayat, M. I. Khan, MHD stagnation point flow of viscoelastic nanofluid with nonlinear radiation effects, J. Mol. Liq. 221 (2016) 1097-1103. [6] M. Sheikholeslami, M. G. Bandpy, R. Ellahi, A. Zeeshan, Simolation of MHD CuO-water nanofluid flow and convective heat transfer considering Lorentz forces, J. Magn Magn Mater 369 (2014) 69-80. [7] R. Ellahi, M. Hassan, A. Zeeshan, Shape effects of nanosize particles in Cu-H2 O nanofluid on entropy generation, Int. J. Heat and Mass Transf. 81 (2015) 449-456. [8] A. Malvandi, A. Ghasemi, D. D. Ganji, Thermal performance analysis of hydomagnetic Al2 O3 -water nanofluid flows inside a concentric microannulus considering nanoparticles migration and asymmetric heating, Int. J. Therm. Sci. 109 (2016) 10-22. [9] T. Hayat, S. Qayyum, A. Alsaedi, Entropy generation in flow with silver and copper nanoparticles,Collid. Surf. A. 539 (2018) 335-346. [10] F. E. Alsaadi,
T. Hayat,
M. I. Khan and F. E. Alsaadi,
Heat trans-
port and entropy optimization in flow of magneto-Williamson nanomaterial with Arrhenius activation energy, Comp. Meth. Prog. Biome. 183 2020 105051 https://doi.org/10.1016/j.cmpb.2019.105051. 13
[11] D. Nouri, M. Pasandideh-Fard, M. J. Oboodi, O. Mahian, A. Z. Sahin, Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden expansion and contraction, Int. J. Heat Mass Transf. 116 (2018) 1036–1043. [12] K. Hosseinzadeh, A. Asadi, A. R. Mogharrebi, J. Khalesi, S. M. Mousavisani and D. D. Ganji, Entropy generation analysis of (CH2OH)2 containing CNTs nanofluid flow under effect of MHD and thermal radiation, Case Stud. Ther. Eng. 14 (2019) 100482 https://doi.org/10.1016/j.csite.2019.100482. [13] S. Ahmad, M. I Khan, T. Hayat, M. I Khan, Entropy generation optimization and unsteady squeezing flow of viscous fluid with five different shapes of nanoparticles, Colliods and Surfaces A 554(2018)197-210. [14] J. Qing, M. M. Bhatti, M. A. Abbas, M. M. Rashidi and M. E. S. Ali, Entropy generation on MHD casson nanofluid flow over a porous/shrinking surface, Entropy, 18 (2016) e18040123. [15] H. sithole, H. Mondal and P. Sibanda, Entropy generation in second grade magnetohydrodynamic nanofluid over a convectively heated stretching sheet with nonlinear thermal radiation and viscous dissipation, Res. Phy. 9 (2018) 1077-1085. [16] D. Nouri, M.P. Fard, M. J. Oboodi, O. Mahian and A. Z. Sahin, Entropy generation analysis of nanofluid fluid flow over a spherical heat source inside a chanel with sudden expansion and contraction , Int. J. Heat Mass Tran. 116 (2018) 1036-1043. [17] M. I. Afridi, M. Qasim, I. Khan and I. Tlili, Entropy generation in MHD mixed Convection Stagnation-point flow in the presence of Joule and frictional heating, Case Stud. Ther. Eng. 12 (2018) 292-300. [18] M. I. Khan, S. A. Khan, T. Hayat, M. F. Javed and M. Waqas, Entropy generation in radiative flow of Ree-Eyring fluid due to due rotating disks, Int. J. Num. Meth. Heat Fluid Flow (2019) https://doi.org/10.1108/HFF-11-2018-0683. [19] F. Shah, M. I. Khan, T. Hayat, M. I. Khan, A. Alsaedi and W. A. Khan, Theoretical and mathematical analysis of entropy generation in fluid flow subject to aluminum and ethylene glycol nanoparticles, Comp. Meth. Prog. Biome. 182 (2019) 105057 https://doi.org/10.1016/j.cmpb.2019.105057. 14
[20] R. Kumar, R. Kumar, M. Sheikholeslami and A. J. Chamkha, Irreversibility analysis of the three dimensional flow of carbon nanotubes due to nonlinear thermal radiation and quartic chemical reactions, J. Mol. Liq. 274 (2019) 379-392. [21] M. A. E. Aziz and A. A. Afify, MHD Casson fluid flow over a stretching sheet with entropy generation analysis and Hall influence, Entropy 21 (2019) 592 https://doi.org/10.3390/e21060592. [22] M. I. Khan, S. A. Khan, T. Hayat and A. Alsaedi, Entropy optimization in magnetohydrodynamic flow of third-grade Nnanofluid with viscous dissipation and chemical reaction, Iran. J. Sci. Technol. Trans. Sci. 43 (2019) 2679–2689. [23] T. Hayat, M. I. Khan, S. Qayyum, A. Alsaedi and M. I. Khan, New thermodynamics of entropy generation minimization with nonlinear thermal radiation and nanomaterials, Phy. Lett. A 382 (2018) 749-760. [24] T. Hayat, M.I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, Int. J. Heat Mass Transf., 99 (2016) 702-710. [25] K. L. Hsiao, Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet, Appl. Thermal Eng., 98 (2016) 850-861. [26] K. L. Hsiao, Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature, Int. J. Heat Mass Transf., 112 (2017) 983-990. [27] M.I. Khan, M. Waqas, T. Hayat and A. Alsaedi, A comparative study of Casson fluid with homogeneous-heterogeneous reactions, Journal of Colloid and Interface Science, 498 (2017) 85-90. [28] K. L. Hsiao, To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method, Energy, 130 (2017) 486-499. [29] K. L. Hsiao, Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects, Appl. Thermal Eng., 112 (2017) 1281-1288. 15
Figure Captions Fig. 1 } curve for }f . Fig. 2 } curves for }g , }θ , and }φ . Fig. 3: f 0 (η) via M . Fig. 4: f (η) via M . Fig. 5: f 0 (η) via β. Fig. 6: f (η) via β. Fig. 7: f 0 (η) via λ. Fig. 8: f (η) via λ. Fig. 9: θ (η) via T r. Fig. 10: θ (η) via Ec. Fig. 11: θ (η) via N t. Fig. 12: θ (η) via β. Fig. 13: θ (η) via Pr. Fig. 14: φ (η) via N t. Fig. 15: φ (η) via N b. Fig. 16: φ (η) via Sc. Fig. 17: φ (η) via K2 . Fig. 18: NG via T r. Fig. 19: Be via T r. Fig. 20: NG via Br. Fig. 21: Be via Br. Fig. 22: NG via L1 . Fig. 23: Be via L1 . Fig. 24: NG via L2 . Fig. 25: Be via L2 .
16
Declaration of Competing Interest The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere.
17
Figure Captions
Fig. 1 } curve for }f .
Fig. 2 } curves for }g , }θ , and }φ .
Fig. 3: f 0 (η) via M .
15
Fig. 4: f (η) via M .
Fig. 5: f 0 (η) via β.
Fig. 6: f (η) via β.
Fig. 7: f 0 (η) via λ.
16
Fig. 8: f (η) via λ.
Fig. 9: θ (η) via T r.
Fig. 10: θ (η) via Ec.
Fig. 11: θ (η) via N t.
17
Fig. 12: θ (η) via β.
Fig. 13: θ (η) via Pr.
Fig. 14: φ (η) via N t.
Fig. 15: φ (η) via N b. 18
Fig. 16: φ (η) via Sc.
Fig. 17: φ (η) via K2 .
Fig. 18: NG via T r.
Fig. 19: Be via T r.
19
Fig. 20: NG via Br.
Fig. 21: Be via Br.
Fig. 22: NG via L1 .
Fig. 23: Be via L1 .
20
Fig. 24: NG via L2 .
Fig. 25: Be via L2 .
21
Journal Pre-proof
Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates M. Ijaz Khan, Mujeeb ur Rahman, Sohail A. Khan, T. Hayat, M. Imran Khan PII: DOI: Reference:
S0169-2607(19)31730-4 https://doi.org/10.1016/j.cmpb.2019.105149 COMM 105149
To appear in:
Computer Methods and Programs in Biomedicine
Received date: Revised date: Accepted date:
7 October 2019 17 October 2019 18 October 2019
Please cite this article as: M. Ijaz Khan, Mujeeb ur Rahman, Sohail A. Khan, T. Hayat, M. Imran Khan, Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105149
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Highlights • Here we have discussed MHD squeezed flow between two plates. • The upper plate squeezed towards the lower plates. • Energy equation is modeled subject to dissipation and Joule heating. • Entropy rate is calculated through thermodynamics law.
1
Evaluation of entropy generation in cubic autocatalytic unsteady squeezing flow of nanofluid between two parallel plates M. Ijaz Khana,1 , Mujeeb ur Rahmanb , Sohail A. Khana , T. Hayata,c and M. Imran Khand,2 a
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
b
Department of Mathematics, Karakoram International University Gilgit 15100, Pakistan c
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University P. O. Box 80207, Jeddah 21589, Saudi Arabia
d
School of Engineering, University of Portsmouth, Winston Churchill Avenue Portsmouth PO1 2UP, United Kingdom Abstract: Background: Nanomaterials have advanced behaviors that make them possibly beneficial in
various applications in mass and heat transports such as engine cooling, pharmaceutical processes, fuel cells, engine cooling and domestic refrigerator etc. Therefore here we deliberated the entropy generation in unsteady magnetohydrodynamic squeezing flow of viscous nanomaterials between two parallel plates. The upper plate is squeezing towards lower plate. The lower plate exhibits porous character. Energy attributes are discussed through heat flux, dissipation and Joule heating. Furthermore the irreversibility analysis with cubic autocatalysis chemical reaction is also accounted.
Methods: Nonlinear differential systems are converted to ordinary differential system by transformations. For convergent series solution the given system are solved by homotopy analysis method (HAM).
Results: Characteristics of various interesting variables on velocity, Bejan number, concentration, entropy optimization and temperature are deliberated through graphs. Gradient of velocity
(Cf x ) and Nusselt number (N ux ) are numerically computed against various physical variables. Entropy generation and Bejan number both quantitatively enhance versus radiation parameter. For larger squeezing parameter the velocity and temperature field are increased.
Conclusions: The obtained results show that for larger squeezing parameter the velocity field boosts up. Velocity have opposite impact For larger magnetic and porosity parameters. Temperature is decreased for higher values of radiation parameter and Prandtl number. Temperature and concentration have same outcome for thermophoresis parameter. Entropy generation and Bejan 1 2
Corresponding author E-mail address: ijazfmg [email protected] (M. I. Khan) Corresponding author E-mail address: [email protected] (M. I. Khan)
2
number both quantitatively enhance versus radiation parameter, while reverse is hold for Brinkman number.
Keywords: Entropy generation; Thermal radiation; Homogenous heterogenous reactions; Viscous dissipation; Squeezing flow and Joule heating.
1
Introduction
Colloidal suspension of nano size solid particles (1-100nm) in traditionally working materials is known as nanoliquids. Most commonly metals are used as nanoparticles like (Cu, Ag, Au, F e), carbon nanotubes, nitride (SiN, SiC, T iC) and metallic oxides (T iO2 , Al2 O3 , CuO). Heat conduction phenomenon of conventionally working materials are improved by enclosure of nanoparticles into it. Nanoparticles have wide range of application in heat exchanger, compressor, combustors, computer processor, hybrid powered engines and air conditioners etc. Khanafer et al. [1] studied heat transfer enhancement through nanoparticles. Sheikholeslami et al. [2] discussed effect of magnetohydrodynamic on flow of copper water nanoliquid with natural convection in an enclosure. Mixed convective flow of Ag and Cu nanoparticles with heat flux is illustrated by Hayat et al. .[3]. Hayat et al. [4] discussed the study of Jeffrey nanomaterials flow with Hall and ion effect. Farooq et al. [5] analyzed viscoelastic nanofluid flow with stagnation point and heat flux. Magnetohydrodynamic effect on CuO-water nanofluid with mix convection is presented by Sheikholeslami et al. [6]. Ellahi et al. [7] examined shape effect of nanoparticles in Cu − H2 O on entropy generation. Thermal behaviors of hydromagnetic Al2 O3 -water nanoliquid in a micro annular cylinder is examined by Malvindi et al. [8]. Hayat et al. [9] examined copper (Cu) and silver (Ag) nanoparticles flow with entropy optimization. Entropy generation is the measurement of energy in a system, which are not used to do significant work. As a significant of thermodynamics second law entropy generates. According to thermodynamics second law entropy optimization of an isolated system does not decays over time. Entropy of a reversible processes never change, while in an irreversible processed entropy always enhances and as a results total entropy increases. These irreversible processes contains radiation, Joule heating, fluid viscosity, chemical reaction, friction between solid surfaces and diffusion in a system. For effectiveness of equipment entropy generation is minimized. Entropy minimization is used in curved pipes, microchannel, reactor, chillers, reg3
ular graded materials, fuel cell and helical coil. Alsaadi et al. [10] discussed the irreversibility effect in Williamson nanoliquids with MHD and activation energy over a stretchable surface. Nouri et al. [11] studied the entropy optimization in nanoliquids flow with heat generation/absorption. Entropy optimization study in magnetohydrodynamic flow of CNTs based nanoliquids with Joule heating and heat flux due to stretchable rotating disks is discussed by Hosseinzadeh et al. [12]. Impact of irreversibility analysis in unsteady MHD squeezing flow of viscous nanomaterial between two parallel plates is highlighted by Ahmad et al. [13]. Qing et al. [14] discussed entropy optimization analysis in magnetohydrodynamic flow of Casson nanoliquids with chemical reaction and radiation over a porous medium. Effect of entropy optimization and heat flux in magnetohydrodynamic flow of second grade nanoliquids is highlighted by Sith et al. [15]. Nouri et al. [16] reported the exploration of entropy optimization in nanoliquids flow inside a channel. Impact of frictional and Joule heating in mixed convective magnetohydrodynamic viscous fluid flow with entropy optimization is explored by Afridi et al. [17]. Physical behaviors of irreversibility in Ree-Eyring nanoliquids with cubic autocatalysis chemical reaction is presented by Khan et al. [18]. Shah et al. [19] studied impact of entropy optimization and heat flux in viscous flow of nanoliquids by a stretchable Sheet. Kumar et al. [20] discussed the entropy generation in CNTs nanoliquids with heat flux and quartic autocatalysis chemical reaction. Aziz et al. [21] reported the irreversibility analysis in magnetohydrodynamic flow of Casson fluids with Hall current. Khan et al. [22] studied the MHD dissipative flow of third grade nanoliquids with chemical reaction and entropy generation. Irreversibility analysis on magnetohydrodynamic flow of viscous liquids with heat generation/absorption and heat flux by a stretching surface is illustrated by Hayat et al. [23]. In this article irreversibility analysis in 3D squeezing flow of nanofluids between two parallel plates is investigated. Energy attributes are discussed through heat flux, dissipation and Joule heating are also considered. Furthermore cubic autocatalysis chemical reactions are also taken into account. Nonlinear system are altered to ordinary one by suitable transformations. For series solution the obtained system are solved HAM [24-29]. Effect of numerous engineering parameters on velocity, Bejan number, concentration, entropy generation and temperature are deliberated through graphs. Gradients of temperature and velocity are numerically computed via various flow parameters.
4
2
Mathematical modeling
Here we discussed incompressible unsteady magnetohydrodynamic squeezing flow of viscous nanoliquid between two parallel plates. The flow is caused due to stretching of sheet. Energy characteristics are taken into account through Joule heating, dissipation and heat flux. Furthermore physical aspects of cubic autocatalysis chemical reactions with entropy optimization are also examined. The upper plate is squeezing towards the lower plate with squeez ing velocity Vh = dh(t) . The fluid is electrically conducting with magnetic field strength dt ax 0 B = √B the stretching velocity with stretching rate . Let us assume that Uw = 1−ct 1−ct q a > 0. h(t) = ν(1−ct) is the distance between the two plates. a Isothermal homogeneous-heterogeneous chemical reaction is given as [25-27]: A∗ + 2B ∗ → 3B ∗ , rate = k1 C1 C22 ,
(1)
On the surface an isothermal chemical reaction is A∗ + B ∗ = 3B ∗ , rate = k2 C2 ,
(2)
in which A∗ and B ∗ represent the chemical reaction species, k1 and k2 the reaction rates and C1 and C2 the concentrations. The boundary layer equations are ∂u ∂u ∂w + + = 0, ∂x ∂y ∂z
∂u ∂u ∂u ∂u Ω0 −1 ∂p +u +v +w +2 w= +ν ∂t ∂x ∂y ∂z 1 − ct ρ ∂x
∂v ∂v ∂v ∂v −1 ∂p +u +v +w = +ν ∂t ∂x ∂y ∂z ρ ∂y
∂w ∂w ∂w ∂w Ω0 +u +v +w −2 u=ν ∂t ∂x ∂y ∂z 1 − ct
(3)
2 σB0 u ∂ 2u ∂ 2u ∂ 2u ν + 2+ 2 − + 0 , 2 ∂x ∂y ∂z ρ k 1 − ct (4)
∂ 2v ∂ 2v ∂ 2v + + ∂x2 ∂y 2 ∂z 2
−
ν v , k 0 1 − ct
(5)
2 ∂ 2w ∂ 2w ∂ 2w σB0 ν w + + − + , (6) ∂x2 ∂y 2 ∂z 2 ρ k 0 1 − ct
5
2 2 2 16σ ∗ T 3 2 kf σB 2 (u +w ) ∂T ∂T ∂ T ∂2T ∂2T + u ∂T + v + w = + 3(ρcp )kh∗ ∂∂yT2 + (ρcp0) (1−ct) + + ∂x ∂y ∂z (ρcp ) ∂x2 ∂y 2 ∂z 2 2 2 2 µnf ∂u 2 ∂v ∂w 2 ∂v ∂u ∂u ∂w 2 ∂v ∂w + (ρcp ) 2 ∂x + 2 ∂y + 2 ∂z + ∂x + ∂y + ∂z + ∂x + ∂z + ∂y 2 DT ∂T ∂C1 ∂T ∂C2 ∂T 2 ∂T ∂T ∂C1 ∂T ∂C1 ∂T ∂C2 ∂T ∂C2 +τ Dc1 ∂x ∂x + ∂y ∂y + ∂z ∂z + τ Dc2 ∂x ∂x + ∂y ∂y + ∂z ∂z + Th + + ∂x ∂y ∂T ∂t
(7)
∂C1 ∂C1 ∂C1 ∂ 2 C1 DT ∂ 2 T ∂C1 +u +v +w = Dc1 + − k1 C1 C22 , ∂t ∂x ∂y ∂z ∂y 2 Th ∂y 2
(8)
∂C2 ∂C2 ∂C2 ∂C2 ∂ 2 C2 DT ∂ 2 T +u +v +w = Dc2 + + k1 C1 C22 , 2 2 ∂t ∂x ∂y ∂x ∂y Th ∂y
(9)
∂T 2 ∂z
with
u = uw (x) =
ax , 1−ct
v=
−v0 , 1−ct
u = 0, v = Vh =
−c 2
∂C2 1 w = 0, T = Tw , Dc1 ∂C = −DC = k C and C = C at y = 0, 2 ∂y 2 1 w ∂y q . ν , w = 0, T → Th , C1 → Ch , C2 → 0 when y → h(t) a(1−ct) (10)
where u, v, w show the velocity component in x−, y−and z−directions, ρ the density, k 0 the porosity of medium, ν the kinematic viscosity, σ ∗ the Stefan Boltzmann constant, µnf
the viscosity, kf the thermal conductivity, k ∗ the mean absorption coefficient, σ nf the electric conductivity, Cp the specific heat, Ω0 angular speed, Dc1 and Dc2 the diffusion coefficients due to concentrations C1 and C2 respectively and DT the thermal diffusion. Considering the transformations u= f (η) , v = − g (η) , (η) , w = y −Th θ (η) = TTw−T , φ (η) = CCh1 , χ (η) = CCh2 , η = h(t) . ax 1−ct
One can write
0
p
0
h
000
f (4) + f f 00 + f f −
aν f 1−ct
ax 1−ct
β 00 000 00 0 3f + ηf − (M + λ) f − 2αg = 0, 2
η 0 0 g + f g − f g − β g + g − (M + λ) g − 2αf = 0, 2 00
0
0
(11)
(12)
(13)
02 02 02 1 β 2 002 00 [1 + T r] θ + f − η θ+Ec 4f + g 2 +Ed g + 2f + M f + g 2 +f θ0 +N tθ0 Pr 2 (14) 6
,
1 00 β 0 1 N t 00 K1 2 φ − ηφ + f φ0 + θ − φχ = 0, Sc 2 Sc N b Sc
(15)
1 N t 00 K1 2 δ 1 00 β 0 χ − ηχ + f χ0 + θ + φχ = 0, Sc 2 Sc N b Sc
(16)
0 f (0) = δ, f 0 (0) = 1, g (0) = 0, θ(0) = 1, φ (0) = K2 φ (0) , δ 1 χ0 (0) = K2 φ (0) , 0 f (1) = β , f (1) = 0, g (1) = 0, θ(1) = 0, φ(1) = 1, χ0 (0) = 0.
(17)
2
σ f B02 where β = ac indicates the squeezing parameter, M = aρ the magnetic paramef ν V0 ter, λ = akf0 the porosity parameter, α = Ωa0 the rotation parameter, δ = ah the suc µ (ρc ) 2 av p tion parameter, Pr = f kf f the Prandtl number, Ec = (cp ) (Twf−Th )h2 the local Eckert f 2 16σ ∗ Th3 Uw number, Ed = (cp ) (T the radiation parameter, the Eckert number, T r = w−Th ) 3kf k∗ f Tw −Th Dc1 Ch DT N t = τ Th the thermophoresis parameter, N b = τ ν f the Brownian moνf ν Dc2 the diffusion coefficients ratio, Sc = Dcf1 the Schmidt number, tion parameter, δ 1 = Dc 1 k C 2 h2 2 h(t) K1 = 1Dch1 the homogeneous reaction parameter and K2 = kDc the heterogeneous 1 reaction parameter.
For equal diffusion coefficients i.e. Dc1 = Dc2 we have
φ (η) + χ (η) = 1,
(18)
1 00 β 0 1 N t 00 K1 φ − ηφ + f φ0 + θ − φ (1 − φ)2 = 0, Sc 2 Sc N b Sc
(19)
From Eqs. (15) and (16) one has
0
φ (0) = K2 φ (0) , φ(1) = 1.
3 3.1
(20)
Physical quantities Skin friction coefficient
Surface drag force (Cf x ) of present flow is defined as
Cf x =
−2τ w , ρnf Uw2 7
(21)
here τ w shear stress is expressed as
τ = µnf
∂v ∂u + ∂x ∂y
,
(22)
Dimensionless form is 1
00
Cf x (Rex ) 2 = −2f (0) .
3.2
(23)
Nusselt number
Gradient of temperature (N ux ) is define as
N ux = −
xqw , knf (Tw − Th )
(24)
in which qw shows the heat flux is given as
qw = − knf
16σ ∗ Th3 + 3k ∗
∂T , ∂y
(25)
One can write
N ux (Rex ) Rex (=
4
xUw ) νf
−1 2
= − (1 + T r) θ0 (0).
(26)
shows the local Reynold number.
Entropy modeling
Mathematically entropy optimization is defined as
EG =
kf Th2
1+ µ
+ Thf
16σ ∗ T3 3knf k∗ h
2
∂u 2 ∂x
1 + RDc C1
∂T 2 ∂x
+ 2
+2 2 ∂C1 ∂y
∂v ∂y
+
∂T ∂y
+2
RDc1 T
2
+
∂w 2 ∂z
∂C1 ∂y
∂T 2 ∂z
+ ∂T ∂y
The final version are
8
+
σ f B02 Th (1−ct)
∂v ∂x
+
∂u ∂y
+
RDc2 C2
2
+
∂C2 ∂y
2
2
(u + w ) + ∂u ∂z
2
+
+
∂w 2 ∂x
RDc2 T
µf Th k0 (1−ct)
+ ∂w ∂y ∂C2 ∂T , ∂y ∂y +
∂v ∂z
{u + v + w } 2 2
2
(27)
2
0 2 02 02 0 0 0 0 φ φ L1 L2 2 NG = (1 + T r) θ + Tc L1 φ + (Tc +θ) θ φ + L2 (1−φ) − (Tc +θ) θ φ 002 02 02 02 02 1 2 2 2 2 Tc Br f + g + 4f + g + M f + g + λ f + Rex f + g
Bejan number is given as
.
(28)
Entropy generation due to heat and mass transfer (29) Total entropy generation Th where Br (= Pr Ec) the Brinkman number, Tc = Tw −Th the temperature difference parameter, L1 = RDckf1 Ch the diffusion parameter with respect to homogeneous reaction, EG Th2 h2 RDc2 Ch EG0 = k (T −T the shows entropy generation rate and L = the diffusion 2 2 2 2 kf ) T h Be =
f
w
h
h
variable with respect to heterogeneous.
5
Solution methodology
Suitable linear operators and initial guesses are f0 (η) = δ + η + (−4 + 3β − 6δ) η + (1 − β + 2δ) η 2 g0 (η) = η − η , θ0 (η) = 1 − η φ0 (η) = η 1 2
2
0000
3
00
00
00
£f = f , £g = g , £θ = θ , £φ = φ ,
(30)
(31)
with h 3 i 2 £f d1 η6 + d2 η2 + d3 η + d4 , £g = d5 η + d6 , £ =d η+d , £ =d η+d , θ
7
8
φ
9
(32)
10
in which di (i = 1 − 10) represents the arbitrary constants.
6
Convergence analysis
Convergence region is precise with the help of auxiliary variables (~f , ~g , ~θ and ~φ ) in HAM. The ~-curves for temperature velocity and concentration expression are sketched in Figs. 1 and 2. The convergence region of auxiliary variables are −1.7 ≤ ~f ≤ −0.2, −1.8 ≤ ~g ≤ 9
−0.2, −0.6 ≤ ~θ ≤ −0.2 and −1.1 ≤ ~φ ≤ −0.1. Table 1 it is scrutinized that our expressions starts convergence at 5th , 10th , 10th and 10th respectively. 00
0
0
0
Table 1: Different order of iterations for f (0), g (0), θ (0) and φ (0).
7
Order of approximations
−f 00 (0)
−g 0 (0)
−θ0 (0)
φ0 (0)
1
5.1969
0.04833
5.1969
0.04833
5
5.1966
0.000916
5.1966
0.000916
10
5.1966
0.000914
5.1966
0.000914
15
5.1966
0.000914
5.1966
0.000914
20
5.1966
0.000914
5.1966
0.000914
25
5.1966
0.000914
5.1966
0.000914
Computational results and discussion
Here features of various engineering variables on entropy optimization, temperature, Bejan number, velocity and concentration are scrutinized in this section.
7.1
Velocity
0 Influences of different variables like (M ), (β) and (λ) on velocity f (η), f (η) are plotted 0 in Figs. (3 − 8). Figs. (3) and (4) show the behaviors of (M ) on velocity f (η), f (η) .
Here velocity components are decreased versus larger (M ). Physically magnetic field creates resistive force, which causes resistance in flow region and thus velocity is diminished. Effect 0
of (β) on f (η) and f (η) is sketched in Figs. (5 and 6). In fact for larger (β) the upper plate moves downward which exerts more force on nanoliquids and therefore velocity components 0 0 f (η), f (η) are enhanced. Figs. (7 and 8) show the effect of (λ) on f (η) and f (η). Clearly for rising values of porosity parameter velocity components is decayed.
7.2
Temperature
Salient behaviors of pertinent variables like (T r), (Ec), (N t), (β) and (Pr) on temperature (θ (η)) is highlighted in Figs. (9 − 13). Fig. 9 illustrates the temperature (θ (η)) versus radiation parameter (T r). Clearly noticed that θ (η) is decaying function of (T r). Influence of Eckert number (Ec) on θ (η) is displayed in Fig. 10. From this Fig it is analyzed that 10
θ (η) boosts up against (Ec). Fig. 11 highlights the result of temperature field through thermophoresis parameter (N t). The augmentation of (N t) boosts up the temperature (θ (η)). Effect of (β) on θ (η) is illustrated in Fig. 12. For larger (β) the upper plate move towards downwards and inter atomic collision of nanoliquids enhances and consequently temperature boosts up. The variation of θ (η) with (Pr) is presented in Fig. 13. In fact for rising values of (Pr) thermal diffusivity decays and therefore θ (η) is diminished.
7.3
Concentration
Salient feature of thermophortic parameter (N t) on concentration (φ (η)) is presented in Fig. 14. In fact for larger (N t) thermophortic force enhances which conveyances nanoparticles from high to low temperature region. Therefore temperature enhances. Fig. 15 indicates the influences of (N b) on concentration (φ (η)). Here one can found that φ (η) decreases versus higher values of (N b). The variation of concentration (φ (η)) via (Sc) is sketched in Fig. 16. Here for larger (Sc) mass diffusivity reduces which leads to reduction in concentration. Behavior of (K2 ) on φ (η) is illustrated in Fig. 17. Here φ (η) decreases for higher values of (K2 ).
7.4
Entropy generation and Bejan number
Figs. (18 and 19) highlight the outcome of radiation parameter (T r) on (NG ) and (Be). From these Figs. one can found that both (NG ) and (Be) quantitatively boost up for larger (T r). Impact of (Br) on (NG ) and (Be) is illustrated in Figs. (20 and 21). For larger (Br) thermal conductivity decays and consequently it boosts up (NG ). While reverse effect is noticed for Be. Figs. (22 and 23) show the effect of (L1 ) on (NG ) and (Be). Clearly for higher estimation of (L1 ) (NG ) has enhancing effect while opposite holds for (Be). Characteristic of (L2 ) on (NG ) and (Be) is displayed in Figs. (24 and 25). Here one can found that both (NG ) and (Be) have increasing impact against (L2 ).
7.5
Physical quantities of interest
Influences of various interesting parameters on gradients of velocity (Cf x ) and temperature (N ux ) are scrutinized in Table 2. From Table 2 clearly observed that for larger estimation of porosity and magnetic parameters the (Cf x ) boosts up. One can found that heat transfer 11
rate increases against larger values of (M ) while opposite is hold for (λ). Table 2: Numerical results for Cf x and N ux . λ
M
Cf x
N ux
0.1
1.0
2.62732
0.865741
0.3
2.98765
0.658967
0.5
3.1467
0.523456
1.0
5.98674
4.65743
2.0
6.09832
4.84968
3.0
6.98364
5.02342
0.2
8
Conclusions
The following observations are observed on the basis of obtained results: • Velocity of fluid particles increases via squeezing variable(β). • Velocity decays via (λ) and (M ). • θ (η) boosts up via rising value of (Ec). • Temperature is increased against (β) and (β). • For larger (N t) and (T r) temperature boosts up. • φ (η) decays versus higher estimation of (N b), (Sc) and (K2 ). • Concentration enhances for rising values of (N t). • NG and Be have opposite impact for (Br) and (L1 ). • NG and Be have similar effect for (T r) and (L2 ). • Cf x increases for larger values of (λ) and (M ). • (λ) and (M ) have opposite impact on N ux .
12
References [1] K. Khanafer, K. Vafai, M. lightstone, Buoyancy-driven heat transfer enhancement in two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 36393653. [2] M. Sheikholeslami, M. G. Bandpy, D. D. Ganji, S. Soheil, Effect of magnetic field on natural convection in an inclined half annulus enclosure filled with Cu-water nanofluid using CVFEM, Adv. Powder technol. 24 (2013) 980-991. [3] T. Hayat, S. Qayyum, M. Imtiaz, A. Alsaedi, Comparative study of silver and copper water nanofluids with mixed convection and nonlinear thermal radiation, Int. J. Heat Mass Transf., 102 (2016) 723-732. [4] T. Hayat, M. Shafique, A. Tanveer, A. Alsaedi, Hall and ion slip effects on peristaltic flow of Jeffery nanofluid with joule heating, J. Magn. Magn. Mater .407 (2016) 51-59. [5] M. Farooq, M. I. Khan, M. Waqas, T. Hayat, M. I. Khan, MHD stagnation point flow of viscoelastic nanofluid with nonlinear radiation effects, J. Mol. Liq. 221 (2016) 1097-1103. [6] M. Sheikholeslami, M. G. Bandpy, R. Ellahi, A. Zeeshan, Simolation of MHD CuO-water nanofluid flow and convective heat transfer considering Lorentz forces, J. Magn Magn Mater 369 (2014) 69-80. [7] R. Ellahi, M. Hassan, A. Zeeshan, Shape effects of nanosize particles in Cu-H2 O nanofluid on entropy generation, Int. J. Heat and Mass Transf. 81 (2015) 449-456. [8] A. Malvandi, A. Ghasemi, D. D. Ganji, Thermal performance analysis of hydomagnetic Al2 O3 -water nanofluid flows inside a concentric microannulus considering nanoparticles migration and asymmetric heating, Int. J. Therm. Sci. 109 (2016) 10-22. [9] T. Hayat, S. Qayyum, A. Alsaedi, Entropy generation in flow with silver and copper nanoparticles,Collid. Surf. A. 539 (2018) 335-346. [10] F. E. Alsaadi,
T. Hayat,
M. I. Khan and F. E. Alsaadi,
Heat trans-
port and entropy optimization in flow of magneto-Williamson nanomaterial with Arrhenius activation energy, Comp. Meth. Prog. Biome. 183 2020 105051 https://doi.org/10.1016/j.cmpb.2019.105051. 13
[11] D. Nouri, M. Pasandideh-Fard, M. J. Oboodi, O. Mahian, A. Z. Sahin, Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden expansion and contraction, Int. J. Heat Mass Transf. 116 (2018) 1036–1043. [12] K. Hosseinzadeh, A. Asadi, A. R. Mogharrebi, J. Khalesi, S. M. Mousavisani and D. D. Ganji, Entropy generation analysis of (CH2OH)2 containing CNTs nanofluid flow under effect of MHD and thermal radiation, Case Stud. Ther. Eng. 14 (2019) 100482 https://doi.org/10.1016/j.csite.2019.100482. [13] S. Ahmad, M. I Khan, T. Hayat, M. I Khan, Entropy generation optimization and unsteady squeezing flow of viscous fluid with five different shapes of nanoparticles, Colliods and Surfaces A 554(2018)197-210. [14] J. Qing, M. M. Bhatti, M. A. Abbas, M. M. Rashidi and M. E. S. Ali, Entropy generation on MHD casson nanofluid flow over a porous/shrinking surface, Entropy, 18 (2016) e18040123. [15] H. sithole, H. Mondal and P. Sibanda, Entropy generation in second grade magnetohydrodynamic nanofluid over a convectively heated stretching sheet with nonlinear thermal radiation and viscous dissipation, Res. Phy. 9 (2018) 1077-1085. [16] D. Nouri, M.P. Fard, M. J. Oboodi, O. Mahian and A. Z. Sahin, Entropy generation analysis of nanofluid fluid flow over a spherical heat source inside a chanel with sudden expansion and contraction , Int. J. Heat Mass Tran. 116 (2018) 1036-1043. [17] M. I. Afridi, M. Qasim, I. Khan and I. Tlili, Entropy generation in MHD mixed Convection Stagnation-point flow in the presence of Joule and frictional heating, Case Stud. Ther. Eng. 12 (2018) 292-300. [18] M. I. Khan, S. A. Khan, T. Hayat, M. F. Javed and M. Waqas, Entropy generation in radiative flow of Ree-Eyring fluid due to due rotating disks, Int. J. Num. Meth. Heat Fluid Flow (2019) https://doi.org/10.1108/HFF-11-2018-0683. [19] F. Shah, M. I. Khan, T. Hayat, M. I. Khan, A. Alsaedi and W. A. Khan, Theoretical and mathematical analysis of entropy generation in fluid flow subject to aluminum and ethylene glycol nanoparticles, Comp. Meth. Prog. Biome. 182 (2019) 105057 https://doi.org/10.1016/j.cmpb.2019.105057. 14
[20] R. Kumar, R. Kumar, M. Sheikholeslami and A. J. Chamkha, Irreversibility analysis of the three dimensional flow of carbon nanotubes due to nonlinear thermal radiation and quartic chemical reactions, J. Mol. Liq. 274 (2019) 379-392. [21] M. A. E. Aziz and A. A. Afify, MHD Casson fluid flow over a stretching sheet with entropy generation analysis and Hall influence, Entropy 21 (2019) 592 https://doi.org/10.3390/e21060592. [22] M. I. Khan, S. A. Khan, T. Hayat and A. Alsaedi, Entropy optimization in magnetohydrodynamic flow of third-grade Nnanofluid with viscous dissipation and chemical reaction, Iran. J. Sci. Technol. Trans. Sci. 43 (2019) 2679–2689. [23] T. Hayat, M. I. Khan, S. Qayyum, A. Alsaedi and M. I. Khan, New thermodynamics of entropy generation minimization with nonlinear thermal radiation and nanomaterials, Phy. Lett. A 382 (2018) 749-760. [24] T. Hayat, M.I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, Int. J. Heat Mass Transf., 99 (2016) 702-710. [25] K. L. Hsiao, Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet, Appl. Thermal Eng., 98 (2016) 850-861. [26] K. L. Hsiao, Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature, Int. J. Heat Mass Transf., 112 (2017) 983-990. [27] M.I. Khan, M. Waqas, T. Hayat and A. Alsaedi, A comparative study of Casson fluid with homogeneous-heterogeneous reactions, Journal of Colloid and Interface Science, 498 (2017) 85-90. [28] K. L. Hsiao, To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method, Energy, 130 (2017) 486-499. [29] K. L. Hsiao, Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects, Appl. Thermal Eng., 112 (2017) 1281-1288. 15
Figure Captions Fig. 1 } curve for }f . Fig. 2 } curves for }g , }θ , and }φ . Fig. 3: f 0 (η) via M . Fig. 4: f (η) via M . Fig. 5: f 0 (η) via β. Fig. 6: f (η) via β. Fig. 7: f 0 (η) via λ. Fig. 8: f (η) via λ. Fig. 9: θ (η) via T r. Fig. 10: θ (η) via Ec. Fig. 11: θ (η) via N t. Fig. 12: θ (η) via β. Fig. 13: θ (η) via Pr. Fig. 14: φ (η) via N t. Fig. 15: φ (η) via N b. Fig. 16: φ (η) via Sc. Fig. 17: φ (η) via K2 . Fig. 18: NG via T r. Fig. 19: Be via T r. Fig. 20: NG via Br. Fig. 21: Be via Br. Fig. 22: NG via L1 . Fig. 23: Be via L1 . Fig. 24: NG via L2 . Fig. 25: Be via L2 .
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Declaration of Competing Interest The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere.
17
Figure Captions
Fig. 1 } curve for }f .
Fig. 2 } curves for }g , }θ , and }φ .
Fig. 3: f 0 (η) via M .
15
Fig. 4: f (η) via M .
Fig. 5: f 0 (η) via β.
Fig. 6: f (η) via β.
Fig. 7: f 0 (η) via λ.
16
Fig. 8: f (η) via λ.
Fig. 9: θ (η) via T r.
Fig. 10: θ (η) via Ec.
Fig. 11: θ (η) via N t.
17
Fig. 12: θ (η) via β.
Fig. 13: θ (η) via Pr.
Fig. 14: φ (η) via N t.
Fig. 15: φ (η) via N b. 18
Fig. 16: φ (η) via Sc.
Fig. 17: φ (η) via K2 .
Fig. 18: NG via T r.
Fig. 19: Be via T r.
19
Fig. 20: NG via Br.
Fig. 21: Be via Br.
Fig. 22: NG via L1 .
Fig. 23: Be via L1 .
20
Fig. 24: NG via L2 .
Fig. 25: Be via L2 .
21