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Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition
Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition M. Ijaz Khan, T. Hayat, M. Waqas, M. Imran Khan, A. Alsaedi PII: DOI: Reference:
S0167-7322(17)36257-8 doi:10.1016/j.molliq.2018.02.012 MOLLIQ 8654
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
31 December 2017 29 January 2018 2 February 2018
Please cite this article as: M. Ijaz Khan, T. Hayat, M. Waqas, M. Imran Khan, A. Alsaedi, Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition, Journal of Molecular Liquids (2018), doi:10.1016/j.molliq.2018.02.012
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Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
b
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M. Ijaz Khana , T. Hayata,b , M. Waqasa , M. Imran Khanc,1 and A. Alsaedib
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah
Heriot Watt University, Edinburgh Campus, Edinburgh EH14 4AS,United Kingdom
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c
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21589, Saudi Arabia
Abstract: Main emphasis here is to investigate the novel characteristics of entropy generation
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in nonlinear mixed convective flow of nanofluid between two stretchable rotating disks. Buongiorno nanofluid model of nanomaterial is implemented in mathematical modeling. Nanofluid aspects for thermophoresis and Brownian movement are considered. Heat transport mechanism is examined
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subject to convective condition and Joule heating. Velocity slip is considered at the lower and upper disks. Total entropy generation rate is discussed. Systems of PDEs is first converted into ODEs
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and then tackled by for convergent solutions. The impacts of Reynold number, Prandtl number, Hartman number, velocity slip parameter, Biot numbers of heat and mass transfer, thermophoresis,
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Brownian motion, nonlinear convection parameters for temperature and concentration, mixed convection parameter and Schmidt number on velocities, temperature, Bejan number, concentration and total entropy generation rate are graphically examined. Our investigation reveal that entropy generation rate and Bejan number have inverse behavior for higher estimation of Hartman number. Moreover velocity and temperature gradients are physically interpreted.
Keywords: Two rotating disks; Entropy generation and Bejan number; Convective boundary conditions; Velocity slip; Joule heating; nanofluid (Buongiorno model).
1
Introduction
Liquids with low thermal conductivity are principal impediment to escalate the heat transport in engineering frameworks. Thus it is essential to have liquids of high thermal conductivity. Nanoliquids are considered important for this purpose. The thermal conductivity of nano1
Corresponding author: Email: [email protected] (M. Imran Khan) [email protected] (M. Ijaz
Khan)
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ACCEPTED MANUSCRIPT liquid is larger than base fluid. It is due to metallic nanomaterial as suspension in liquid which significantly enhances the thermal conductivity. Choi [1] introduced the word nanoliquid. He noticed that insertion of metallic nanomertials in conventional heat transport
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liquids enhances the thermal conductivities and heat transfer performance. Xuan and Li
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[2] accomplished pure copper particles in convective heat transfer and flow characteristics of nanoliquids. Here volume fraction and material characteristics play remarkable role to prevail a considerable intensification of heat transport and viscosity. Hsiao [3] investigated MHD
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nanomaterial flow of micropolar fluid with viscous dissipation. Convective flow of viscoelastic nanoliquid with chemical reaction and variable thickness is examined by Qayyum et al. [4].
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MHD two phase nanomaterial flow in the presence of melting is analyzed by Sheikholeslami and Rokni [5]. Some recent studies on this topic can be mentioned in Refs. [6 − 15].
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Fluid flow by rotating disk is very attractive problem at present. Flow by a rotating disk in theoretical chastisements has been carried out by various researchers. It is due to numerous practical applications in different areas like rotating machinery, computer storage
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devices, electronic devices, engineering processes, medical equipment and also very important in industrial processes. Von Karman [16] initially scrutinized hydrodynamic flow due to
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a rotating disk. He introduced a new class of similarity transformation which reduced the system of PDEs into ODEs. MHD three dimensional flow due to rotating disk with uniform
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vertical magnetic field is examined by Turkyilmazoglu [17]. In this study accurate spectral computational integration method is implemented to obtain numerical result. The obtained outcomes show that shear stresses and stagnation velocities are depend upon the stretching, rotation and magnetic variables. Heat transport in Bodewadt flow due to stationary stretching disk is investigated by Turkyilmazoglu 18]. Convective flow of viscoelastic nanoliquid by a two rotating stretching disks is examined by Hayat et al. [19]. Statistical declaration for surface drag force and heat transfer rate for radiative (two phase flow utilizing silver-water and copper-water nanomaterials) is also discussed by Hayat et al. [20]. Few latest investigations on rotating disks can be quoted through the analyses [21 − 30]. Entropy generation minimization (thermodynamic approach) is used to optimize the thermal devices for larger effectiveness. Engineering devices performance within the sight of irreversibilities is diminished. Entropy generation is a measure of the level of accessible irreversibilities in a procedure. In the past decade, numerous investigators and researchers have been inspired to utilize thermodynamics second law consideration in thermal devices systems 2
ACCEPTED MANUSCRIPT with the aim of EGM (Entropy Generation Minimization). Entropy generation minimization is used in many systems such as fuel cells, microchannels, reactors, chillers, curved pipes, gas turbines, air separators, chemical and electrochemical, evaporative cooling, natural convec-
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tion, solar thermal, regular and functionally graded materials, rarefied gaseous slip flow and
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helical coils and so forth (see refs.[31 − 40]).
Present work aims to model entropy generation in three dimensional nonlinear mixed convective flow between two rotating disks. Our inspiration in present attempt is based through
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five novel features. Firstly to model and examine three-dimensional flow of viscous liquid between two rotating stretchable disks. Secondly to analyze entropy generation rate and
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Bejan number. Thirdly to examine Brownian movement and thermophoresis effects. Fourth to explore heat transfer characteristics with convective boundary conditions and Joule heat-
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ing. Fifth to derive convergent series solutions for velocity, temperature and concentration distributions through homotopy analysis method (HAM) (41 − 50). Graphical outcomes are utilized to elaborate the impacts of different variables. Furthermore velocity and temperature
Problem description
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2
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gradients are discussed and analyzed through Tables 2 and 3.
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Consider three-dimensional nonlinear mixed convective flow of viscous nanoliquid between two stretchable rotating disks. Thermophoressis and Brownian motion attributes are present. Applied magnetic field in z−direction is with strength B0 . Induced magnetic field for low magnetic Reynolds number is neglected. Flow in the presence of heat and mass transsfer is analyzed with convective conditions and Joule heating. Velocity slip is considered at the lower and upper disks. Lower disk at z = 0 is rotating with angular velocity Ω 1 in axial direction and stretching rate a1 while upper disk (at distant h apart) rotates with angular velocity Ω2 and stretching rate a2 (see Fig. 1). The problems statements are [38, 39]: ∂u u ∂w + + = 0, ∂r r ∂z 2 ∂ u = − ρ1 ∂p + ν + 1r ∂u + ∂r ∂r 2 ∂r
(1)
− σρ B02 u +g [λ1 (T − T2 ) + λ2 (T − T2 )2 ] + g [λ3 (C − C2 ) + λ4 (C − C2 )2 ] , 2 ∂v v σ ∂v uv ∂ v 1 ∂v ∂ 2 v u + +w + =ν + 2 − 2 − B02 v, 2 ∂r ∂z r ∂r r ∂r ∂z r ρ u ∂u + w ∂u − ∂r ∂z
v2 r
3
∂2u ∂z 2
−
u r2
(2)
(3)
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Consider Von Karman transformations [16]:
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0
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2 1 ∂p ∂w ∂ w 1 ∂w ∂ 2 w ∂w +u =− +ν + , w + ∂r ∂z ρ ∂z ∂r 2 r ∂r ∂z 2 ∂T k 1 ∂T ∂2T ∂2T + + w + + = u ∂T 2 2 ∂r ∂z (ρc)f r ∂r ∂r ∂z h n o i ∂C ∂T σB02 ∂T 2 ∂T 2 ∂C ∂T τ DT2T + + D + (u2 + v 2 ), + B ∂r ∂z ∂r ∂r ∂z ∂z (ρc)f ∂C ∂2T 1 ∂C ∂ 2 C ∂ 2 C ∂C DT 1 ∂T ∂ 2T u , + +w = + 2 + 2 + DB + T2 r ∂r ∂r ∂z ∂r ∂z r ∂r ∂r2 ∂z 2 ∂T ∂u ∂v u = ra1 + λ5 ∂z , v = rΩ1 + λ6 ∂z , w = 0, k ∂z = −β 1 (T1 − T ), ∂C −DB ∂z = β 2 (C1 − C) at z = 0, u = ra2 − λ5 ∂u , v = rΩ2 − λ6 ∂v , kf ∂T = −β 3 (T − T2 ), ∂z ∂z ∂z ∂C −DB ∂z = β 4 (C − C2 ) at z = h, T −T2 , T1 −T2
u = rΩ1 f (ξ), v = rΩ1 g(ξ), w = −2hΩ1 f (ξ), θ = 1 r2 2 , p = ρΩ ν P (ξ) + , ξ = hz , φ = CC−C 1 2 h2 1 −C2
(4)
(5)
(6)
(6)
(7)
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in which p denotes the pressure, (u, v, w) the velocity components, (r, θ, z) the cylindrical coordinates, ρ the density, σ the electrical conductivity, B0 the magnetic field strength, g the
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gravitational acceleration, T the temperature, T1 and T2 the temperatures for convection, C the concentration, C1 and C2 the concentrations for convection, a1 and a2 the stretching
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rates at lower and upper disks, DB the coefficient of Brownian diffusion, DT coefficient of thermal expansion, Ω 1 the angular velocity at lower disk, Ω 2 the angular velocity at upper disk, k the thermal conductivity, λ1 and λ2 the linear and nonlinear coefficients of thermal expansion, h the distance between two disks, λ3 and λ4 the linear and nonlinear coefficients (ρc ) p of concentration expansion, τ = (ρcp )p the ratio between the effective nanoparticle material f
heat capacity and the base fluid heat capacity, ρp the particle density, ρf the fluid density,
k thermal conductivity, ν the kinematic viscosity, λ5 and λ6 the velocity slip coefficients at lower and upper disks, β 1 and β 2 the coefficients of convective heat transfer and convective mass transfer at lower disk and β 3 and β 4 the coefficients of convective heat transfer and convective mass transfer at upper disk. Applying the above transformation expression (1) is identically verified and remaining Eqs. (2-7) yield f 000 + Re 2f f 00 − f 02 + g 2 − M f 0 + α1 θ(1 + β t θ) + α1 N ∗ φ(1 + β c φ) − = 0, g 00 + Re (2f g 0 − 2f 0 g − M g) = 0, 4
(8) (9)
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(10)
1 00 θ + 2 Re f θ 0 + N tθ 02 + N bθ 0 φ0 + Re M Ec(f 02 + g 2 ) = 0, Pr N t 00 θ = 0, φ00 + 2f φ0 Sc Re + Nb
(11)
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5
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(12) 0 00 0 00 0 f (0) = 0, f (1) = 0, f (0) = A1 + γ 1 f (0), f (1) = A2 − γ 1 f (1), g(0) = 1 + γ 2 g (0), 0 0 0 g(1) = Ω − γ 2 g (1), θ (0) = −γ 3 (1 − θ(0)), θ (1) = −γ 4 θ(1), 0 0 φ (0) = −γ (1 − φ(0)), φ (1) = −γ φ(1), P (0) = 0. 2
(13) 2 B σ
upper disks.
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elucidates Reynolds number, Pr = (ρckp )ν the Prandtl number, M = ρΩ0 1 the Hartman number, A1 = Ωa11 and A2 = Ωa21 the scaled stretching rate variables, 1 −T2 ) γ 1 = λd5 and γ 2 = λd6 the velocity slip variables, α1 = gλ1 (T the mixed convection (or rΩ21 2 2 r Ω1 λ3 (C1 −C2 ) ∗ thermal buoyancy) variable, Ec = cp (T1 −T2 ) the Eckert number, N = λ1 (T1 −T2 ) the ratio of concentration to thermal buoyancy forces, β t = λ2 (Tλ11−T2 ) and β c = λ4 (Cλ13−C2 ) the non (ρc)p DT (T1 −T2 ) the linear convection variables due to temperature and concentration, N t = (ρc)f νT2 ν B (C1 −C2 ) thermophoresis variable, N b = (ρc)p D(ρc) the Brownian motion variable, Sc = DB ν f β1h β3h the Schmidt number, γ 3 = k and γ 4 = k the Biot numbers of heat transfer for lower and upper disks and γ 5 = βD2Bh and γ 6 = βD4Bh Biot numbers of mass transfer for lower and Ω1 h ν
Here Re =
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We now differentiated Eq. (9) with respect to ξ and obtain f iv + Re (2f f 000 + 2gg 0 − M f 00 + α1 θ0 + 2α1 β t θθ 0 + α1 N ∗ φ0 + 2α1 N ∗ β c φ0 φ) = 0.
(14)
Through Eqs. (9) and boundary conditions one has 2 = f 000 (0)+Re (g(0))2 − (f 0 (0)) − M f 0 (0) + α1 θ(0)(1 + β t θ(0)) + α1 N ∗ φ(0)(1 + β c φ(0)) . (15)
Integration of Eq. (11) from 0 to ξ yields P = −2 Re(f 2 ) + f 0 − f 0 (0) .
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(16)
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Declaration of curiosity
3.1
Skin friction coefficients (surface drag forces)
Cf 2 =
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Cf 1 =
τ w |z=0 , ρ(rΩ1 )2 τ w |z=h , ρ(rΩ1 )2
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Mathematically at lower and upper disks we have (17)
∂u μrΩ1 f 00 (0) ∂v μrΩ1 g 0 (0) =μ = = μ = , τ . zθ ∂z z=0 h ∂z z=0 h
Total shear stress is defined as
q
τ 2zr + τ 2zθ .
Putting Eq. (17) in Eq. (16) we have
+ (g (0)) ] , [(f 00 (1))2 + (g 0 (1))2 ]1/2 .
Cf 2 =
1 Rer
0
3.2
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Cf 1 =
1 [(f 00 (0))2 Rer
(18)
(19)
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τw =
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τ zr
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where shear stresses in radial (τ zr ) and tangential (τ zθ ) directions are
2 1/2
(20)
Nusselt numbers (heat transfer rates)
where
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Mathematically for lower and upper disks we write hqw hq w , N ux2 = N ux1 = , k(T1 − T2 ) z=0 k(T1 − T2 ) z=h qw |z=0 = −
qw |z=h
k ∂T ∂z z=0
= − k ∂T
∂z z=h
= =
0 − k(T1 −Th2 )θ (0) , 0 − k(T1 −Th2 )θ (1) .
Invoking Eq. (21) in Eq. (20) one has
N ux1 = −θ0 (0), N ux2 = −θ0 (1).
(21)
(22)
(23)
Here (τ w ) denotes the shear stress, (qw ) the heat flux and Rer (= rΩ1 h/ν) the local Reynolds number.
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Series solutions
Initial guesses (f0 (ξ), g0 (ξ), θ 0 (ξ), φ0 (ξ)) and auxiliary linear operators (Lf , Lg , Lθ , Lφ ) are taken as follows
−6γ 1 A1 ξ 2 + A1 ξ 3 + A2 ξ 3 + 2A2 γ 1 ξ 3 + 2γ 1 A1 ξ 3
g0 (ξ) =
,
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(1 + 2γ 1 )(1 + 6γ 1 )
1 + γ 2 + γ 2 Ω + (Ω − 1)ξ , 1 + 2γ 2
(24)
(25)
γ 3 + γ 3 γ 4 (1 − ξ) , γ 3 + γ 4 + γ 3γ 4
(26)
γ 5 + γ 5 γ 6 (1 − ξ) , γ 5 + γ 6 + γ 5γ 6
(27)
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θ0 (ξ) =
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f0 (ξ) =
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A1 ξ + 4γ 1 A1 ξ − 2γ 1 A2 ξ − 2A1 ξ 2 − A2 ξ 2
φ0 (ξ) =
Lf = f 0000 , Lg = g 00 , Lθ = θ00 , Lφ = φ00 ,
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(28)
with
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Lf˜ c∗1 + c∗2 ξ + c∗3 ξ 2 + c∗4 ξ 3 = 0,
(29)
Lg˜[c∗5 + c∗6 ξ] = 0,
(30)
L˜θ [c∗7 + c∗8 ξ] = 0,
(31)
Lφ˜ [c∗9 + c∗10 ξ] = 0,
(32)
in which c∗i (i = 1 − 10) indicate arbitrary constants.
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Inspection of entropy generation
Dimensional form of entropy generation is defined as [38, 39]: h 2 i ∂T 2 + + Tμm Φ SG = Tk2 ∂T ∂r ∂z m h 2 i RD ∂T ∂φ ∂φ ∂φ 2 + Tσm B02 (u2 + v 2 ) + RD + + Tm ∂r ∂r + ∂r ∂z Cm
where
Φ=
2
h
∂u 2 ∂r
+
∂w ∂r
2 i ∂v 2 + ∂w + ∂z ∂z 2 2 ∂ v + ∂u + r ∂r , ∂z r
+
2u2 r2
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∂T ∂φ ∂z ∂z
,
(33)
(34)
ACCEPTED MANUSCRIPT Putting Eq. (33) in Eq. (32) we have SG =
where Cm =
+
+
C1 +C2 2
{z
∂T ∂z
2 # }
RD Cm |
∂φ ∂r
∂φ ∂z
+
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Thermal irreversibility ∂u 2 2u2 ∂w 2 ∂v 2 + r2 + 2 ∂z + ∂z σ 2 2 ∂r + B0 (u + v 2 ) 2 2 2 ∂w ∂u ∂v ∂ v T m + ∂z + ∂r + r ∂r r | {z } ∂r {z } Joule dissipation irreversibility friction irreversibility "Fluid 2 2 #
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|
2
+
RD ∂T ∂φ ∂T ∂φ , + Tm ∂r ∂r ∂z ∂z {z } +
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+ Tμm
|
∂T ∂r
2
Diffusive irreversibility
and Tm =
T1 +T2 2
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k 2 Tm
"
(35)
represent the mean concentration and temperature
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respectively, D the mass diffusivity and R the constant.
Re
represents the dimensionless temperature ratio parameter, μΩ21 h2 4C C1 −C2 = the dimensionless concentration ratio parameter, Br = the Cm Cm k4T 2 T m SG ν Brinkman number, NG = k4T the entropy generation rate, A = hr 2 the dimensionless Ω1 RD(C1 −C2 ) the diffusive parameter. radial parameter and L = k
=
4T Tm
2 Re
(36)
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=
T1 −T2 Tm
1
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where α∗2 =
α∗1
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Dimensionless number NG (ξ) for the entropy generation is defined as 02 ∗ 1 Br 02 ∗ 02 ∗ 002 NG (ξ) = θ α1 Re + Re (12f + A g + A f ) +M BrA∗ (f 02 + g 2 ) + Lα∗ α∗ 1 φ02 + L 1 θ0 φ0 ,
Bejan number (Be) in dimensionless form is defined as
Entropy generation subject to heat and mass transfer Total entropy generation 02 ∗ 1 1 0 0 ∗ ∗ 1 02 θ α1 Re + Lα1 α2 Re φ + L Re θ φ Be = 1 θ02 α∗1 Re + Br (12f 02 + A∗ g 02 + A∗ f 002 ) + Re 1 0 0 ∗ 02 2 ∗ ∗ 1 02 M BrA (f + g ) + Lα1 α2 Re φ + L Re θ φ
Be =
6
(37)
(38)
Convergence analysis
Selection of auxiliary variables }f , }g , }θ and }φ provide us opportunity to adjust the convergence of nonlinear expressions. To show the convergence of nonlinear expressions graphically form Fig. 2 is plotted. From Fig. 2 it is observed that the permissible values of auxiliary 8
ACCEPTED MANUSCRIPT variables lie in the ranges −2.0 ≤ }f ≤ −0.1, −1.9 ≤ }g ≤ −0.1, −1.8 ≤ }θ ≤ −0.2 and
−2.3 ≤ }φ ≤ −0.1. Table 1 communicates the convergence of series solutions of concentration, temperature and momentum for the given flow analysis correct upto four decimal
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places. Table 1 designates that f 00 (0), g 0 (0), θ 0 (0) and φ0 (0) converge at 5th , 10th , 35th and
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45th order of approximations.
Table 1: Different order of approximations when Re = 0.1, Pr = 1.0, M = 0.5, A1 = 0.4, A2 = 0.9, γ 1 = 0.5, γ 2 = 0.7, γ 3 = 0.5, γ 4 = 0.7, γ 5 = 0.01, γ 6 = 0.01, α1 = α2 = 0.4,
−f 00 (0)
−g 00 (0)
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−θ0 (0)
−φ0 (0)
1
0.7221
0.30627
0.2205
0.0098
5
0.7222
0.3061
0.2188
0.0099
10
0.7222
0.3060
0.2194
0.0100
0.7222
0.3060
0.2196
0.0100
0.7222
0.3060
0.2196
0.0100
0.7222
0.3060
0.2196
0.0100
0.7222
0.3060
0.2195
0.0099
35
0.7222
0.3060
0.2194
0.0099
40
0.7222
0.3060
0.2194
0.0099
45
0.7222
0.3060
0.2194
0.0100
50
0.7222
0.3060
0.2194
0.0100
55
0.7222
0.3060
0.2194
0.0100
60
0.7222
0.3060
0.2194
0.0100
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Order of approximation
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Ω = 0.3, Bt = 0.5, Bc = 0.7, N t = 0.1, N b = 1.0, Ec = 0.4 and Sc = 1.5.
15
25
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30
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20
7
Physical analysis
This section illuminates the impact of relevant variables on velocities axial f (ξ), radial f 0 (ξ), tangential g(ξ), concentration (φ(ξ)), temperature (θ(ξ)), entropy generation NG (ξ), Bejan number (Be), skin friction coefficients (Cf 1 , Cf 2 ) and heat transfer rates (N ux1 , N ux2 ).
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ACCEPTED MANUSCRIPT 7.1 7.1.1
Velocity distribution Axial (f (ξ)) , radial (f 0 (ξ)) and tangential (g(ξ)) velocity field
Figs. (3 − 16) are depicted to display the behavior of involved parameters on axial ( f (ξ)),
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radial (f 0 (ξ)) and tangential (g(ξ)) velocity fields. This section comprises the graphs and
related investigations for various physical variables. Figs. (3 − 5) depict the impact of
Reynold number (Re) on axial (f (ξ)), radial (f 0 (ξ)) and tangential (g(ξ)) velocity fields.
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Here magnitude of axial (f (ξ)) and tangential (g(ξ)) velocities increase for larger estimation of Reynold number near the lower disk. Physically for larger Reynold number the viscous
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effects decays and therefore less resistance occurs to liquid motion. Magnitude of axial velocity initially increases at lower disk and then decreases as the value of (Re) approache
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to 6. Figs. (6 − 8) illustrate the Hartman number (M ) impact on axial (f (ξ)), radial (f 0 (ξ)) and tangential (g(ξ)) velocity fields. Axial, radial and tangential are decreased by Hartman number (M ). Since Lorentz force is resistive force so it resist the motion of liquid particles
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and therefore velocities (f (ξ)), (f 0 (ξ)) and (g(ξ)) decay. Figs. (9 − 11) are portrayed to
demonstrate the essential impact of velocity slip variable (γ 1 ) on axial (f (ξ)), radial (f 0 (ξ))
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and tangential (g(ξ)) velocity fields. Here magnitude of axial (f (ξ)) and radial (f 0 (ξ)) velocity fields have monotonic behavior for larger (γ 1 ) . Moreover tangential velocity decays for higher
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velocity slip variable (see Fig. 11). Fig. 12 illustrates that axial velocity is decreasing function of mixed convection (or thermal buoyancy) variable (α1 ). However dual behavior is observed for radial velocity (see Fig. 13). Similar impact is obtained for nonlinear convection variable (β t ) due to temperature on axial and radial velocities (see Figs.14 and 15) . Fig. 16 is sketched to show that tangential velocity profile g(ξ) decays at lower disk for larger estimation of slip parameter (γ 2 ). Near upper disk the tangential velocity enhances.
7.2
Temperature distribution
Figs. (17 − 22) are sketched to show the impact of various variable like Hartman number (M ), Biot number of heat transfer for lower disk (γ 3 ), Biot number of heat transfer for upper disk (γ 4 ), Eckert number (Ec), thermophoresis (N t) and Prandtl number (Pr) on temperature field (θ(ξ)) . Fig. 17 is sketched to examine temperature field for Hartman number. Here temperature is enhanced by Hartman number. As Hartman number ( M ) is linked with Lorentz force which is a resistive force. Clearlyt an enhancement in Hartman 10
ACCEPTED MANUSCRIPT number provides more resistance and therefore temperature field increases. From Fig. 18 it is clearly examined that stronger temperature field (θ(ξ)) is generated by larger estimation of Biot number of heat transfer for lower disk (γ 3 ). Physically heat transfer coefficient enhances
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for larger Biot number (γ 3 ). Fig. 19 is sketched to examine temperature (θ(ξ)) for thermal
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Biot number of upper disk (γ 4 ). An increment in (γ 4 ) shows reduction in temperature field (θ(ξ)). Since purpose of Biot number is to determine the variation in temperature profile. It is noticed from Figs. 18 and 19 that liquid temperature varies considerably. Fig. 20 displayed
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the graphs of (θ(ξ)) for (Ec). Here temperature is increasing function of Eckert number (Ec). Physically inside friction of molecules the conversion of mechanical energy to thermal energy
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is the reason for temperature field. Fig. 21 presents the variation in temperature profile for thermophoresis variable (N t). It is examined that higher estimation of thermophoresis
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variable provides more (θ(ξ)) . Physically an increment in thermophoresis variable leads to more thermophoretic force which allows deeper movement of nanomaterials in liquid. Fig. 22 is drawn for the impact of Prandtl number on (θ(ξ)) . By increasing (Pr) less temperature
Concentration distribution
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7.3
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(θ(ξ)) is noticed.
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Figs. (23 − 27) are portrayed here to show the impact of Schmidt number (Sc), Biot number for mass transfer at lower disk (γ 5 ) , Biot number for mass transfer at upper disk (γ 6 ) , thermophoresis (N t) and Brownian motion variable (N b) on concentration field. Fig. 23 addresses the variation of Schmidt number (Sc) on concentration field (φ(ξ)) . An increment in Schmidt number (Sc) shows decrease in magnitude of (φ(ξ)) . The ratio of momentum diffusivity and mass diffusivity is called Schmidt number (Sc). Physically mass diffusivity decreases for more Schmidt number. Therefore concentration field reduces. Behavior of concentration Biot numbers (γ 5 ) and (γ 6 ) is presented in Figs. 24 and 25. For larger values of (γ 5 ) the concentration profile enhances. It is due to an increase in internal energy of liquid particles for larger (γ 5 ). Opposite behavior is observed for concentration Biot number of upper disk (γ 6 ). From these Figs. we can conclude that concentration varies considerably with the increase in (γ 5 ) and (γ 6 ) . Fig. 26 portrays the effect of N t on (φ(ξ)) . Enhancement in magnitude of concentration (φ(ξ)) is noticed for larger N t. Physically for larger N t the temperature difference between ambient and surface increases. That is why concentration of liquid enhances. For larger Brownian motion parameter (N b) the magnitude of concentration 11
ACCEPTED MANUSCRIPT decreases (see Fig. 27).
7.4
Entropy generation rate and Bejan number
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Behaviors of Hartman number (M ), slip parameter (γ 1 ), thermal Biot number (γ 3 ), Brinkman
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number (Br), temperature ratio parameter (α∗1 ), thermophoretic parameter (N t), Eckert number (Ec), Schmidt number (Sc) and diffusive parameter (L) on local entropy generation
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NG (ξ) and Bejan number Be is scrutinized in Figs. (28 − 41).
Local entropy generation (NG (ξ)) and Bejan number (Be) for higher estimation of Hart-
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man number (M ) are graphically presented in Figs. 28 and 29. Here entropy generation rate and Bejan number have inverse behavior for higher estimation of Hartman number ( M ). Physically larger Lorentz force offers more resistance to the liquid particle motion. Thus
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more heat is produced and therefore entropy generation rate enhances. It is noticed that Bejan number is decreasing function of Hartman number. Figs. 30 and 31 are drawn to
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show the impacts of slip parameter (γ 1 ) on entropy generation rate and Bejan number. Here entropy generation rate and Bejan number show opposite behavior for rising values of ( γ 1 ).
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Impacts regarding thermal Biot number of lower disk (γ 3 ) on entropy generation rate NG (ξ) and Bejan number (Be) are displayed by Figs. (32 and 33). Here both entropy generation
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rate and Bejan number increase for higher estimation of (γ 3 ). Figs. (34 and 35) show entropy generation (NG (ξ)) and Bejan number (Be) for higher estimation of Brinkman number (Br) . Brinkman number determines the releasing of heat by viscous heating in relation to heat transfer by molecular conduction. Brinkman number is directly related to entropy generation near the disk. Heat transfer by molecular conduction is greater than heat released by viscous effects near the disk. Quite large amount of heat is generated between the layers of moving fluid particles. It increases the entropy and disorderness of the system also enhances. Fig. 35 shows that (Be) decays for Brinkman number. Figs. (36 and 37) portray impact of temperature ratio parameter (α∗1 ) on entropy generation rate NG (ξ) and Bejan number (Be). Here entropy generation rate and Bejan number are enhanced for larger estimations of ( α∗1 ). Figs. (38 and 39) depict the influence of thermophoresis variable (N t) and Eckert number (Ec) on Bejan number (Be) . Here Bejan number decreases for both thermophoresis variable (N t) and Eckert number (Ec). It is due to the fact that an enhancement in (N t) and (Ec) increase the internal source of energy. Furthermore higher concentration is achieved through larger thermal conductivity which decays the Bejan number. Fig. 40 elucidates analysis of 12
ACCEPTED MANUSCRIPT (Sc) on Bejan number. Fig. 40 shows Bejan number is increasing function of ( Sc). Fig. 41 displays behavior of diffusive parameter (L) on Bejan number. Here we noticed that Bejan
Drag forces (Cf 1 , Cf 2 ) and rates of heat transfer (N ux1 , N ux2 )
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number enhances for larger (L) .
Tables 2 and 3 are presented to show the computational outcomes for surface drag forces
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(Cf 1 , Cf 2 ) and heat transfer rates (N ux1 , N ux2 ). From Table 2 it is observed that surface drag force decays for different estimations of (λ1 ), (β c ) and (γ 1 ) near the lower disk while it
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shows reverse trend for Hartman number (M ). Surface drag force also decreases via (β 2 ), (γ 1 ) and (M ) close to upper disk and it increases for larger (λ1 ). Table 3 reveals that magnitude of heat transfer rate decays near the lower disk for higher values of (Ec) and (Pr) while it
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enhances for (γ 3 ). Also magnitude of rate of heat transfer enhances closed to upper disk for higher estimations of (γ 3 ), (Ec) and (Pr).
λ1
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Table 2: Numerical values of surface drag forces (Cf 1 , Cf 2 ). βc
γ1
M
Cf 1
Cf 2
0.7
0.5
0.5
0.7844334
1.258870
0.6
0.7842177
1.259100
0.8
0.7840075
1.259322
0.9
0.7844297
1.258874
1.1
0.7844289
1.258869
0.7
0.6204424
0.9996764
0.9
0.5273426
0.8368111
0.6
0.7856426
1.258344
0.7
0.7868482
1.257836
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0.4
0.4
0.7
0.5
Table 3: Computational results for heat transfer rates (N u1 , N u2 ).
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Ec
Pr
N ux1
N ux2
0.5
0.4
1
-0.21802
-0.23565
0.6
-0.23543
-0.25379
0.7
-0.24961
-0.26869
0.5
-0.21673
-0.23722
0.6
-0.21548
-0.23877
1.1
-0.21735
-0.23649
1.2
-0.21650
0.4
Conclusions
-0.23758
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Entropy generation in mixed convective flow of nanofluid between two stretchable rotating disks is considered. Main conclusions of the present flow analysis are given below: • Magnitude of axial (f (ξ)) and tangential (g(ξ)) velocities increase for larger estimation
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of Reynold number near lower disk.
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• Magnitude of axial velocity initially increases at lower disk and then decreases when (Re) approaches to 6.
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• Axial, radial and tangential velocities are decreasing function of Hartman number ( M ). • Temperature is an increasing function of Hartman and Eckert numbers. • Higher estimation of thermophoresis variable (N t) provides larger temperature (θ(ξ)) . • An increment in Schmidt number (Sc) shows decay in magnitude of (φ(ξ)) . • Magnitude of concentration decays for via Brownian motion parameter ( N b). • Entropy generation rate and Bejan number have inverse behavior of Hartman number (M ). • Rate of entropy generation NG (ξ) enhances for larger Brinkman number whereas reverse trend is observed for Bejan number (Be).
14
ACCEPTED MANUSCRIPT References [1] S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME, FED
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231/Md, USA, (1995) 99-105.
ids, Journal of Heat Transfer, 125 (2003) 151-155.
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[2] Y. Xuan and Q. Li, Investigation on convective heat transfer and flow features of nanoflu-
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[3] K. L. Hsiao, Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature, International Journal of Heat and Mass
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Transfer, 112 (2017) 983-990.
[4] S. Qayyum, T. Hayat, A. Alsaedi and B. Ahmad, Magnetohydrodynamic (MHD) nonlin-
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ear convective flow of Jeffrey nanofluid over a nonlinear stretching surface with variable thickness and chemical reaction, International Journal of Mechanical Sciences, (2017)
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(In press).
[5] M. Sheikholeslami and H. B. Rokni, Influence of melting surface on MHD nanofluid flow
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by means of two phase model, Chinese Journal of Physics, 55 (2017) 1352-1360. [6] M. Farooq, M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, MHD stag-
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nation point flow of viscoelastic nanofluid with non-linear radiation effects, Journal of Molecular Liquids, 221 (2016) 1097-1103. [7] B. Mahanthesh, B. J. Gireesha, B. C. Prasannakumara and N. S. Shashikumar, Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source, Nuclear Engineering and Technology, 49 (2017) 1660-1668. [8] S. Qayyum, M. I. Khan, T. Hayat and A. Alsaedi, Comparative investigation of five nanoparticles in flow of viscous fluid with Joule heating and slip due to rotating disk, Physica B, (2018) Inpress. [9] G. K. Ramesh, B. C. Prasannakumara, B. J. Gireesha, S. A. Shehzad and F. M. Abbasi, Three dimensional flow of Maxwell fluid with suspended nanoparticles past a bidirectional porous stretching surface with thermal radiation, Thermal Science and Engineering Progress, 1 (2017) 6-14.
15
ACCEPTED MANUSCRIPT [10] M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, Theoretical investigation of doubly stratified flow of Eyring-Powell nanomaterial via heat generation/absorption, European Physical Journal Plus, 132 (2017) 489.
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[11] B. C. Prasannakumara, B. J. Gireesha, M. R. Krishnamurthy and R. S. R. Gorla, Slip
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flow and nonlinear radiative heat transfer of suspended nanoparticles due to a rotating disk in the presence of convective boundary condition, International Journal of Nanopar-
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ticles, 9 (2017) 180-200.
[12] T. Hayat, S. Ullah, M. I. Khan and A. Alsaedi, On framing potential features of SWCNTs
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and MWCNTs in mixed convective flow, Results in Physics, 8 (2018) 357–364. [13] K. G. Kumar, B. J. Gireesha, B. C. Prasannakumara and O. D. Makinde, Impact of
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chemical reaction on Marangoni boundary layer flow of a Casson nano liquid in the presence of uniform heat source sink, Diffusion Foundations, 11 (2017) 22-32.
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[14] T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, Modern developments about statistical declaration and probable error for skin friction and Nusselt number with copper and
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silver nanoparticles, Chinese Journal of Physics, 55 (2017) 2501-2513.
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[15] B. C. Prasannakumara, B. J. Gireesha, M. R. Krishnamurthy and K. G. Kumar, MHD flow and nonlinear radiative heat transfer of Sisko nanofluid over a nonlinear stretching sheet, Informatics in Medicine Unlocked, 9 (2017) 123-132. ¨ [16] V. Karman. Uber laminare und turbulente Reibung, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 1 (1921) 233-252. [17] M. Turkyilmazoglu, Three dimensional MHD stagnation flow due to a stretchable rotating disk, International Journal of Heat and Mass Transfer, 55 (2012) 6959-6965. [18] M. Turkyilmazoglu, B¨odewadt flow and heat transfer over a stretching stationary disk, International Journal of Mechanical Sciences, 90 (2015) 246-250. [19] T. Hayat, M. Javed, M. Imtiaz and A. Alsaedi, Convective flow of Jeffrey nanofluid due to two stretchable rotating disks, Journal of Molecular Liquids, 240 (2017) 291-302.
16
ACCEPTED MANUSCRIPT [20] T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, Current progresses about probable error and statistical declaration for radiative two phase flow using Ag − H2 O and Cu − H2 O nanomaterials, International Journal of Hydrogen Energy, 42 (2017) 29107-29120.
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[21] T. Hayat, M. Rashid, M. Imtiaz and A. Alsaedi, Nanofluid flow due to rotating disk with
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variable thickness and homogeneous-heterogeneous reactions, International Journal of Heat and Mass Transfer, 113 (2017) 96-105.
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[22] T. Hayat, M. I. Khan, A. Alsaedi and M. I. Khan, Joule heating and viscous dissipation
Mass Transfer, 89 (2017) 190-197.
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in flow of nanomaterial by a rotating disk, International Communications in Heat and
[23] Y. H. Liu, G. J. Peng, W. C. Lai, C. Y. Huang and J. H. Liang, Flow field investigation
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in a rotating disk chemical vapor deposition chamber with a perforated showerhead, Experimental Thermal and Fluid Science, 88 (2017) 389-399.
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[24] P. T. Griffiths, Flow of a generalised Newtonian fluid due to a rotating disk, Journal of
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Non-Newtonian Fluid Mechanics, 221 (2015) 9-17. [25] C. Yin, L. Zheng, C. Zhang and X. Zhang, Flow and heat transfer of nanofluids over a
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rotating disk with uniform stretching rate in the radial direction, Propulsion and Power Research, 6 (2017) 25-30. [26] B. S. Dandapat and S. K. Singh, Two-layer film flow over a rotating disk, Communications in Nonlinear Science and Numerical Simulation, 17 (2012) 2854-2863. [27] S. Xun, J. Zhao, L. Zheng, X. Chen and X. Zhang, Flow and heat transfer of Ostwald-de Waele fluid over a variable thickness rotating disk with index decreasing, International Journal of Heat and Mass Transfer, 103 (2016) 1214-1224. [28] T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, Homogeneous-heterogeneous reactions in nonlinear radiative flow of Jeffrey fluid between two stretchable rotating disks, Results in Physics, 7 (2017) 2557-2567. [29] T. Hayat, S. Qayyum, A. Alsaedi and B. Ahmad, Significant consequences of heat generation/absorption and homogeneous-heterogeneous reactions in second grade fluid due to rotating disk, Results in Physics, 8 (2018) 223-230. 17
ACCEPTED MANUSCRIPT [30] C. Ming, L. Zheng, X. Zhang, F. Liu and V. Anh, Flow and heat transfer of power-law fluid over a rotating disk with generalized diffusion, International Communications in Heat and Mass Transfer, 79 (2016) 81-88.
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[31] A. Bejan, Second law analysis in heat transfer, Energy, 5 (1980) 720-732. [32] H. Abbasi, Entropy generation analysis in a uniformly heated microchannel heat sink,
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Energy, 32 (2007) 1932-1947.
[33] D. Srinivasacharya and K. H. Bindu, Entropy generation in a porous annulus due to
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micropolar fluid flow with slip and convective boundary conditions, Energy, 111 (2016) 165-177.
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[34] A. Ebrahimi, F. Rikhtegar, A. Sabaghan and E. Roohi, Heat transfer and entropy generation in a microchannel with longitudinal vortex generators using nanofluids, Energy,
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101, (2016) 190-201.
[35] Z. Y. Xie and Y. J. Jian, Entropy generation of two-layer magnetohydrodynamic elec-
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troosmotic flow through microparallel channels, Energy 139 (2017) 1080-1093. [36] M. W. A. Khan, M. I. Khan, T. Hayat and A. Alsaedi, Entropy generation minimization
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(EGM) of nanofluid flow by a thin moving needle with nonlinear thermal radiation, Physica B, (2018) Inpress. [37] M. I. Khan, T. Hayat and A. Alsaedi, Numerical investigation for entropy generation in hydromagnetic flow of fluid with variable properties and slip, AIP, Physics of Fluid, (2018) Inpress.
[38] T. Hayat, M. I. Khan, S. Qayyum and A. Alsaedi, Entropy generation in flow with silver and copper nanoparticles, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 539 (2018) 335-346. [39] T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, Entropy generation in magnetohydrodynamic radiative flow due to rotating disk in presence of viscous dissipation and Joule heating, AIP, Physics of Fluids 30 (2018) 017101. [40] D. Nouri, M. P. Fard, M. J. Oboodi, O. Mahian and A. Z. Sahin, Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden 18
ACCEPTED MANUSCRIPT expansion and contraction, International Journal of Heat and Mass Transfer, 116 (2018) 1036-1043. [41] S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman
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& Hall/CRC (2003).
[42] 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
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a variable thicked surface, International Journal of Heat and Mass Transfer, 99 (2016)
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702-710.
[43] L. Zheng, C. Zhang, X. Zhang and J. Zhang, Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous
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medium, Journal of The Franklin Institute, 350 (2013) 990-1007. [44] T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Stagnation point flow
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with Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions, Journal of
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Molecular Liquids, 220 (2016) 49-55. [45] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations
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arising in heat transfer, Physics Letters A 360 (2006) 109-113. [46] 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.
[47] A. Alseadi, M. I. Khan, M. Farooq, N. Gull and T. Hayat, Magnetohydrodynamic (MHD) stratified bioconvective flow of nanofluid due to gyrotactic microorganisms, Advance Powder Technology 28 (2017) 288-298. [48] M. I. Khan, T. Hayat, M. I. Khan and A. Alsaedi, A modified homogeneousheterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating, International Journal of Heat and Mass Transfer 113 (2017) 310-317. [49] M. I. Khan, M. I. Khan, M. Waqas, T. Hayat and A. Alsaedi, Chemically reactive flow of Maxwell liquid due to variable thicked surface, International Communications in Heat and Mass Transfer 86 (2017) 231-238. 19
ACCEPTED MANUSCRIPT [50] T. Hayat, M. I. Khan, M. Waqas and A. Alsaedi, Effectiveness of magnetic nanoparticles in radiative flow of Eyring-Powell fluid, Journal of Molecular Liquids 231 (2017) 126-133. Figure Captions
Fig. 3: f (ξ) via Re .
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Fig. 4: f 0 (ξ) via Re . Fig. 5: g(ξ) via Re .
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Fig. 6: f (ξ) via M. Fig. 7: f 0 (ξ) via M. Fig. 8: g(ξ) via M.
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Fig. 9: f (ξ) via γ 1 . Fig. 10: f 0 (ξ) via γ 1 . Fig. 12: f (ξ) via α1 .
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Fig. 13: f 0 (ξ) via α1 .
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Fig. 11: g(ξ) via γ 1 .
Fig. 14: f (ξ) via β t .
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Fig. 2: ~−curves for f 00 (0), g 0 (0), θ 0 (0) and φ0 (0).
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Fig. 1: Flow analysis and coordinate system.
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Fig. 15: f 0 (ξ) via β t . Fig. 16: g(ξ) via γ 2 .
Fig. 17: θ(ξ) via M. Fig. 18: θ(ξ) via γ 3 . Fig. 19: θ(ξ) via γ 4 .
Fig. 20: θ(ξ) via Ec. Fig. 21: θ(ξ) via N t. Fig. 22: θ(ξ) via Pr . Fig. 23: φ(ξ) via Sc. Fig. 24: φ(ξ) via γ 5 . Fig. 25: φ(ξ) via γ 6 . Fig. 26: φ(ξ) via N t. Fig. 27: φ(ξ) via N b. Fig. 28: NG (ξ) via M. Fig. 29: Be via M. 20
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Fig. 33: Be via γ 3 .
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Fig. 34: NG (ξ) via Br. Fig. 35: Be via Br. Fig. 36: NG (ξ) via α∗1 .
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Fig. 37: Be via α∗1 . Fig. 38: Be via N t.
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Fig. 39: Be via Ec. Fig. 40: Be via Sc.
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Fig. 41: Be via L.
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ACCEPTED MANUSCRIPT Highlights • Characteristics of entropy generation in nonlinear mixed convective flow of nanofluid
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between two stretchable rotating disks are considered.
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• Buongiorno nanofluid model of nanomaterial is implemented in mathematical modeling. • Nanofluid aspects for thermophoresis and Brownian movement are considered.
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• Heat transport mechanism is examined subject to convective condition and Joule heating.
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• Velocity slip is considered at the lower and upper disks.
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• Total entropy generation rate is discussed.
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S0167-7322(17)36257-8 doi:10.1016/j.molliq.2018.02.012 MOLLIQ 8654
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
31 December 2017 29 January 2018 2 February 2018
Please cite this article as: M. Ijaz Khan, T. Hayat, M. Waqas, M. Imran Khan, A. Alsaedi, Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition, Journal of Molecular Liquids (2018), doi:10.1016/j.molliq.2018.02.012
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Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
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M. Ijaz Khana , T. Hayata,b , M. Waqasa , M. Imran Khanc,1 and A. Alsaedib
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah
Heriot Watt University, Edinburgh Campus, Edinburgh EH14 4AS,United Kingdom
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c
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21589, Saudi Arabia
Abstract: Main emphasis here is to investigate the novel characteristics of entropy generation
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in nonlinear mixed convective flow of nanofluid between two stretchable rotating disks. Buongiorno nanofluid model of nanomaterial is implemented in mathematical modeling. Nanofluid aspects for thermophoresis and Brownian movement are considered. Heat transport mechanism is examined
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subject to convective condition and Joule heating. Velocity slip is considered at the lower and upper disks. Total entropy generation rate is discussed. Systems of PDEs is first converted into ODEs
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and then tackled by for convergent solutions. The impacts of Reynold number, Prandtl number, Hartman number, velocity slip parameter, Biot numbers of heat and mass transfer, thermophoresis,
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Brownian motion, nonlinear convection parameters for temperature and concentration, mixed convection parameter and Schmidt number on velocities, temperature, Bejan number, concentration and total entropy generation rate are graphically examined. Our investigation reveal that entropy generation rate and Bejan number have inverse behavior for higher estimation of Hartman number. Moreover velocity and temperature gradients are physically interpreted.
Keywords: Two rotating disks; Entropy generation and Bejan number; Convective boundary conditions; Velocity slip; Joule heating; nanofluid (Buongiorno model).
1
Introduction
Liquids with low thermal conductivity are principal impediment to escalate the heat transport in engineering frameworks. Thus it is essential to have liquids of high thermal conductivity. Nanoliquids are considered important for this purpose. The thermal conductivity of nano1
Corresponding author: Email: [email protected] (M. Imran Khan) [email protected] (M. Ijaz
Khan)
1
ACCEPTED MANUSCRIPT liquid is larger than base fluid. It is due to metallic nanomaterial as suspension in liquid which significantly enhances the thermal conductivity. Choi [1] introduced the word nanoliquid. He noticed that insertion of metallic nanomertials in conventional heat transport
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liquids enhances the thermal conductivities and heat transfer performance. Xuan and Li
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[2] accomplished pure copper particles in convective heat transfer and flow characteristics of nanoliquids. Here volume fraction and material characteristics play remarkable role to prevail a considerable intensification of heat transport and viscosity. Hsiao [3] investigated MHD
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nanomaterial flow of micropolar fluid with viscous dissipation. Convective flow of viscoelastic nanoliquid with chemical reaction and variable thickness is examined by Qayyum et al. [4].
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MHD two phase nanomaterial flow in the presence of melting is analyzed by Sheikholeslami and Rokni [5]. Some recent studies on this topic can be mentioned in Refs. [6 − 15].
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Fluid flow by rotating disk is very attractive problem at present. Flow by a rotating disk in theoretical chastisements has been carried out by various researchers. It is due to numerous practical applications in different areas like rotating machinery, computer storage
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devices, electronic devices, engineering processes, medical equipment and also very important in industrial processes. Von Karman [16] initially scrutinized hydrodynamic flow due to
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a rotating disk. He introduced a new class of similarity transformation which reduced the system of PDEs into ODEs. MHD three dimensional flow due to rotating disk with uniform
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vertical magnetic field is examined by Turkyilmazoglu [17]. In this study accurate spectral computational integration method is implemented to obtain numerical result. The obtained outcomes show that shear stresses and stagnation velocities are depend upon the stretching, rotation and magnetic variables. Heat transport in Bodewadt flow due to stationary stretching disk is investigated by Turkyilmazoglu 18]. Convective flow of viscoelastic nanoliquid by a two rotating stretching disks is examined by Hayat et al. [19]. Statistical declaration for surface drag force and heat transfer rate for radiative (two phase flow utilizing silver-water and copper-water nanomaterials) is also discussed by Hayat et al. [20]. Few latest investigations on rotating disks can be quoted through the analyses [21 − 30]. Entropy generation minimization (thermodynamic approach) is used to optimize the thermal devices for larger effectiveness. Engineering devices performance within the sight of irreversibilities is diminished. Entropy generation is a measure of the level of accessible irreversibilities in a procedure. In the past decade, numerous investigators and researchers have been inspired to utilize thermodynamics second law consideration in thermal devices systems 2
ACCEPTED MANUSCRIPT with the aim of EGM (Entropy Generation Minimization). Entropy generation minimization is used in many systems such as fuel cells, microchannels, reactors, chillers, curved pipes, gas turbines, air separators, chemical and electrochemical, evaporative cooling, natural convec-
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tion, solar thermal, regular and functionally graded materials, rarefied gaseous slip flow and
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helical coils and so forth (see refs.[31 − 40]).
Present work aims to model entropy generation in three dimensional nonlinear mixed convective flow between two rotating disks. Our inspiration in present attempt is based through
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five novel features. Firstly to model and examine three-dimensional flow of viscous liquid between two rotating stretchable disks. Secondly to analyze entropy generation rate and
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Bejan number. Thirdly to examine Brownian movement and thermophoresis effects. Fourth to explore heat transfer characteristics with convective boundary conditions and Joule heat-
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ing. Fifth to derive convergent series solutions for velocity, temperature and concentration distributions through homotopy analysis method (HAM) (41 − 50). Graphical outcomes are utilized to elaborate the impacts of different variables. Furthermore velocity and temperature
Problem description
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2
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gradients are discussed and analyzed through Tables 2 and 3.
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Consider three-dimensional nonlinear mixed convective flow of viscous nanoliquid between two stretchable rotating disks. Thermophoressis and Brownian motion attributes are present. Applied magnetic field in z−direction is with strength B0 . Induced magnetic field for low magnetic Reynolds number is neglected. Flow in the presence of heat and mass transsfer is analyzed with convective conditions and Joule heating. Velocity slip is considered at the lower and upper disks. Lower disk at z = 0 is rotating with angular velocity Ω 1 in axial direction and stretching rate a1 while upper disk (at distant h apart) rotates with angular velocity Ω2 and stretching rate a2 (see Fig. 1). The problems statements are [38, 39]: ∂u u ∂w + + = 0, ∂r r ∂z 2 ∂ u = − ρ1 ∂p + ν + 1r ∂u + ∂r ∂r 2 ∂r
(1)
− σρ B02 u +g [λ1 (T − T2 ) + λ2 (T − T2 )2 ] + g [λ3 (C − C2 ) + λ4 (C − C2 )2 ] , 2 ∂v v σ ∂v uv ∂ v 1 ∂v ∂ 2 v u + +w + =ν + 2 − 2 − B02 v, 2 ∂r ∂z r ∂r r ∂r ∂z r ρ u ∂u + w ∂u − ∂r ∂z
v2 r
3
∂2u ∂z 2
−
u r2
(2)
(3)
ACCEPTED MANUSCRIPT
Consider Von Karman transformations [16]:
MA
0
NU
SC
RI P
T
2 1 ∂p ∂w ∂ w 1 ∂w ∂ 2 w ∂w +u =− +ν + , w + ∂r ∂z ρ ∂z ∂r 2 r ∂r ∂z 2 ∂T k 1 ∂T ∂2T ∂2T + + w + + = u ∂T 2 2 ∂r ∂z (ρc)f r ∂r ∂r ∂z h n o i ∂C ∂T σB02 ∂T 2 ∂T 2 ∂C ∂T τ DT2T + + D + (u2 + v 2 ), + B ∂r ∂z ∂r ∂r ∂z ∂z (ρc)f ∂C ∂2T 1 ∂C ∂ 2 C ∂ 2 C ∂C DT 1 ∂T ∂ 2T u , + +w = + 2 + 2 + DB + T2 r ∂r ∂r ∂z ∂r ∂z r ∂r ∂r2 ∂z 2 ∂T ∂u ∂v u = ra1 + λ5 ∂z , v = rΩ1 + λ6 ∂z , w = 0, k ∂z = −β 1 (T1 − T ), ∂C −DB ∂z = β 2 (C1 − C) at z = 0, u = ra2 − λ5 ∂u , v = rΩ2 − λ6 ∂v , kf ∂T = −β 3 (T − T2 ), ∂z ∂z ∂z ∂C −DB ∂z = β 4 (C − C2 ) at z = h, T −T2 , T1 −T2
u = rΩ1 f (ξ), v = rΩ1 g(ξ), w = −2hΩ1 f (ξ), θ = 1 r2 2 , p = ρΩ ν P (ξ) + , ξ = hz , φ = CC−C 1 2 h2 1 −C2
(4)
(5)
(6)
(6)
(7)
ED
in which p denotes the pressure, (u, v, w) the velocity components, (r, θ, z) the cylindrical coordinates, ρ the density, σ the electrical conductivity, B0 the magnetic field strength, g the
PT
gravitational acceleration, T the temperature, T1 and T2 the temperatures for convection, C the concentration, C1 and C2 the concentrations for convection, a1 and a2 the stretching
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rates at lower and upper disks, DB the coefficient of Brownian diffusion, DT coefficient of thermal expansion, Ω 1 the angular velocity at lower disk, Ω 2 the angular velocity at upper disk, k the thermal conductivity, λ1 and λ2 the linear and nonlinear coefficients of thermal expansion, h the distance between two disks, λ3 and λ4 the linear and nonlinear coefficients (ρc ) p of concentration expansion, τ = (ρcp )p the ratio between the effective nanoparticle material f
heat capacity and the base fluid heat capacity, ρp the particle density, ρf the fluid density,
k thermal conductivity, ν the kinematic viscosity, λ5 and λ6 the velocity slip coefficients at lower and upper disks, β 1 and β 2 the coefficients of convective heat transfer and convective mass transfer at lower disk and β 3 and β 4 the coefficients of convective heat transfer and convective mass transfer at upper disk. Applying the above transformation expression (1) is identically verified and remaining Eqs. (2-7) yield f 000 + Re 2f f 00 − f 02 + g 2 − M f 0 + α1 θ(1 + β t θ) + α1 N ∗ φ(1 + β c φ) − = 0, g 00 + Re (2f g 0 − 2f 0 g − M g) = 0, 4
(8) (9)
ACCEPTED MANUSCRIPT P 0 = −4 Re f f 0 − 2f 00 ,
(10)
1 00 θ + 2 Re f θ 0 + N tθ 02 + N bθ 0 φ0 + Re M Ec(f 02 + g 2 ) = 0, Pr N t 00 θ = 0, φ00 + 2f φ0 Sc Re + Nb
(11)
6
SC
5
RI P
T
(12) 0 00 0 00 0 f (0) = 0, f (1) = 0, f (0) = A1 + γ 1 f (0), f (1) = A2 − γ 1 f (1), g(0) = 1 + γ 2 g (0), 0 0 0 g(1) = Ω − γ 2 g (1), θ (0) = −γ 3 (1 − θ(0)), θ (1) = −γ 4 θ(1), 0 0 φ (0) = −γ (1 − φ(0)), φ (1) = −γ φ(1), P (0) = 0. 2
(13) 2 B σ
upper disks.
PT
ED
MA
NU
elucidates Reynolds number, Pr = (ρckp )ν the Prandtl number, M = ρΩ0 1 the Hartman number, A1 = Ωa11 and A2 = Ωa21 the scaled stretching rate variables, 1 −T2 ) γ 1 = λd5 and γ 2 = λd6 the velocity slip variables, α1 = gλ1 (T the mixed convection (or rΩ21 2 2 r Ω1 λ3 (C1 −C2 ) ∗ thermal buoyancy) variable, Ec = cp (T1 −T2 ) the Eckert number, N = λ1 (T1 −T2 ) the ratio of concentration to thermal buoyancy forces, β t = λ2 (Tλ11−T2 ) and β c = λ4 (Cλ13−C2 ) the non (ρc)p DT (T1 −T2 ) the linear convection variables due to temperature and concentration, N t = (ρc)f νT2 ν B (C1 −C2 ) thermophoresis variable, N b = (ρc)p D(ρc) the Brownian motion variable, Sc = DB ν f β1h β3h the Schmidt number, γ 3 = k and γ 4 = k the Biot numbers of heat transfer for lower and upper disks and γ 5 = βD2Bh and γ 6 = βD4Bh Biot numbers of mass transfer for lower and Ω1 h ν
Here Re =
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We now differentiated Eq. (9) with respect to ξ and obtain f iv + Re (2f f 000 + 2gg 0 − M f 00 + α1 θ0 + 2α1 β t θθ 0 + α1 N ∗ φ0 + 2α1 N ∗ β c φ0 φ) = 0.
(14)
Through Eqs. (9) and boundary conditions one has 2 = f 000 (0)+Re (g(0))2 − (f 0 (0)) − M f 0 (0) + α1 θ(0)(1 + β t θ(0)) + α1 N ∗ φ(0)(1 + β c φ(0)) . (15)
Integration of Eq. (11) from 0 to ξ yields P = −2 Re(f 2 ) + f 0 − f 0 (0) .
5
(16)
ACCEPTED MANUSCRIPT 3
Declaration of curiosity
3.1
Skin friction coefficients (surface drag forces)
Cf 2 =
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Cf 1 =
τ w |z=0 , ρ(rΩ1 )2 τ w |z=h , ρ(rΩ1 )2
T
Mathematically at lower and upper disks we have (17)
∂u μrΩ1 f 00 (0) ∂v μrΩ1 g 0 (0) =μ = = μ = , τ . zθ ∂z z=0 h ∂z z=0 h
Total shear stress is defined as
q
τ 2zr + τ 2zθ .
Putting Eq. (17) in Eq. (16) we have
+ (g (0)) ] , [(f 00 (1))2 + (g 0 (1))2 ]1/2 .
Cf 2 =
1 Rer
0
3.2
PT
ED
Cf 1 =
1 [(f 00 (0))2 Rer
(18)
(19)
MA
τw =
NU
τ zr
SC
where shear stresses in radial (τ zr ) and tangential (τ zθ ) directions are
2 1/2
(20)
Nusselt numbers (heat transfer rates)
where
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Mathematically for lower and upper disks we write hqw hq w , N ux2 = N ux1 = , k(T1 − T2 ) z=0 k(T1 − T2 ) z=h qw |z=0 = −
qw |z=h
k ∂T ∂z z=0
= − k ∂T
∂z z=h
= =
0 − k(T1 −Th2 )θ (0) , 0 − k(T1 −Th2 )θ (1) .
Invoking Eq. (21) in Eq. (20) one has
N ux1 = −θ0 (0), N ux2 = −θ0 (1).
(21)
(22)
(23)
Here (τ w ) denotes the shear stress, (qw ) the heat flux and Rer (= rΩ1 h/ν) the local Reynolds number.
6
ACCEPTED MANUSCRIPT 4
Series solutions
Initial guesses (f0 (ξ), g0 (ξ), θ 0 (ξ), φ0 (ξ)) and auxiliary linear operators (Lf , Lg , Lθ , Lφ ) are taken as follows
−6γ 1 A1 ξ 2 + A1 ξ 3 + A2 ξ 3 + 2A2 γ 1 ξ 3 + 2γ 1 A1 ξ 3
g0 (ξ) =
,
SC
(1 + 2γ 1 )(1 + 6γ 1 )
1 + γ 2 + γ 2 Ω + (Ω − 1)ξ , 1 + 2γ 2
(24)
(25)
γ 3 + γ 3 γ 4 (1 − ξ) , γ 3 + γ 4 + γ 3γ 4
(26)
γ 5 + γ 5 γ 6 (1 − ξ) , γ 5 + γ 6 + γ 5γ 6
(27)
MA
θ0 (ξ) =
NU
f0 (ξ) =
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T
A1 ξ + 4γ 1 A1 ξ − 2γ 1 A2 ξ − 2A1 ξ 2 − A2 ξ 2
φ0 (ξ) =
Lf = f 0000 , Lg = g 00 , Lθ = θ00 , Lφ = φ00 ,
ED
(28)
with
AC CE
PT
Lf˜ c∗1 + c∗2 ξ + c∗3 ξ 2 + c∗4 ξ 3 = 0,
(29)
Lg˜[c∗5 + c∗6 ξ] = 0,
(30)
L˜θ [c∗7 + c∗8 ξ] = 0,
(31)
Lφ˜ [c∗9 + c∗10 ξ] = 0,
(32)
in which c∗i (i = 1 − 10) indicate arbitrary constants.
5
Inspection of entropy generation
Dimensional form of entropy generation is defined as [38, 39]: h 2 i ∂T 2 + + Tμm Φ SG = Tk2 ∂T ∂r ∂z m h 2 i RD ∂T ∂φ ∂φ ∂φ 2 + Tσm B02 (u2 + v 2 ) + RD + + Tm ∂r ∂r + ∂r ∂z Cm
where
Φ=
2
h
∂u 2 ∂r
+
∂w ∂r
2 i ∂v 2 + ∂w + ∂z ∂z 2 2 ∂ v + ∂u + r ∂r , ∂z r
+
2u2 r2
7
∂T ∂φ ∂z ∂z
,
(33)
(34)
ACCEPTED MANUSCRIPT Putting Eq. (33) in Eq. (32) we have SG =
where Cm =
+
+
C1 +C2 2
{z
∂T ∂z
2 # }
RD Cm |
∂φ ∂r
∂φ ∂z
+
T
Thermal irreversibility ∂u 2 2u2 ∂w 2 ∂v 2 + r2 + 2 ∂z + ∂z σ 2 2 ∂r + B0 (u + v 2 ) 2 2 2 ∂w ∂u ∂v ∂ v T m + ∂z + ∂r + r ∂r r | {z } ∂r {z } Joule dissipation irreversibility friction irreversibility "Fluid 2 2 #
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|
2
+
RD ∂T ∂φ ∂T ∂φ , + Tm ∂r ∂r ∂z ∂z {z } +
SC
+ Tμm
|
∂T ∂r
2
Diffusive irreversibility
and Tm =
T1 +T2 2
NU
k 2 Tm
"
(35)
represent the mean concentration and temperature
MA
respectively, D the mass diffusivity and R the constant.
Re
represents the dimensionless temperature ratio parameter, μΩ21 h2 4C C1 −C2 = the dimensionless concentration ratio parameter, Br = the Cm Cm k4T 2 T m SG ν Brinkman number, NG = k4T the entropy generation rate, A = hr 2 the dimensionless Ω1 RD(C1 −C2 ) the diffusive parameter. radial parameter and L = k
=
4T Tm
2 Re
(36)
PT
=
T1 −T2 Tm
1
AC CE
where α∗2 =
α∗1
ED
Dimensionless number NG (ξ) for the entropy generation is defined as 02 ∗ 1 Br 02 ∗ 02 ∗ 002 NG (ξ) = θ α1 Re + Re (12f + A g + A f ) +M BrA∗ (f 02 + g 2 ) + Lα∗ α∗ 1 φ02 + L 1 θ0 φ0 ,
Bejan number (Be) in dimensionless form is defined as
Entropy generation subject to heat and mass transfer Total entropy generation 02 ∗ 1 1 0 0 ∗ ∗ 1 02 θ α1 Re + Lα1 α2 Re φ + L Re θ φ Be = 1 θ02 α∗1 Re + Br (12f 02 + A∗ g 02 + A∗ f 002 ) + Re 1 0 0 ∗ 02 2 ∗ ∗ 1 02 M BrA (f + g ) + Lα1 α2 Re φ + L Re θ φ
Be =
6
(37)
(38)
Convergence analysis
Selection of auxiliary variables }f , }g , }θ and }φ provide us opportunity to adjust the convergence of nonlinear expressions. To show the convergence of nonlinear expressions graphically form Fig. 2 is plotted. From Fig. 2 it is observed that the permissible values of auxiliary 8
ACCEPTED MANUSCRIPT variables lie in the ranges −2.0 ≤ }f ≤ −0.1, −1.9 ≤ }g ≤ −0.1, −1.8 ≤ }θ ≤ −0.2 and
−2.3 ≤ }φ ≤ −0.1. Table 1 communicates the convergence of series solutions of concentration, temperature and momentum for the given flow analysis correct upto four decimal
T
places. Table 1 designates that f 00 (0), g 0 (0), θ 0 (0) and φ0 (0) converge at 5th , 10th , 35th and
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45th order of approximations.
Table 1: Different order of approximations when Re = 0.1, Pr = 1.0, M = 0.5, A1 = 0.4, A2 = 0.9, γ 1 = 0.5, γ 2 = 0.7, γ 3 = 0.5, γ 4 = 0.7, γ 5 = 0.01, γ 6 = 0.01, α1 = α2 = 0.4,
−f 00 (0)
−g 00 (0)
SC
−θ0 (0)
−φ0 (0)
1
0.7221
0.30627
0.2205
0.0098
5
0.7222
0.3061
0.2188
0.0099
10
0.7222
0.3060
0.2194
0.0100
0.7222
0.3060
0.2196
0.0100
0.7222
0.3060
0.2196
0.0100
0.7222
0.3060
0.2196
0.0100
0.7222
0.3060
0.2195
0.0099
35
0.7222
0.3060
0.2194
0.0099
40
0.7222
0.3060
0.2194
0.0099
45
0.7222
0.3060
0.2194
0.0100
50
0.7222
0.3060
0.2194
0.0100
55
0.7222
0.3060
0.2194
0.0100
60
0.7222
0.3060
0.2194
0.0100
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Order of approximation
MA
Ω = 0.3, Bt = 0.5, Bc = 0.7, N t = 0.1, N b = 1.0, Ec = 0.4 and Sc = 1.5.
15
25
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PT
30
ED
20
7
Physical analysis
This section illuminates the impact of relevant variables on velocities axial f (ξ), radial f 0 (ξ), tangential g(ξ), concentration (φ(ξ)), temperature (θ(ξ)), entropy generation NG (ξ), Bejan number (Be), skin friction coefficients (Cf 1 , Cf 2 ) and heat transfer rates (N ux1 , N ux2 ).
9
ACCEPTED MANUSCRIPT 7.1 7.1.1
Velocity distribution Axial (f (ξ)) , radial (f 0 (ξ)) and tangential (g(ξ)) velocity field
Figs. (3 − 16) are depicted to display the behavior of involved parameters on axial ( f (ξ)),
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T
radial (f 0 (ξ)) and tangential (g(ξ)) velocity fields. This section comprises the graphs and
related investigations for various physical variables. Figs. (3 − 5) depict the impact of
Reynold number (Re) on axial (f (ξ)), radial (f 0 (ξ)) and tangential (g(ξ)) velocity fields.
SC
Here magnitude of axial (f (ξ)) and tangential (g(ξ)) velocities increase for larger estimation of Reynold number near the lower disk. Physically for larger Reynold number the viscous
NU
effects decays and therefore less resistance occurs to liquid motion. Magnitude of axial velocity initially increases at lower disk and then decreases as the value of (Re) approache
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to 6. Figs. (6 − 8) illustrate the Hartman number (M ) impact on axial (f (ξ)), radial (f 0 (ξ)) and tangential (g(ξ)) velocity fields. Axial, radial and tangential are decreased by Hartman number (M ). Since Lorentz force is resistive force so it resist the motion of liquid particles
ED
and therefore velocities (f (ξ)), (f 0 (ξ)) and (g(ξ)) decay. Figs. (9 − 11) are portrayed to
demonstrate the essential impact of velocity slip variable (γ 1 ) on axial (f (ξ)), radial (f 0 (ξ))
PT
and tangential (g(ξ)) velocity fields. Here magnitude of axial (f (ξ)) and radial (f 0 (ξ)) velocity fields have monotonic behavior for larger (γ 1 ) . Moreover tangential velocity decays for higher
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velocity slip variable (see Fig. 11). Fig. 12 illustrates that axial velocity is decreasing function of mixed convection (or thermal buoyancy) variable (α1 ). However dual behavior is observed for radial velocity (see Fig. 13). Similar impact is obtained for nonlinear convection variable (β t ) due to temperature on axial and radial velocities (see Figs.14 and 15) . Fig. 16 is sketched to show that tangential velocity profile g(ξ) decays at lower disk for larger estimation of slip parameter (γ 2 ). Near upper disk the tangential velocity enhances.
7.2
Temperature distribution
Figs. (17 − 22) are sketched to show the impact of various variable like Hartman number (M ), Biot number of heat transfer for lower disk (γ 3 ), Biot number of heat transfer for upper disk (γ 4 ), Eckert number (Ec), thermophoresis (N t) and Prandtl number (Pr) on temperature field (θ(ξ)) . Fig. 17 is sketched to examine temperature field for Hartman number. Here temperature is enhanced by Hartman number. As Hartman number ( M ) is linked with Lorentz force which is a resistive force. Clearlyt an enhancement in Hartman 10
ACCEPTED MANUSCRIPT number provides more resistance and therefore temperature field increases. From Fig. 18 it is clearly examined that stronger temperature field (θ(ξ)) is generated by larger estimation of Biot number of heat transfer for lower disk (γ 3 ). Physically heat transfer coefficient enhances
T
for larger Biot number (γ 3 ). Fig. 19 is sketched to examine temperature (θ(ξ)) for thermal
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Biot number of upper disk (γ 4 ). An increment in (γ 4 ) shows reduction in temperature field (θ(ξ)). Since purpose of Biot number is to determine the variation in temperature profile. It is noticed from Figs. 18 and 19 that liquid temperature varies considerably. Fig. 20 displayed
SC
the graphs of (θ(ξ)) for (Ec). Here temperature is increasing function of Eckert number (Ec). Physically inside friction of molecules the conversion of mechanical energy to thermal energy
NU
is the reason for temperature field. Fig. 21 presents the variation in temperature profile for thermophoresis variable (N t). It is examined that higher estimation of thermophoresis
MA
variable provides more (θ(ξ)) . Physically an increment in thermophoresis variable leads to more thermophoretic force which allows deeper movement of nanomaterials in liquid. Fig. 22 is drawn for the impact of Prandtl number on (θ(ξ)) . By increasing (Pr) less temperature
Concentration distribution
PT
7.3
ED
(θ(ξ)) is noticed.
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Figs. (23 − 27) are portrayed here to show the impact of Schmidt number (Sc), Biot number for mass transfer at lower disk (γ 5 ) , Biot number for mass transfer at upper disk (γ 6 ) , thermophoresis (N t) and Brownian motion variable (N b) on concentration field. Fig. 23 addresses the variation of Schmidt number (Sc) on concentration field (φ(ξ)) . An increment in Schmidt number (Sc) shows decrease in magnitude of (φ(ξ)) . The ratio of momentum diffusivity and mass diffusivity is called Schmidt number (Sc). Physically mass diffusivity decreases for more Schmidt number. Therefore concentration field reduces. Behavior of concentration Biot numbers (γ 5 ) and (γ 6 ) is presented in Figs. 24 and 25. For larger values of (γ 5 ) the concentration profile enhances. It is due to an increase in internal energy of liquid particles for larger (γ 5 ). Opposite behavior is observed for concentration Biot number of upper disk (γ 6 ). From these Figs. we can conclude that concentration varies considerably with the increase in (γ 5 ) and (γ 6 ) . Fig. 26 portrays the effect of N t on (φ(ξ)) . Enhancement in magnitude of concentration (φ(ξ)) is noticed for larger N t. Physically for larger N t the temperature difference between ambient and surface increases. That is why concentration of liquid enhances. For larger Brownian motion parameter (N b) the magnitude of concentration 11
ACCEPTED MANUSCRIPT decreases (see Fig. 27).
7.4
Entropy generation rate and Bejan number
T
Behaviors of Hartman number (M ), slip parameter (γ 1 ), thermal Biot number (γ 3 ), Brinkman
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number (Br), temperature ratio parameter (α∗1 ), thermophoretic parameter (N t), Eckert number (Ec), Schmidt number (Sc) and diffusive parameter (L) on local entropy generation
SC
NG (ξ) and Bejan number Be is scrutinized in Figs. (28 − 41).
Local entropy generation (NG (ξ)) and Bejan number (Be) for higher estimation of Hart-
NU
man number (M ) are graphically presented in Figs. 28 and 29. Here entropy generation rate and Bejan number have inverse behavior for higher estimation of Hartman number ( M ). Physically larger Lorentz force offers more resistance to the liquid particle motion. Thus
MA
more heat is produced and therefore entropy generation rate enhances. It is noticed that Bejan number is decreasing function of Hartman number. Figs. 30 and 31 are drawn to
ED
show the impacts of slip parameter (γ 1 ) on entropy generation rate and Bejan number. Here entropy generation rate and Bejan number show opposite behavior for rising values of ( γ 1 ).
PT
Impacts regarding thermal Biot number of lower disk (γ 3 ) on entropy generation rate NG (ξ) and Bejan number (Be) are displayed by Figs. (32 and 33). Here both entropy generation
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rate and Bejan number increase for higher estimation of (γ 3 ). Figs. (34 and 35) show entropy generation (NG (ξ)) and Bejan number (Be) for higher estimation of Brinkman number (Br) . Brinkman number determines the releasing of heat by viscous heating in relation to heat transfer by molecular conduction. Brinkman number is directly related to entropy generation near the disk. Heat transfer by molecular conduction is greater than heat released by viscous effects near the disk. Quite large amount of heat is generated between the layers of moving fluid particles. It increases the entropy and disorderness of the system also enhances. Fig. 35 shows that (Be) decays for Brinkman number. Figs. (36 and 37) portray impact of temperature ratio parameter (α∗1 ) on entropy generation rate NG (ξ) and Bejan number (Be). Here entropy generation rate and Bejan number are enhanced for larger estimations of ( α∗1 ). Figs. (38 and 39) depict the influence of thermophoresis variable (N t) and Eckert number (Ec) on Bejan number (Be) . Here Bejan number decreases for both thermophoresis variable (N t) and Eckert number (Ec). It is due to the fact that an enhancement in (N t) and (Ec) increase the internal source of energy. Furthermore higher concentration is achieved through larger thermal conductivity which decays the Bejan number. Fig. 40 elucidates analysis of 12
ACCEPTED MANUSCRIPT (Sc) on Bejan number. Fig. 40 shows Bejan number is increasing function of ( Sc). Fig. 41 displays behavior of diffusive parameter (L) on Bejan number. Here we noticed that Bejan
Drag forces (Cf 1 , Cf 2 ) and rates of heat transfer (N ux1 , N ux2 )
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7.5
T
number enhances for larger (L) .
Tables 2 and 3 are presented to show the computational outcomes for surface drag forces
SC
(Cf 1 , Cf 2 ) and heat transfer rates (N ux1 , N ux2 ). From Table 2 it is observed that surface drag force decays for different estimations of (λ1 ), (β c ) and (γ 1 ) near the lower disk while it
NU
shows reverse trend for Hartman number (M ). Surface drag force also decreases via (β 2 ), (γ 1 ) and (M ) close to upper disk and it increases for larger (λ1 ). Table 3 reveals that magnitude of heat transfer rate decays near the lower disk for higher values of (Ec) and (Pr) while it
MA
enhances for (γ 3 ). Also magnitude of rate of heat transfer enhances closed to upper disk for higher estimations of (γ 3 ), (Ec) and (Pr).
λ1
ED
Table 2: Numerical values of surface drag forces (Cf 1 , Cf 2 ). βc
γ1
M
Cf 1
Cf 2
0.7
0.5
0.5
0.7844334
1.258870
0.6
0.7842177
1.259100
0.8
0.7840075
1.259322
0.9
0.7844297
1.258874
1.1
0.7844289
1.258869
0.7
0.6204424
0.9996764
0.9
0.5273426
0.8368111
0.6
0.7856426
1.258344
0.7
0.7868482
1.257836
AC CE
PT
0.4
0.4
0.7
0.5
Table 3: Computational results for heat transfer rates (N u1 , N u2 ).
13
Ec
Pr
N ux1
N ux2
0.5
0.4
1
-0.21802
-0.23565
0.6
-0.23543
-0.25379
0.7
-0.24961
-0.26869
0.5
-0.21673
-0.23722
0.6
-0.21548
-0.23877
1.1
-0.21735
-0.23649
1.2
-0.21650
0.4
Conclusions
-0.23758
NU
8
SC
0.5
RI P
γ3
T
ACCEPTED MANUSCRIPT
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Entropy generation in mixed convective flow of nanofluid between two stretchable rotating disks is considered. Main conclusions of the present flow analysis are given below: • Magnitude of axial (f (ξ)) and tangential (g(ξ)) velocities increase for larger estimation
ED
of Reynold number near lower disk.
PT
• Magnitude of axial velocity initially increases at lower disk and then decreases when (Re) approaches to 6.
AC CE
• Axial, radial and tangential velocities are decreasing function of Hartman number ( M ). • Temperature is an increasing function of Hartman and Eckert numbers. • Higher estimation of thermophoresis variable (N t) provides larger temperature (θ(ξ)) . • An increment in Schmidt number (Sc) shows decay in magnitude of (φ(ξ)) . • Magnitude of concentration decays for via Brownian motion parameter ( N b). • Entropy generation rate and Bejan number have inverse behavior of Hartman number (M ). • Rate of entropy generation NG (ξ) enhances for larger Brinkman number whereas reverse trend is observed for Bejan number (Be).
14
ACCEPTED MANUSCRIPT References [1] S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME, FED
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231/Md, USA, (1995) 99-105.
ids, Journal of Heat Transfer, 125 (2003) 151-155.
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[2] Y. Xuan and Q. Li, Investigation on convective heat transfer and flow features of nanoflu-
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[3] K. L. Hsiao, Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature, International Journal of Heat and Mass
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Transfer, 112 (2017) 983-990.
[4] S. Qayyum, T. Hayat, A. Alsaedi and B. Ahmad, Magnetohydrodynamic (MHD) nonlin-
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ear convective flow of Jeffrey nanofluid over a nonlinear stretching surface with variable thickness and chemical reaction, International Journal of Mechanical Sciences, (2017)
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(In press).
[5] M. Sheikholeslami and H. B. Rokni, Influence of melting surface on MHD nanofluid flow
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by means of two phase model, Chinese Journal of Physics, 55 (2017) 1352-1360. [6] M. Farooq, M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, MHD stag-
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nation point flow of viscoelastic nanofluid with non-linear radiation effects, Journal of Molecular Liquids, 221 (2016) 1097-1103. [7] B. Mahanthesh, B. J. Gireesha, B. C. Prasannakumara and N. S. Shashikumar, Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source, Nuclear Engineering and Technology, 49 (2017) 1660-1668. [8] S. Qayyum, M. I. Khan, T. Hayat and A. Alsaedi, Comparative investigation of five nanoparticles in flow of viscous fluid with Joule heating and slip due to rotating disk, Physica B, (2018) Inpress. [9] G. K. Ramesh, B. C. Prasannakumara, B. J. Gireesha, S. A. Shehzad and F. M. Abbasi, Three dimensional flow of Maxwell fluid with suspended nanoparticles past a bidirectional porous stretching surface with thermal radiation, Thermal Science and Engineering Progress, 1 (2017) 6-14.
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ACCEPTED MANUSCRIPT [10] M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, Theoretical investigation of doubly stratified flow of Eyring-Powell nanomaterial via heat generation/absorption, European Physical Journal Plus, 132 (2017) 489.
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[11] B. C. Prasannakumara, B. J. Gireesha, M. R. Krishnamurthy and R. S. R. Gorla, Slip
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flow and nonlinear radiative heat transfer of suspended nanoparticles due to a rotating disk in the presence of convective boundary condition, International Journal of Nanopar-
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ticles, 9 (2017) 180-200.
[12] T. Hayat, S. Ullah, M. I. Khan and A. Alsaedi, On framing potential features of SWCNTs
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and MWCNTs in mixed convective flow, Results in Physics, 8 (2018) 357–364. [13] K. G. Kumar, B. J. Gireesha, B. C. Prasannakumara and O. D. Makinde, Impact of
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chemical reaction on Marangoni boundary layer flow of a Casson nano liquid in the presence of uniform heat source sink, Diffusion Foundations, 11 (2017) 22-32.
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[14] T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, Modern developments about statistical declaration and probable error for skin friction and Nusselt number with copper and
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silver nanoparticles, Chinese Journal of Physics, 55 (2017) 2501-2513.
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[15] B. C. Prasannakumara, B. J. Gireesha, M. R. Krishnamurthy and K. G. Kumar, MHD flow and nonlinear radiative heat transfer of Sisko nanofluid over a nonlinear stretching sheet, Informatics in Medicine Unlocked, 9 (2017) 123-132. ¨ [16] V. Karman. Uber laminare und turbulente Reibung, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 1 (1921) 233-252. [17] M. Turkyilmazoglu, Three dimensional MHD stagnation flow due to a stretchable rotating disk, International Journal of Heat and Mass Transfer, 55 (2012) 6959-6965. [18] M. Turkyilmazoglu, B¨odewadt flow and heat transfer over a stretching stationary disk, International Journal of Mechanical Sciences, 90 (2015) 246-250. [19] T. Hayat, M. Javed, M. Imtiaz and A. Alsaedi, Convective flow of Jeffrey nanofluid due to two stretchable rotating disks, Journal of Molecular Liquids, 240 (2017) 291-302.
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ACCEPTED MANUSCRIPT [20] T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, Current progresses about probable error and statistical declaration for radiative two phase flow using Ag − H2 O and Cu − H2 O nanomaterials, International Journal of Hydrogen Energy, 42 (2017) 29107-29120.
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[21] T. Hayat, M. Rashid, M. Imtiaz and A. Alsaedi, Nanofluid flow due to rotating disk with
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variable thickness and homogeneous-heterogeneous reactions, International Journal of Heat and Mass Transfer, 113 (2017) 96-105.
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[22] T. Hayat, M. I. Khan, A. Alsaedi and M. I. Khan, Joule heating and viscous dissipation
Mass Transfer, 89 (2017) 190-197.
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in flow of nanomaterial by a rotating disk, International Communications in Heat and
[23] Y. H. Liu, G. J. Peng, W. C. Lai, C. Y. Huang and J. H. Liang, Flow field investigation
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in a rotating disk chemical vapor deposition chamber with a perforated showerhead, Experimental Thermal and Fluid Science, 88 (2017) 389-399.
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[24] P. T. Griffiths, Flow of a generalised Newtonian fluid due to a rotating disk, Journal of
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Non-Newtonian Fluid Mechanics, 221 (2015) 9-17. [25] C. Yin, L. Zheng, C. Zhang and X. Zhang, Flow and heat transfer of nanofluids over a
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rotating disk with uniform stretching rate in the radial direction, Propulsion and Power Research, 6 (2017) 25-30. [26] B. S. Dandapat and S. K. Singh, Two-layer film flow over a rotating disk, Communications in Nonlinear Science and Numerical Simulation, 17 (2012) 2854-2863. [27] S. Xun, J. Zhao, L. Zheng, X. Chen and X. Zhang, Flow and heat transfer of Ostwald-de Waele fluid over a variable thickness rotating disk with index decreasing, International Journal of Heat and Mass Transfer, 103 (2016) 1214-1224. [28] T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, Homogeneous-heterogeneous reactions in nonlinear radiative flow of Jeffrey fluid between two stretchable rotating disks, Results in Physics, 7 (2017) 2557-2567. [29] T. Hayat, S. Qayyum, A. Alsaedi and B. Ahmad, Significant consequences of heat generation/absorption and homogeneous-heterogeneous reactions in second grade fluid due to rotating disk, Results in Physics, 8 (2018) 223-230. 17
ACCEPTED MANUSCRIPT [30] C. Ming, L. Zheng, X. Zhang, F. Liu and V. Anh, Flow and heat transfer of power-law fluid over a rotating disk with generalized diffusion, International Communications in Heat and Mass Transfer, 79 (2016) 81-88.
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[31] A. Bejan, Second law analysis in heat transfer, Energy, 5 (1980) 720-732. [32] H. Abbasi, Entropy generation analysis in a uniformly heated microchannel heat sink,
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Energy, 32 (2007) 1932-1947.
[33] D. Srinivasacharya and K. H. Bindu, Entropy generation in a porous annulus due to
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micropolar fluid flow with slip and convective boundary conditions, Energy, 111 (2016) 165-177.
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[34] A. Ebrahimi, F. Rikhtegar, A. Sabaghan and E. Roohi, Heat transfer and entropy generation in a microchannel with longitudinal vortex generators using nanofluids, Energy,
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101, (2016) 190-201.
[35] Z. Y. Xie and Y. J. Jian, Entropy generation of two-layer magnetohydrodynamic elec-
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troosmotic flow through microparallel channels, Energy 139 (2017) 1080-1093. [36] M. W. A. Khan, M. I. Khan, T. Hayat and A. Alsaedi, Entropy generation minimization
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(EGM) of nanofluid flow by a thin moving needle with nonlinear thermal radiation, Physica B, (2018) Inpress. [37] M. I. Khan, T. Hayat and A. Alsaedi, Numerical investigation for entropy generation in hydromagnetic flow of fluid with variable properties and slip, AIP, Physics of Fluid, (2018) Inpress.
[38] T. Hayat, M. I. Khan, S. Qayyum and A. Alsaedi, Entropy generation in flow with silver and copper nanoparticles, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 539 (2018) 335-346. [39] T. Hayat, S. Qayyum, M. I. Khan and A. Alsaedi, Entropy generation in magnetohydrodynamic radiative flow due to rotating disk in presence of viscous dissipation and Joule heating, AIP, Physics of Fluids 30 (2018) 017101. [40] D. Nouri, M. P. Fard, M. J. Oboodi, O. Mahian and A. Z. Sahin, Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden 18
ACCEPTED MANUSCRIPT expansion and contraction, International Journal of Heat and Mass Transfer, 116 (2018) 1036-1043. [41] S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman
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& Hall/CRC (2003).
[42] 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
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a variable thicked surface, International Journal of Heat and Mass Transfer, 99 (2016)
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702-710.
[43] L. Zheng, C. Zhang, X. Zhang and J. Zhang, Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous
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medium, Journal of The Franklin Institute, 350 (2013) 990-1007. [44] T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Stagnation point flow
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with Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions, Journal of
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Molecular Liquids, 220 (2016) 49-55. [45] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations
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arising in heat transfer, Physics Letters A 360 (2006) 109-113. [46] 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.
[47] A. Alseadi, M. I. Khan, M. Farooq, N. Gull and T. Hayat, Magnetohydrodynamic (MHD) stratified bioconvective flow of nanofluid due to gyrotactic microorganisms, Advance Powder Technology 28 (2017) 288-298. [48] M. I. Khan, T. Hayat, M. I. Khan and A. Alsaedi, A modified homogeneousheterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating, International Journal of Heat and Mass Transfer 113 (2017) 310-317. [49] M. I. Khan, M. I. Khan, M. Waqas, T. Hayat and A. Alsaedi, Chemically reactive flow of Maxwell liquid due to variable thicked surface, International Communications in Heat and Mass Transfer 86 (2017) 231-238. 19
ACCEPTED MANUSCRIPT [50] T. Hayat, M. I. Khan, M. Waqas and A. Alsaedi, Effectiveness of magnetic nanoparticles in radiative flow of Eyring-Powell fluid, Journal of Molecular Liquids 231 (2017) 126-133. Figure Captions
Fig. 3: f (ξ) via Re .
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Fig. 4: f 0 (ξ) via Re . Fig. 5: g(ξ) via Re .
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Fig. 6: f (ξ) via M. Fig. 7: f 0 (ξ) via M. Fig. 8: g(ξ) via M.
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Fig. 9: f (ξ) via γ 1 . Fig. 10: f 0 (ξ) via γ 1 . Fig. 12: f (ξ) via α1 .
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Fig. 13: f 0 (ξ) via α1 .
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Fig. 11: g(ξ) via γ 1 .
Fig. 14: f (ξ) via β t .
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Fig. 2: ~−curves for f 00 (0), g 0 (0), θ 0 (0) and φ0 (0).
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Fig. 1: Flow analysis and coordinate system.
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Fig. 15: f 0 (ξ) via β t . Fig. 16: g(ξ) via γ 2 .
Fig. 17: θ(ξ) via M. Fig. 18: θ(ξ) via γ 3 . Fig. 19: θ(ξ) via γ 4 .
Fig. 20: θ(ξ) via Ec. Fig. 21: θ(ξ) via N t. Fig. 22: θ(ξ) via Pr . Fig. 23: φ(ξ) via Sc. Fig. 24: φ(ξ) via γ 5 . Fig. 25: φ(ξ) via γ 6 . Fig. 26: φ(ξ) via N t. Fig. 27: φ(ξ) via N b. Fig. 28: NG (ξ) via M. Fig. 29: Be via M. 20
ACCEPTED MANUSCRIPT Fig. 30: NG (ξ) via γ 1 . Fig. 31: Be via γ 1 . Fig. 32: NG (ξ) via γ 3 .
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Fig. 33: Be via γ 3 .
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Fig. 34: NG (ξ) via Br. Fig. 35: Be via Br. Fig. 36: NG (ξ) via α∗1 .
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Fig. 37: Be via α∗1 . Fig. 38: Be via N t.
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Fig. 39: Be via Ec. Fig. 40: Be via Sc.
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Fig. 41: Be via L.
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ACCEPTED MANUSCRIPT Highlights • Characteristics of entropy generation in nonlinear mixed convective flow of nanofluid
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between two stretchable rotating disks are considered.
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• Buongiorno nanofluid model of nanomaterial is implemented in mathematical modeling. • Nanofluid aspects for thermophoresis and Brownian movement are considered.
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• Heat transport mechanism is examined subject to convective condition and Joule heating.
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• Velocity slip is considered at the lower and upper disks.
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• Total entropy generation rate is discussed.
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