- Email: [email protected]
sink on entropy generation minimization rate
Accepted Manuscript Nonlinear radiative heat flux and heat source/sink on entropy generation minimization rate T. Hayat, M. Waleed Ahmed Khan, M. Ijaz Khan, A. Alsaedi PII:
S0921-4526(18)30084-X
DOI:
10.1016/j.physb.2018.01.054
Reference:
PHYSB 310701
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 17 January 2018 Revised Date:
25 January 2018
Accepted Date: 25 January 2018
Please cite this article as: T. Hayat, M.W.A. Khan, M. Ijaz Khan, A. Alsaedi, Nonlinear radiative heat flux and heat source/sink on entropy generation minimization rate, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.01.054. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Nonlinear radiative heat flux and heat source/sink on entropy generation minimization rate T. Hayat M. W. A. Khan , M. Ijaz Khan1 and A. Alsaedi Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
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Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia
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Abstract: Entropy generation minimization in nonlinear radiative mixed convective flow towards a variable thicked surface is addressed. Entropy generation for momentum and temperature
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is carried out. The source for this flow analysis is stretching velocity of sheet. Transformations are used to reduce system of partial differential equations into ordinary ones. Total entropy generation rate is determined. Series solutions for the zeroth and mth order deformation systems are computed. Domain of convergence for obtained solutions is identified. Velocity, temperature and concentration fields are plotted and interpreted. Entropy equation is studied through nonlinear mixed convection and radiative heat flux. Velocity and temperature gradients are discussed through graphs.
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Meaningful results are concluded in the final remarks.
Keywords: Entropy generation; Heat source/sink; Nonlinear mixed convection; Nonlinear thermal radiation; Bejan number. 1
Corresponding author:
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email address: [email protected]
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Fig. 1. Flow geometry
1
Introduction
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Boundary layer flows over a stretched surface with heat transfer analysis has greatly attracted many researchers for its vast applications in many industrial and technological processes for example glass fiber manufacturing, drawing of plastic thin films, cooling and drying of papers and polymer industries etc. Crane [1] initiated this work by providing exact solution
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for governing equations of flow analysis over a stretched surface. Non-Fourier heat flux in chemically reactive flow in the presence of temperature dependent thermal conductivity
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towards a stretched surface is presented by Hayat et. al. [2] Homotopy analysis technique is implemented to contract the series solution of nonlinear expressions. Their obtained outcomes highlight that temperature field is decays for higher estimation of thermal relaxation factor. It is also investigated that surface drag force is increases for higher ratio parameter and Deborah number in term of retardation time. Total residual errors are presented for velocity, concentration and temperature expressions. Impact of homogeneous-heterogeneous reactions in stagnation point flow with non-Fourier heat flux model is studied by Hayat et al. [3]. They discussed variable thicked surface first time with non-Fourier heat flux. The results indicate that temperature and concentration fields decays via thermal relaxation factor and chemical reaction variable. Furthermore series solutions are obtained through HAM. Heat and 2
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mass transport in non-Newtonian liquid through isothermal chemical reaction is examined by Khan et al. [4] MHD convective flow of rate type nanomaterial towards exponential stretched surface with slip conditions. Cross diffusion impact in radiative flow of viscous fluid with inclined magnetic field over a stretched surface is scrutinized by Raju et al. [5].
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Some latest analyses through this aspects referred through Refs. [6 − 15].
From a thermodynamic perspective, heat transport is always related with a least change of entropy. Entropy is process of irreversibility in system. Entropy minimization is the optimization process used to increase the efficiency of machines. Second law of thermodynamics
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is the main cause for entropy minimization process like internal molecular friction, vibration, spin moment and kinetic energy etc. In real life processes such loss of energy cannot be
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regained without extra work done. Therefore entropy is considered to be the measure of irreversibility. This process of entropy minimization is used by researchers in many systems such as fuel cells, gas turbines, natural convection and evaporative cooling etc. Few resent articles through this direction are given in Refs [16 − 25]
The phenomenon of stretching is very helpful in many industrial applications such as extrusion of plastic sheets, paper production, wire drawing, hot rolling, cooling of metallic
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sheets, glass blowing and polymer engineering etc. In such applications the quality of products mainly depends on rate of heat transport. Therefore choice of suitable cooling/heating liquid is very important because of its direct impact on the heat transfer rate. MHD convective
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nanomaterial flow of viscoelastic liquid subject to chemical reaction and variable thicked surface is examined by Qayyum et. [26]. Haq et al. [27] investigated MHD flow of carbon2 nanomaterial towards a stretchable surface. Heat transport mechanism in viscoelastic
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fluid flow with non-Fourier heat flux is studied by Li et al. [28]. Waqas et al. [29] presented natural convective flow of non-Newtonian liquid with convective boundary constraint. Farooq et al. [30] scrutinized MHD nanoliquid flow of Jeffrey model near a stagnation point with nonlinear thermal radiation. Hayat et al. [31] explored nanomaterial flow of Walters-B fluid in the presence of convective boundary conditions and variable thicked surface. Hsiao [32] worked on MHD nanoliquid flow of micropolar fluid with dissipation effects. Qayyum et al. [33] scrutinized melting heat transport and tangent hyperbolic liquid flow with isothermal chemical reaction and inclined MHD over a nonlinear stretched surface. Prasannakumara et al. [34] studied nonlinear thermal radiative flow of Sisko fluid. Thammanna et al. [35] examined MHD Casson fluid flow with chemical reaction towards a stretchable surface. 3
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The purpose of this research is to analyze nonlinear mixed convective flow of viscous material towards a variable thicked surface. Heat transport properties have been analyzed in the presence of nonlinear radiative heat flux and heat source/sink. Thickness of sheet is taken to be variable. Simple isothermal chemical reaction is implemented. Total entropy
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generation has been investigated for different flow variables. Homotopy analysis technique [36 − 50] is implemented to contract the series solution of nonlinear ordinary ones. Results are presented graphically.
Formulation
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2
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Here steady, incompressible 2 mixed convective flow of viscous liquid towards a variable thicked surface is considered. A uniform magnetic field of strength 0 is applied perpendicular to the plane of sheet. Smaller Reynolds number implies no induced magnetic field. Nonlinear thermal radiation and heat source/sink effects are considered. Furthermore effect of nonlinear mixed convection and simple chemical reaction are examined. Sheet lies parallel to -axis with stretching velocity = 0 ( + ) (see Fig. 1). Here is wall temperature while ∞
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being ambient temperature The relevant flow problems are [15]: + = 0
(1)
⎫ + = + (1 ( − ∞ ) + 2 ( − ∞ ) ) ⎬ ⎭ + (3 ( − ∞ ) + 4 ( − ∞ )2 ) µ ¶ 2 1 ˆ + ˆ = + + ( − ∞ ) 2
2 2
2
(2)
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ˆ
(3)
2 + ˆ = 2 − 1 ( − ∞ )
= = 0 ( + ) = 0 = = at = 1 ( + ) = 0 = 0 = ∞ = ∞ at → ∞
(4) 1− 2
⎫ ⎬ ⎭
(5)
Here indicates velocity components, kinematic viscosity, gravitational acceleration, 1 and 2 linear and nonlinear coefficients of thermal expansion, 3 and 4 linear and nonlinear coefficients of solutal expansion, temperature, ∞ ambient temperature, concentration,∞ ambient concentration, thermal conductivity, radiative heat flux, density, specific heat, coefficient of heat source/sink, diffusion coefficient, 1 chemical 4
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reaction rate, stretching velocity, dimensional constant, 1 small variable regarding surface is sufficient thin and power law index.
Θ () =
−∞ −∞
Φ=
−∞ −∞
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Equation of continuity is identically zero and the remaining equations one has ⎫ ¡ 2 ¢ 02 ¡ 2 ¢ 21 000 00 0 0 02 ⎬ + − +1 − +1 + +1 (Θ + Θ ) ∗ ⎭ + 21 (Φ0 + Φ02 ) = 0 +1
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⎫ Θ00 + 43 (Θ( − 1) + 1)2 [3Θ02 ( − 1) + (Θ( − 1) + 1) Θ00 ] ⎬ ¡ 2 ¢ ⎭ + Pr Θ0 + +1 Pr Θ = 0 µ ¶ 2 00 0 Φ + 2 Re Φ − 1 Φ = 0 +1 ⎫ ¡ 1− ¢ 0 0 ⎪ () = 1+ , () = 0 (∞) = 0 ⎪ ⎪ ⎬
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Θ() = 1 Θ(∞) = 0 Φ() = 1 Φ(∞) = 0
Applying the following transformations
⎫ ⎪ () = ( − ) = () ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
⎪ ⎪ ⎪ Φ () = ( − ) = () ⎭
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000 + 00 −
00
+
¡
¡ 2 ¢ ¡ 0 ¢ ⎫ 0 21 2 − +1 0 + +1 + 02 ⎬ ¢ ∗ ¡ 0 ⎭ 1 + 2+1 + 02 = 0
2 +1
¢
£ 02 ¤ ⎫ 00 ⎬ − 1) + 1) 3 ( − 1) + (( − 1) + 1) ¡ 2 ¢ ⎭ + Pr 0 + +1 Pr = 0 ¶ µ 2 00 0 + 2 Re − 1 = 0 +1 ⎫ ¡ 1− ¢ 0 0 ⎪ (0) = 1+ , (0) = 1 (∞) = 0 ⎪ ⎪ ⎬
4 (( 3
⎪ ⎪ ⎪ ⎭
(6)
(7)
(8)
(9)
(10)
(11)
Θ () = ( − ) = ()
Equations (7 − 10) one has
⎫ ⎪ ⎪ ⎪ ⎬
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Considering the following transformations q q −1 +1 = (+1)02(+) = 20 (+) () +1 q ¢ −1 ¡ = 0 ( + ) 0 () = − (+1)02(+) 0 () () + −1 +1
(12)
2
(0) = 1 (∞) = 0
(0) = 1 (0) = 0 5
⎪ ⎪ ⎪ ⎭
(13)
(14)
(15)
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02 0 (+)−1
´
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´ ³ ¡ ¢ shows magnetic parameter, Pr = Prandtl number, = 0 (+) −1 ³ ´ ³ ´ ( −∞ ) 2 ( −∞ )2 heat generation parameter 1 = 21(+) linear mixed convection parameter, = 2−1 02 (+)2−1 0 ³ ´ ( − ) ∞ nonlinear mixed convection variable for temperature, ∗ = ( −∞ ) 3 the ratio of ³ ´ concentration to thermal buoyancy forces, = ∞ temperature ratio parameter, ´ ´ ³ ³ ¡ ¢ ∗ 3 16 ∞ (+) radiation parameter, = Reynolds num = 3∗ Schmidt number , Re ³ ´ ³ ´ 2 −∞ ) 4 ber, 1 10 ( + )1− chemical reaction parameter and = (( nonlinear mixed −∞ ) ³ Here =
convection variable for concentration.
3.1
Surface drag force
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Quantities of interest
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3
Mathematical expression for surface drag force is 1 Re05 = 2
Heat transfer rate
+1 2
¶12
˜0 (0)
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3.2
µ
(17)
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Mathematical expression of heat transfer rate is ¶ ¶µ µ 4 +1 −05 3 1 + (1 + ( − 1)(0)) 0 (0) =− Re 2 3
(16)
3.3
Mass transfer rate
4
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The expression for Sherwood number is defined as Re−05 = −0 (0)
(18)
Exploration of entropy generation
Equation for entropy generation in dimensional form is given as µ ¶2 µ ¶2 2 2 = 2 + + (∇)2 + (∇ · ∇ ) + ( ) ∞ ∞ ∞ ∞ ∞ 0 where
6
(19)
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´ ³ ´ ⎫ ⎬ ´ ³ ´ ³ ∇ · ∇ = ⎭ (∇)2 =
³
(20)
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After using boundary layer assumptions, it can be written as
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Joule dissipation irreversibility
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(21)
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µ ¶2 µ ¶2 + = 2 ∞ | {z } | ∞ {z } Thermal irreversibility Fluid friction irreversibility ³ ´2 ³ ´ + + ∞ ∞ 2 2 ( + 2 ) + ∞ 0 | {z }
Equation (21) indicates that there are three main sources of entropy generation. First term is due to heat transfer with radiation effects. Second term occurs because of fluid friction irreversibility and last term is caused by Joule dissipation irreversibility. Dimensionless number for entropy generation is defined as
⎫ ⎪ = 1 + − 1) + 1] 2 ⎪ ⎪ 2 ⎬ £¡ +1 ¢ ¡ ¢ ¤ 0 0 02 +1 + + 1 2 2 2 ⎪ ⎪ ¡ +1 ¢ ⎪ ⎭ 002 02 + 2 + 3
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4 [ ( 3
where
02
⎫ = 1 = − ∞ ) ⎬ ⎭ = ∆ ∞(+) −1
−∞
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2 =
¡ +1 ¢
∆
(
(22)
(23)
0
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Here 2 denotes dimensionless temperature difference, Brinkman number and ³ ´ ∞ = ∆ (+)−1 entropy generation rate. 0
Irreversibility due to heat transfer is prominent when À 05. On the other hand when
¿ 05 the viscous effects dominates. For = 05 both effects are equal. Bejan number () in dimensionless form is defined as Entropy generation due to heat transfer Total entropy generation ¡ ¢ 02 2 1 + 43 [ ( − 1) + 1]3 +1 2 ⎫ = 3 ¡ +1 ¢ 02 4 ⎪ 1 + 3 [ ( − 1) + 1] 2 ⎪ ⎪ 2 ⎬ £¡ +1 ¢ ¡ ¢ ¤ 0 0 02 +1 + + 1 2 2 2 ⎪ ⎪ ¡ +1 ¢ ⎪ ⎭ 002 02 + 2 +
=
7
(24) (25)
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5
Homotopy procedure
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This procedure provide a greater opportunity to choose initial approximations along with linear operators. So initial guesses and linear operators are defined as ⎫ ⎪ 0 () = exp(−) ⎪ ⎪ ⎬ 0 () = exp(−)
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⎪ ⎪ ⎪ 0 () = exp(−) ⎭
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L = 00 − L = 00 − L = 00 − with
£ ¤ L 2 + 3 − = 0 £ ¤ L 6 + 7 − = 0 £ ¤ L 8 + 9 − = 0
6
⎪ ⎪ ⎪ ⎭
(27)
(28)
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where ( = 1 − 9) denotes the arbitrary constants.
⎫ ⎪ ⎪ ⎪ ⎬
(26)
Convergence analysis
Auxiliary parameters } } and } are chosen in a wide range which provides us the advantage
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to adjust the convergence of nonlinear system of equations. Fig. 2 displays the ~-curves for momentum, thermal and concentration equations at different order of approximations.
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Suitable estimations of these variables are established in the ranges −15 ≤ } ≤ −02 −13 ≤ } ≤ −03 and −17 ≤ } ≤ −03 Table 1 highlights convergence of series solutions of momentum, energy and concentration constraints for the concerned flow problem. Table 0
0
0
1 indicates that (0) (0) and (0) converge at 15 , 25 and 19 order of approximations
8
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respectively. f''(0) = q '(0) = f'(0) =
0.0
-0.5
-1.0
-1.5
- 1.5
- 1.0
Ñ f ,Ñq , Ñf
- 0.5
0.0
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- 2.0
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f '' 0 ,q ' 0 , f ' 0
0.5
Fig. 2. ~−curves for and
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Table 1: Different order of approximations when 1 = 03 = = 05 = 07 = 07 Pr = 19 = 01 = 03 ∗ = 02 = 05 = 12 Re = 10 and 1 = 05 0 (0)
Order of approximation 1 11
−0 (0)
0.1610 0.8295 1.55556
0.2859 0.5038 1.26243 0.2857 0.5233 1.25899
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15
−0 (0)
0.2857 0.5341 1.25894
25
0.2857 0.5350 1.25894
30
0.2857 0.5350 1.25894
35
0.2857 0.5350 1.25894
40
0.2857 0.5350 1.25894
50
0.2857 0.5350 1.25894
7
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19
Discussion
The influence of dimensionless parameters on MHD flow of viscous liquid towards a variable thicked surface. Figs. (3 − 22) are plotted for outcomes of velocity 0 () temperature (), ¡ ¢ ¡ ¢ concentration (), skin friction coefficient Re05 heat transfer rate Re−05 and ¢ ¡ Sherwood number Re−05 . Behavior of wall thickness variable on velocity, temperature and concentration are shown in Figs. 3 − 3 Characteristics of wall thickness variable on
velocity is sketched in Fig. 3. As wall thickness increases, less force is transferred to fluid due to stretching of sheet. That is why velocity tends to decrease. Effect of wall thickness 9
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parameter on temperature is given in Fig. 3. Here () is a decreasing function of wall thickness variable. Physically less heat is transfer to system, so temperature decreases. Effects of mixed convection parameter (1 ) on velocity is displayed via Fig. 4. Magnitude of velocity enhances for larger (1 ). Fig. 4 shows that for larger values of 1 higher thermal
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buoyancy force results an increase in velocity. Silent features of heat generation parameter is sketched through Fig. 5. More heat is produced in the system by introducing heat generation factor. That is why temperature profile increases. Fig. 6 is portrayed to show the behavior of mixed convection parameter due to temperature ( ). It reveals that velocity
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is an increasing function of Figs. 7 and 8 are displayed to show the impact of magnetic parameter on velocity field As magnetic parameter is related with Lorentz force (resistive
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force) so an increase in provides more resistance to liquid flow and as a result velocity decays. Influence of magnetic parameter on temperature ˜() is shown in Fig. 8. Here an increase in temperature profile is observed for larger This is due to the fact that higher Lorentz force provides more resistance to fluid particles motion. Thus more heat is produced and it enhances the temperature profile. Variation of velocity field for different estimation of power law index is shown through Figs. (9 − 11). These figures obviously reveal that
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motion of particles enhances for larger values of Physically, enhancing values of results into decreasing the viscosity which increases velocity of fluid. Fig. 12 highlights the impact of ratio parameter for thermal to buoyancy force ( ∗ ) on velocity field. This parameter
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assists to increase the velocity field. Impact of Prandtl number on velocity and temperature distributions is sketched through Figs. 13 and 14 By increasing values of Prandtl number, momentum diffusivity is enhanced which results an increase in velocity profile. Similarly
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thermal diffusivity is decreased for higher Prandtl number. That is why thermal boundary layer decays and as a result temperature field is decayed. Fig. 15 is displayed to present behavior of temperature profile ˜() for different variation of radiation parameter Temperature profile increases with higher values of The reason for this increase is enhance in internal energy of molecules to thermal energy. So more thermal energy is provided to the system and it enhances the temperature profile. Similarly Fig. 16 is displayed to show impact of temperature ratio parameter on ˜() Form this Fig. it is examined that temperature field increases in the presence of higher estimation of Characteristics of Schmidt number on concentration profile is sketched in Fig. 17. Physically Schmidt number is the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and is utilized 10
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to differentiate liquid flows in which there are instantaneous momentum and mass diffusion convection procedures. Therefore larger estimations of decays the concentration profile. Skin friction coefficient pertaining to magnetic and mixed convection variables and are shown in Figs. (18 − 20) Surface drag force enhances for larger , and . Similarly
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Fig. (21) is displayed to present the impact of magnetic parameter Pr on Nusselt number. From this Fig. it is investigate that Nusselt number enhances for rising estimation of Prandtl
1.0
1.0
0.8
0.8
a = 0.1,
0.2, 0.3, 0.4, 0.5, 0.6
0.6
q x
f 'x
0.6
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a = 0.1, 0.2, 0.3, 0.4, 0.5
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number.
0.4
0.4
0.2
0.2
0.0
0.0
0
2
4 x
6
8
0
1.0 0.8
4 x
0.8 d = 0.1, 0.4, 0.7, 1.0, 1.2, 1.5
0.6
0.0 0
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0.2
1
2
3
4
qx
EP
f ' x 0.4
8
1.0
a1 = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.6
6
Fig. 3b: () via
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Fig. 3a: 0() via
2
0.4 0.2
5
6
7
x
Fig. 4: 0() via 1
0.0 0
1
2
3 x
4
Fig. 5: () via
11
5
6
1.0
1.0
0.8
0.8
bt = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2
0.6
0.4
0.2
0.2
1
2
3 x
4
5
0.0
6
0
3
4
5
6
7
Fig. 7: 0() via
1.0
0.8
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0.8 M = 0.1, 0.4, 0.7, 1.0, 1.2, 1.5
n=0.2, 0.4, 0.6, 0.8, 1.0, 1.2
0.6
f ' x
q x
0.6
2
x
Fig. 6: 0() via
1.0
1
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0
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0.4
0.0
M = 0.1, 0.4, 0.7, 1.0, 1.2, 1.5
0.6
f'x
f 'x
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0.4
0.4
0.2 0.0
0.2
0
1
2
3
4
5
0.0
7
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x
6
0
Fig. 8: () via 1.0 0.8
0.0 0
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0.2
1
2
3
3 x
4
5
6
1.0 0.8 n = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.6
f x
EP
q x 0.4
2
Fig. 9: 0() via
n = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.6
1
0.4 0.2 0.0
4
5
6
7
x
Fig. 10: () via
0
1
2 x
Fig. 11: () via
12
3
4
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1.0
1.0
0.8
0.8 N * = 0.1, 0.3, 0.5, 0.7, 0.9, 1.
0.6 Pr = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.4
0.4
0.2
0.2
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f' x
f 'x
0.6
0.0
0.0 0
1
2
3
4
5
0
6
2
x
Fig. 12: 0() via ∗ 0.8
SC
0.8
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1.0
Pr = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
6
8
Fig. 13: 0() via Pr
1.0
R = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.6
qx
q x
0.6
4 x
0.4
0.4
0.2
0.2
0.0
0.0
0
1
2
3
4
5
6
0
7
1
2
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Fig. 14: () via Pr 1.0 0.8
0.0 0
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0.2
1
2
3
5
6
7
1.0 0.8 Sc = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.6
f x
EP
q x 0.4
4
Fig. 15: () via
qw = 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
0.6
3 x
x
0.4 0.2
4
5
6
7
x
Fig. 16: () via
0.0 0
2
4 x
Fig. 17: () via
13
6
8
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- 0.50 M = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
- 0.55 1
- 0.70
- 0.60 - 0.65 - 0.70
- 0.75
- 0.75 0
1
2 N*
3
0
4
4
SC
1
NuRe 2
1.036 1.034 1.032
- 0.70
1.030
- 0.75
1.028
0.0
0.5
2.0
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1.0 1.5 * N Fig. 20: Skin friction via ∗ and
8
3
Pr = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
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1
CfRe 2
- 0.65
- 0.80
1.040 1.038
- 0.60
2 N*
1.042
bt = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
- 0.55
1
Fig. 19: Skin friction via ∗ and
Fig. 18: Skin friction via ∗ and - 0.45 - 0.50
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bc = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
- 0.65
CfRe 2
1
CfRe 2
- 0.60
0.0
0.2
0.4
0.6
0.8
a
Fig. 21: Nusselt number via Pr and
Entropy generation rate (()) and Bejan number
EP
()
Impacts of different variables like dimensionless temperature difference parameter 2 radia-
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tion parameter and Brinkman number on local entropy generation number () and Bejan number are graphically displayed through Figs. (22 − 27) Figs. 22 and 23 exhibit the characteristics of local entropy generation and Bejan number for increasing values of temperature difference parameter 2 Both entropy generation and Bejan number increase through 2 It is obvious that () goes to zero far away from the surface. For larger estimation of 2 , impact of heat transfer rate is more prominent as compare to fluid friction and magnetic impacts. That is the reason for increase in number. Figs. 24 and 25 are displayed to show impact of entropy generation () and Bejan number for larger values of Brinkman number Brinkman number decides the emission of heat by viscous heating with heat exchange by molecular conduction. Brinkman number 14
1.0
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is specifically identified near the surface. Heat transfer rate by conduction is more than heat released due to viscous effects near the sheet. More heat is produced inside the layers of the moving particles, which grows the entropy and disorderliness in the system will increase. Fig. 25 presents that shows decreasing behavior for larger . Impact of on () and
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is demonstrated through Figs. 26 and 27 It has been noticed that entropy generation rate and Bejan number enhances for higher . Physically for larger more heat to the system is provided. That is why heat transfer rate increases and thus more entropy generation is
0.7
1.0
0.6
a2 = 0.1,
0.2, 0.3, 0.4, 0.5, 0.6
0.6
0.2
0.2
1.0
1.5
2.0
2.5
3.0
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0.5
0.1
0
20
40
EP
1.0 0.8
0.0
0.5
1.0
1.5
100
0.6 0.5 0.4 Be
N G x
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0.0
80
Fig. 23: Bejan number via 2
Br = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.6
60 x
Fig. 22: Entropy generation via 2
0.2
0.2, 0.3, 0.4, 0.5, 0.6
0.4
x
0.4
a2 = 0.1,
0.3
0.4
0.0 0.0
0.5
Be
0.8 N G x
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1.2
SC
noted.
Br = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.3 0.2 0.1
2.0
2.5
3.0
0
2
4
6
x
x
Fig. 24: Entropy generation via
Fig. 25: Bejan number via
15
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0.35
0.4 0.3
R = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.25
0.2
0.20 0.15 0.10
0.1
0.05 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
20
x
40
60
80
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x
Fig. 26: Entropy generation via
Fig. 27: Bejan number via
Conclusions
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Be
N G x
0.30
R = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
Optimization of entropy generation in nonlinear mixed convective flow of viscous liquid towards a variable thicked surface is considered. Main points are as follows: • Velocities in radial, axial and tangential direction increase for higher values of power
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law index.
• Temperature ˜() grows for larger estimation of and . • Higher skin friction coefficient is observed for larger magnetic parameter.
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• Temperature gradient decays through magnetic parameter
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• Entropy generation rate () enhances in the presence of and • Bejan number diminishes for while increasing behavior is observed for
References
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thickness and chemical reaction, Int. J. Mechanical Sci. 134 (2017) 306-314. [27] R. U. Haq, Z. H. Khan and W. A. Khan, Thermophysical effects of carbon nanotubes on MHD flow over a stretching surface, Physica E., 63 (2014) 215-222.
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[28] J. Li, L. Zheng and L. Liu, MHD viscoelastic flow and heat transfer over a vertical
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stretching sheet with Cattaneo-Christov heat flux effects, J. Mol. Liq. 221 (2016) 19-25. [29] M. Waqas, M. Farooq, M. I. Khan, A. Alsaedi, T. Hayat and T. Yasmeen, Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, Int. J. Heat Mass Transf. 102 (2016) 766-772. [30] M. Farooq, M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects, J. Mol. Liq. 221 (2016) 1097-1103. [31] T. Hayat, S. Qayyum, A. Alsaedi and B. Ahmad, Magnetohydrodynamic (MHD) nonlinear convective flow of Walters-B nanofluid over a nonlinear stretching sheet with variable thickness, Int. J. Heat Mass Transf. 110 (2017) 506-514. 19
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inclined magnetic field flow of tangent hyperbolic fluid over a nonlinear stretching surface with homogeneous—heterogeneous reactions, Int. J. Mech. Sci. 133 (2017) 1-10.
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heating, Int. J. Heat Mass Transf. 113 (2017) 310-317.
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convected Maxwell fluid, Int. J. Eng. Sci. 45 (2007) 393-401.
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S0921-4526(18)30084-X
DOI:
10.1016/j.physb.2018.01.054
Reference:
PHYSB 310701
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 17 January 2018 Revised Date:
25 January 2018
Accepted Date: 25 January 2018
Please cite this article as: T. Hayat, M.W.A. Khan, M. Ijaz Khan, A. Alsaedi, Nonlinear radiative heat flux and heat source/sink on entropy generation minimization rate, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.01.054. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Nonlinear radiative heat flux and heat source/sink on entropy generation minimization rate T. Hayat M. W. A. Khan , M. Ijaz Khan1 and A. Alsaedi Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
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Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia
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Abstract: Entropy generation minimization in nonlinear radiative mixed convective flow towards a variable thicked surface is addressed. Entropy generation for momentum and temperature
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is carried out. The source for this flow analysis is stretching velocity of sheet. Transformations are used to reduce system of partial differential equations into ordinary ones. Total entropy generation rate is determined. Series solutions for the zeroth and mth order deformation systems are computed. Domain of convergence for obtained solutions is identified. Velocity, temperature and concentration fields are plotted and interpreted. Entropy equation is studied through nonlinear mixed convection and radiative heat flux. Velocity and temperature gradients are discussed through graphs.
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Meaningful results are concluded in the final remarks.
Keywords: Entropy generation; Heat source/sink; Nonlinear mixed convection; Nonlinear thermal radiation; Bejan number. 1
Corresponding author:
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email address: [email protected]
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Fig. 1. Flow geometry
1
Introduction
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Boundary layer flows over a stretched surface with heat transfer analysis has greatly attracted many researchers for its vast applications in many industrial and technological processes for example glass fiber manufacturing, drawing of plastic thin films, cooling and drying of papers and polymer industries etc. Crane [1] initiated this work by providing exact solution
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for governing equations of flow analysis over a stretched surface. Non-Fourier heat flux in chemically reactive flow in the presence of temperature dependent thermal conductivity
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towards a stretched surface is presented by Hayat et. al. [2] Homotopy analysis technique is implemented to contract the series solution of nonlinear expressions. Their obtained outcomes highlight that temperature field is decays for higher estimation of thermal relaxation factor. It is also investigated that surface drag force is increases for higher ratio parameter and Deborah number in term of retardation time. Total residual errors are presented for velocity, concentration and temperature expressions. Impact of homogeneous-heterogeneous reactions in stagnation point flow with non-Fourier heat flux model is studied by Hayat et al. [3]. They discussed variable thicked surface first time with non-Fourier heat flux. The results indicate that temperature and concentration fields decays via thermal relaxation factor and chemical reaction variable. Furthermore series solutions are obtained through HAM. Heat and 2
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mass transport in non-Newtonian liquid through isothermal chemical reaction is examined by Khan et al. [4] MHD convective flow of rate type nanomaterial towards exponential stretched surface with slip conditions. Cross diffusion impact in radiative flow of viscous fluid with inclined magnetic field over a stretched surface is scrutinized by Raju et al. [5].
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Some latest analyses through this aspects referred through Refs. [6 − 15].
From a thermodynamic perspective, heat transport is always related with a least change of entropy. Entropy is process of irreversibility in system. Entropy minimization is the optimization process used to increase the efficiency of machines. Second law of thermodynamics
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is the main cause for entropy minimization process like internal molecular friction, vibration, spin moment and kinetic energy etc. In real life processes such loss of energy cannot be
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regained without extra work done. Therefore entropy is considered to be the measure of irreversibility. This process of entropy minimization is used by researchers in many systems such as fuel cells, gas turbines, natural convection and evaporative cooling etc. Few resent articles through this direction are given in Refs [16 − 25]
The phenomenon of stretching is very helpful in many industrial applications such as extrusion of plastic sheets, paper production, wire drawing, hot rolling, cooling of metallic
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sheets, glass blowing and polymer engineering etc. In such applications the quality of products mainly depends on rate of heat transport. Therefore choice of suitable cooling/heating liquid is very important because of its direct impact on the heat transfer rate. MHD convective
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nanomaterial flow of viscoelastic liquid subject to chemical reaction and variable thicked surface is examined by Qayyum et. [26]. Haq et al. [27] investigated MHD flow of carbon2 nanomaterial towards a stretchable surface. Heat transport mechanism in viscoelastic
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fluid flow with non-Fourier heat flux is studied by Li et al. [28]. Waqas et al. [29] presented natural convective flow of non-Newtonian liquid with convective boundary constraint. Farooq et al. [30] scrutinized MHD nanoliquid flow of Jeffrey model near a stagnation point with nonlinear thermal radiation. Hayat et al. [31] explored nanomaterial flow of Walters-B fluid in the presence of convective boundary conditions and variable thicked surface. Hsiao [32] worked on MHD nanoliquid flow of micropolar fluid with dissipation effects. Qayyum et al. [33] scrutinized melting heat transport and tangent hyperbolic liquid flow with isothermal chemical reaction and inclined MHD over a nonlinear stretched surface. Prasannakumara et al. [34] studied nonlinear thermal radiative flow of Sisko fluid. Thammanna et al. [35] examined MHD Casson fluid flow with chemical reaction towards a stretchable surface. 3
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The purpose of this research is to analyze nonlinear mixed convective flow of viscous material towards a variable thicked surface. Heat transport properties have been analyzed in the presence of nonlinear radiative heat flux and heat source/sink. Thickness of sheet is taken to be variable. Simple isothermal chemical reaction is implemented. Total entropy
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generation has been investigated for different flow variables. Homotopy analysis technique [36 − 50] is implemented to contract the series solution of nonlinear ordinary ones. Results are presented graphically.
Formulation
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2
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Here steady, incompressible 2 mixed convective flow of viscous liquid towards a variable thicked surface is considered. A uniform magnetic field of strength 0 is applied perpendicular to the plane of sheet. Smaller Reynolds number implies no induced magnetic field. Nonlinear thermal radiation and heat source/sink effects are considered. Furthermore effect of nonlinear mixed convection and simple chemical reaction are examined. Sheet lies parallel to -axis with stretching velocity = 0 ( + ) (see Fig. 1). Here is wall temperature while ∞
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being ambient temperature The relevant flow problems are [15]: + = 0
(1)
⎫ + = + (1 ( − ∞ ) + 2 ( − ∞ ) ) ⎬ ⎭ + (3 ( − ∞ ) + 4 ( − ∞ )2 ) µ ¶ 2 1 ˆ + ˆ = + + ( − ∞ ) 2
2 2
2
(2)
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ˆ
(3)
2 + ˆ = 2 − 1 ( − ∞ )
= = 0 ( + ) = 0 = = at = 1 ( + ) = 0 = 0 = ∞ = ∞ at → ∞
(4) 1− 2
⎫ ⎬ ⎭
(5)
Here indicates velocity components, kinematic viscosity, gravitational acceleration, 1 and 2 linear and nonlinear coefficients of thermal expansion, 3 and 4 linear and nonlinear coefficients of solutal expansion, temperature, ∞ ambient temperature, concentration,∞ ambient concentration, thermal conductivity, radiative heat flux, density, specific heat, coefficient of heat source/sink, diffusion coefficient, 1 chemical 4
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reaction rate, stretching velocity, dimensional constant, 1 small variable regarding surface is sufficient thin and power law index.
Θ () =
−∞ −∞
Φ=
−∞ −∞
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Equation of continuity is identically zero and the remaining equations one has ⎫ ¡ 2 ¢ 02 ¡ 2 ¢ 21 000 00 0 0 02 ⎬ + − +1 − +1 + +1 (Θ + Θ ) ∗ ⎭ + 21 (Φ0 + Φ02 ) = 0 +1
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⎫ Θ00 + 43 (Θ( − 1) + 1)2 [3Θ02 ( − 1) + (Θ( − 1) + 1) Θ00 ] ⎬ ¡ 2 ¢ ⎭ + Pr Θ0 + +1 Pr Θ = 0 µ ¶ 2 00 0 Φ + 2 Re Φ − 1 Φ = 0 +1 ⎫ ¡ 1− ¢ 0 0 ⎪ () = 1+ , () = 0 (∞) = 0 ⎪ ⎪ ⎬
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Θ() = 1 Θ(∞) = 0 Φ() = 1 Φ(∞) = 0
Applying the following transformations
⎫ ⎪ () = ( − ) = () ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
⎪ ⎪ ⎪ Φ () = ( − ) = () ⎭
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000 + 00 −
00
+
¡
¡ 2 ¢ ¡ 0 ¢ ⎫ 0 21 2 − +1 0 + +1 + 02 ⎬ ¢ ∗ ¡ 0 ⎭ 1 + 2+1 + 02 = 0
2 +1
¢
£ 02 ¤ ⎫ 00 ⎬ − 1) + 1) 3 ( − 1) + (( − 1) + 1) ¡ 2 ¢ ⎭ + Pr 0 + +1 Pr = 0 ¶ µ 2 00 0 + 2 Re − 1 = 0 +1 ⎫ ¡ 1− ¢ 0 0 ⎪ (0) = 1+ , (0) = 1 (∞) = 0 ⎪ ⎪ ⎬
4 (( 3
⎪ ⎪ ⎪ ⎭
(6)
(7)
(8)
(9)
(10)
(11)
Θ () = ( − ) = ()
Equations (7 − 10) one has
⎫ ⎪ ⎪ ⎪ ⎬
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Considering the following transformations q q −1 +1 = (+1)02(+) = 20 (+) () +1 q ¢ −1 ¡ = 0 ( + ) 0 () = − (+1)02(+) 0 () () + −1 +1
(12)
2
(0) = 1 (∞) = 0
(0) = 1 (0) = 0 5
⎪ ⎪ ⎪ ⎭
(13)
(14)
(15)
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02 0 (+)−1
´
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´ ³ ¡ ¢ shows magnetic parameter, Pr = Prandtl number, = 0 (+) −1 ³ ´ ³ ´ ( −∞ ) 2 ( −∞ )2 heat generation parameter 1 = 21(+) linear mixed convection parameter, = 2−1 02 (+)2−1 0 ³ ´ ( − ) ∞ nonlinear mixed convection variable for temperature, ∗ = ( −∞ ) 3 the ratio of ³ ´ concentration to thermal buoyancy forces, = ∞ temperature ratio parameter, ´ ´ ³ ³ ¡ ¢ ∗ 3 16 ∞ (+) radiation parameter, = Reynolds num = 3∗ Schmidt number , Re ³ ´ ³ ´ 2 −∞ ) 4 ber, 1 10 ( + )1− chemical reaction parameter and = (( nonlinear mixed −∞ ) ³ Here =
convection variable for concentration.
3.1
Surface drag force
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Quantities of interest
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3
Mathematical expression for surface drag force is 1 Re05 = 2
Heat transfer rate
+1 2
¶12
˜0 (0)
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3.2
µ
(17)
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Mathematical expression of heat transfer rate is ¶ ¶µ µ 4 +1 −05 3 1 + (1 + ( − 1)(0)) 0 (0) =− Re 2 3
(16)
3.3
Mass transfer rate
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The expression for Sherwood number is defined as Re−05 = −0 (0)
(18)
Exploration of entropy generation
Equation for entropy generation in dimensional form is given as µ ¶2 µ ¶2 2 2 = 2 + + (∇)2 + (∇ · ∇ ) + ( ) ∞ ∞ ∞ ∞ ∞ 0 where
6
(19)
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´ ³ ´ ⎫ ⎬ ´ ³ ´ ³ ∇ · ∇ = ⎭ (∇)2 =
³
(20)
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After using boundary layer assumptions, it can be written as
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Joule dissipation irreversibility
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(21)
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µ ¶2 µ ¶2 + = 2 ∞ | {z } | ∞ {z } Thermal irreversibility Fluid friction irreversibility ³ ´2 ³ ´ + + ∞ ∞ 2 2 ( + 2 ) + ∞ 0 | {z }
Equation (21) indicates that there are three main sources of entropy generation. First term is due to heat transfer with radiation effects. Second term occurs because of fluid friction irreversibility and last term is caused by Joule dissipation irreversibility. Dimensionless number for entropy generation is defined as
⎫ ⎪ = 1 + − 1) + 1] 2 ⎪ ⎪ 2 ⎬ £¡ +1 ¢ ¡ ¢ ¤ 0 0 02 +1 + + 1 2 2 2 ⎪ ⎪ ¡ +1 ¢ ⎪ ⎭ 002 02 + 2 + 3
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4 [ ( 3
where
02
⎫ = 1 = − ∞ ) ⎬ ⎭ = ∆ ∞(+) −1
−∞
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2 =
¡ +1 ¢
∆
(
(22)
(23)
0
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Here 2 denotes dimensionless temperature difference, Brinkman number and ³ ´ ∞ = ∆ (+)−1 entropy generation rate. 0
Irreversibility due to heat transfer is prominent when À 05. On the other hand when
¿ 05 the viscous effects dominates. For = 05 both effects are equal. Bejan number () in dimensionless form is defined as Entropy generation due to heat transfer Total entropy generation ¡ ¢ 02 2 1 + 43 [ ( − 1) + 1]3 +1 2 ⎫ = 3 ¡ +1 ¢ 02 4 ⎪ 1 + 3 [ ( − 1) + 1] 2 ⎪ ⎪ 2 ⎬ £¡ +1 ¢ ¡ ¢ ¤ 0 0 02 +1 + + 1 2 2 2 ⎪ ⎪ ¡ +1 ¢ ⎪ ⎭ 002 02 + 2 +
=
7
(24) (25)
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Homotopy procedure
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This procedure provide a greater opportunity to choose initial approximations along with linear operators. So initial guesses and linear operators are defined as ⎫ ⎪ 0 () = exp(−) ⎪ ⎪ ⎬ 0 () = exp(−)
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⎪ ⎪ ⎪ 0 () = exp(−) ⎭
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L = 00 − L = 00 − L = 00 − with
£ ¤ L 2 + 3 − = 0 £ ¤ L 6 + 7 − = 0 £ ¤ L 8 + 9 − = 0
6
⎪ ⎪ ⎪ ⎭
(27)
(28)
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where ( = 1 − 9) denotes the arbitrary constants.
⎫ ⎪ ⎪ ⎪ ⎬
(26)
Convergence analysis
Auxiliary parameters } } and } are chosen in a wide range which provides us the advantage
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to adjust the convergence of nonlinear system of equations. Fig. 2 displays the ~-curves for momentum, thermal and concentration equations at different order of approximations.
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Suitable estimations of these variables are established in the ranges −15 ≤ } ≤ −02 −13 ≤ } ≤ −03 and −17 ≤ } ≤ −03 Table 1 highlights convergence of series solutions of momentum, energy and concentration constraints for the concerned flow problem. Table 0
0
0
1 indicates that (0) (0) and (0) converge at 15 , 25 and 19 order of approximations
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respectively. f''(0) = q '(0) = f'(0) =
0.0
-0.5
-1.0
-1.5
- 1.5
- 1.0
Ñ f ,Ñq , Ñf
- 0.5
0.0
SC
- 2.0
RI PT
f '' 0 ,q ' 0 , f ' 0
0.5
Fig. 2. ~−curves for and
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Table 1: Different order of approximations when 1 = 03 = = 05 = 07 = 07 Pr = 19 = 01 = 03 ∗ = 02 = 05 = 12 Re = 10 and 1 = 05 0 (0)
Order of approximation 1 11
−0 (0)
0.1610 0.8295 1.55556
0.2859 0.5038 1.26243 0.2857 0.5233 1.25899
TE D
15
−0 (0)
0.2857 0.5341 1.25894
25
0.2857 0.5350 1.25894
30
0.2857 0.5350 1.25894
35
0.2857 0.5350 1.25894
40
0.2857 0.5350 1.25894
50
0.2857 0.5350 1.25894
7
AC C
EP
19
Discussion
The influence of dimensionless parameters on MHD flow of viscous liquid towards a variable thicked surface. Figs. (3 − 22) are plotted for outcomes of velocity 0 () temperature (), ¡ ¢ ¡ ¢ concentration (), skin friction coefficient Re05 heat transfer rate Re−05 and ¢ ¡ Sherwood number Re−05 . Behavior of wall thickness variable on velocity, temperature and concentration are shown in Figs. 3 − 3 Characteristics of wall thickness variable on
velocity is sketched in Fig. 3. As wall thickness increases, less force is transferred to fluid due to stretching of sheet. That is why velocity tends to decrease. Effect of wall thickness 9
ACCEPTED MANUSCRIPT
parameter on temperature is given in Fig. 3. Here () is a decreasing function of wall thickness variable. Physically less heat is transfer to system, so temperature decreases. Effects of mixed convection parameter (1 ) on velocity is displayed via Fig. 4. Magnitude of velocity enhances for larger (1 ). Fig. 4 shows that for larger values of 1 higher thermal
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buoyancy force results an increase in velocity. Silent features of heat generation parameter is sketched through Fig. 5. More heat is produced in the system by introducing heat generation factor. That is why temperature profile increases. Fig. 6 is portrayed to show the behavior of mixed convection parameter due to temperature ( ). It reveals that velocity
SC
is an increasing function of Figs. 7 and 8 are displayed to show the impact of magnetic parameter on velocity field As magnetic parameter is related with Lorentz force (resistive
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force) so an increase in provides more resistance to liquid flow and as a result velocity decays. Influence of magnetic parameter on temperature ˜() is shown in Fig. 8. Here an increase in temperature profile is observed for larger This is due to the fact that higher Lorentz force provides more resistance to fluid particles motion. Thus more heat is produced and it enhances the temperature profile. Variation of velocity field for different estimation of power law index is shown through Figs. (9 − 11). These figures obviously reveal that
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motion of particles enhances for larger values of Physically, enhancing values of results into decreasing the viscosity which increases velocity of fluid. Fig. 12 highlights the impact of ratio parameter for thermal to buoyancy force ( ∗ ) on velocity field. This parameter
EP
assists to increase the velocity field. Impact of Prandtl number on velocity and temperature distributions is sketched through Figs. 13 and 14 By increasing values of Prandtl number, momentum diffusivity is enhanced which results an increase in velocity profile. Similarly
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thermal diffusivity is decreased for higher Prandtl number. That is why thermal boundary layer decays and as a result temperature field is decayed. Fig. 15 is displayed to present behavior of temperature profile ˜() for different variation of radiation parameter Temperature profile increases with higher values of The reason for this increase is enhance in internal energy of molecules to thermal energy. So more thermal energy is provided to the system and it enhances the temperature profile. Similarly Fig. 16 is displayed to show impact of temperature ratio parameter on ˜() Form this Fig. it is examined that temperature field increases in the presence of higher estimation of Characteristics of Schmidt number on concentration profile is sketched in Fig. 17. Physically Schmidt number is the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and is utilized 10
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to differentiate liquid flows in which there are instantaneous momentum and mass diffusion convection procedures. Therefore larger estimations of decays the concentration profile. Skin friction coefficient pertaining to magnetic and mixed convection variables and are shown in Figs. (18 − 20) Surface drag force enhances for larger , and . Similarly
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Fig. (21) is displayed to present the impact of magnetic parameter Pr on Nusselt number. From this Fig. it is investigate that Nusselt number enhances for rising estimation of Prandtl
1.0
1.0
0.8
0.8
a = 0.1,
0.2, 0.3, 0.4, 0.5, 0.6
0.6
q x
f 'x
0.6
M AN U
a = 0.1, 0.2, 0.3, 0.4, 0.5
SC
number.
0.4
0.4
0.2
0.2
0.0
0.0
0
2
4 x
6
8
0
1.0 0.8
4 x
0.8 d = 0.1, 0.4, 0.7, 1.0, 1.2, 1.5
0.6
0.0 0
AC C
0.2
1
2
3
4
qx
EP
f ' x 0.4
8
1.0
a1 = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.6
6
Fig. 3b: () via
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Fig. 3a: 0() via
2
0.4 0.2
5
6
7
x
Fig. 4: 0() via 1
0.0 0
1
2
3 x
4
Fig. 5: () via
11
5
6
1.0
1.0
0.8
0.8
bt = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2
0.6
0.4
0.2
0.2
1
2
3 x
4
5
0.0
6
0
3
4
5
6
7
Fig. 7: 0() via
1.0
0.8
M AN U
0.8 M = 0.1, 0.4, 0.7, 1.0, 1.2, 1.5
n=0.2, 0.4, 0.6, 0.8, 1.0, 1.2
0.6
f ' x
q x
0.6
2
x
Fig. 6: 0() via
1.0
1
SC
0
RI PT
0.4
0.0
M = 0.1, 0.4, 0.7, 1.0, 1.2, 1.5
0.6
f'x
f 'x
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0.4
0.4
0.2 0.0
0.2
0
1
2
3
4
5
0.0
7
TE D
x
6
0
Fig. 8: () via 1.0 0.8
0.0 0
AC C
0.2
1
2
3
3 x
4
5
6
1.0 0.8 n = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.6
f x
EP
q x 0.4
2
Fig. 9: 0() via
n = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.6
1
0.4 0.2 0.0
4
5
6
7
x
Fig. 10: () via
0
1
2 x
Fig. 11: () via
12
3
4
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1.0
1.0
0.8
0.8 N * = 0.1, 0.3, 0.5, 0.7, 0.9, 1.
0.6 Pr = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
0.4
0.4
0.2
0.2
RI PT
f' x
f 'x
0.6
0.0
0.0 0
1
2
3
4
5
0
6
2
x
Fig. 12: 0() via ∗ 0.8
SC
0.8
M AN U
1.0
Pr = 0.1, 0.3, 0.5, 0.7, 0.9, 1.2
6
8
Fig. 13: 0() via Pr
1.0
R = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.6
qx
q x
0.6
4 x
0.4
0.4
0.2
0.2
0.0
0.0
0
1
2
3
4
5
6
0
7
1
2
TE D
Fig. 14: () via Pr 1.0 0.8
0.0 0
AC C
0.2
1
2
3
5
6
7
1.0 0.8 Sc = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.6
f x
EP
q x 0.4
4
Fig. 15: () via
qw = 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
0.6
3 x
x
0.4 0.2
4
5
6
7
x
Fig. 16: () via
0.0 0
2
4 x
Fig. 17: () via
13
6
8
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- 0.50 M = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
- 0.55 1
- 0.70
- 0.60 - 0.65 - 0.70
- 0.75
- 0.75 0
1
2 N*
3
0
4
4
SC
1
NuRe 2
1.036 1.034 1.032
- 0.70
1.030
- 0.75
1.028
0.0
0.5
2.0
TE D
1.0 1.5 * N Fig. 20: Skin friction via ∗ and
8
3
Pr = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
M AN U
1
CfRe 2
- 0.65
- 0.80
1.040 1.038
- 0.60
2 N*
1.042
bt = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
- 0.55
1
Fig. 19: Skin friction via ∗ and
Fig. 18: Skin friction via ∗ and - 0.45 - 0.50
RI PT
bc = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
- 0.65
CfRe 2
1
CfRe 2
- 0.60
0.0
0.2
0.4
0.6
0.8
a
Fig. 21: Nusselt number via Pr and
Entropy generation rate (()) and Bejan number
EP
()
Impacts of different variables like dimensionless temperature difference parameter 2 radia-
AC C
tion parameter and Brinkman number on local entropy generation number () and Bejan number are graphically displayed through Figs. (22 − 27) Figs. 22 and 23 exhibit the characteristics of local entropy generation and Bejan number for increasing values of temperature difference parameter 2 Both entropy generation and Bejan number increase through 2 It is obvious that () goes to zero far away from the surface. For larger estimation of 2 , impact of heat transfer rate is more prominent as compare to fluid friction and magnetic impacts. That is the reason for increase in number. Figs. 24 and 25 are displayed to show impact of entropy generation () and Bejan number for larger values of Brinkman number Brinkman number decides the emission of heat by viscous heating with heat exchange by molecular conduction. Brinkman number 14
1.0
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is specifically identified near the surface. Heat transfer rate by conduction is more than heat released due to viscous effects near the sheet. More heat is produced inside the layers of the moving particles, which grows the entropy and disorderliness in the system will increase. Fig. 25 presents that shows decreasing behavior for larger . Impact of on () and
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is demonstrated through Figs. 26 and 27 It has been noticed that entropy generation rate and Bejan number enhances for higher . Physically for larger more heat to the system is provided. That is why heat transfer rate increases and thus more entropy generation is
0.7
1.0
0.6
a2 = 0.1,
0.2, 0.3, 0.4, 0.5, 0.6
0.6
0.2
0.2
1.0
1.5
2.0
2.5
3.0
TE D
0.5
0.1
0
20
40
EP
1.0 0.8
0.0
0.5
1.0
1.5
100
0.6 0.5 0.4 Be
N G x
AC C
0.0
80
Fig. 23: Bejan number via 2
Br = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.6
60 x
Fig. 22: Entropy generation via 2
0.2
0.2, 0.3, 0.4, 0.5, 0.6
0.4
x
0.4
a2 = 0.1,
0.3
0.4
0.0 0.0
0.5
Be
0.8 N G x
M AN U
1.2
SC
noted.
Br = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.3 0.2 0.1
2.0
2.5
3.0
0
2
4
6
x
x
Fig. 24: Entropy generation via
Fig. 25: Bejan number via
15
8
ACCEPTED MANUSCRIPT
0.35
0.4 0.3
R = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
0.25
0.2
0.20 0.15 0.10
0.1
0.05 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
20
x
40
60
80
SC
x
Fig. 26: Entropy generation via
Fig. 27: Bejan number via
Conclusions
M AN U
9
RI PT
Be
N G x
0.30
R = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6
Optimization of entropy generation in nonlinear mixed convective flow of viscous liquid towards a variable thicked surface is considered. Main points are as follows: • Velocities in radial, axial and tangential direction increase for higher values of power
TE D
law index.
• Temperature ˜() grows for larger estimation of and . • Higher skin friction coefficient is observed for larger magnetic parameter.
EP
• Temperature gradient decays through magnetic parameter
AC C
• Entropy generation rate () enhances in the presence of and • Bejan number diminishes for while increasing behavior is observed for
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TE D
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