Physica Scripta

ACCEPTED MANUSCRIPT

Optimization of entropy generation in nonlinear stratified Powell-Eyring

fluid with convective boundary conditions
To cite this article before publication: Iffat Jabeen et al 2019 Phys. Scr. in press https://doi.org/10.1088/1402-4896/ab4ffb

Manuscript version: Accepted Manuscript

Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process,
and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted
Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors”

This Accepted Manuscript is © 2019 IOP Publishing Ltd.

During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully
protected by copyright and cannot be reused or reposted elsewhere.
As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse
under a CC BY-NC-ND 3.0 licence after the 12 month embargo period.

After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they
adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0

Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content
within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this
article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be
required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record.

View the article online for updates and enhancements.

This content was downloaded from IP address 150.216.68.200 on 09/11/2019 at 16:47

Page 1 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3
4 Optimization of Entropy Generation in nonlinear

pt
5
6
7
8
stratified Powell-Eyring fluid with convective
9

cri
10
11
boundary conditions
12
13 Iffat Jabeen 𝑎, 𝑏, M. Farooq 𝑎, N. A. Mir 𝑎, S. Ahmad 𝑎, Aisha Anjum 𝑎
14
15 𝑎 Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan
16 𝑏 Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan

us
17
18
19 Abstract:
20
21 Entropy generation phenomenon is applied in reactors, turbines, natural convection, chillers,
22
23
24
25
26
27
28
an
functional and regular graded materials. Irreversibility of a system and surroundings in powell-
Eyring fluid is analyzed in this research article. Flow is deformed by an inclined sheet. Measure
of irreversibility is also called Entropy. Rate of Entropy generation is calculated by considering
heat transfer, radiation and viscous dissipation phenomena. Thermal stratification (nonlinear) and
dM
29 heat generation/absorption are incorporated in heat transport while solutal stratification (nonlinear)
30
31 and chemical reaction have taken part in mass transport. Mixed convection phenomenon is also
32
33 accounted here. To obtain the analytical solutions of nonlinear and non-dimensional governing
34
35
equations homotopic analysis method is applied. The behaviour of emerging parameters is
36 discussed comprehensively via velocity, temperature and concentration profile. Entropy
37
38 generation can be minimized by small values of Prandtl number and Eckert number. Thus, these
pte

39
40 physical phenomena can be used as a cooling agent in various industral processes.
41
Key-words: Powell-Eyring, Inclined sheet, dual stratification (nonlinear), Linear stretching,
42
43 entropy generation, heat generation/absorption, Radiation, chemical reaction, convective boundary
44
45 conditions.
46
1 Introduction
ce

47
48
49 For best utilization of energy during fluid flow entropy generation is introduced and it is used to
50
51 find out the destruction in the performance of the system. Only in irreversible process entropy
52
Ac

53 production is applied not for reversible processes e.g chemical reaction, joule heating, diffusion,
54 friction between fluid surfaces. To increase the efficiency of thermal equipments minimization of
55
56 entropy generation is must. Entropy generation processes are applied in gas turbines, reactors,
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 2 of 20

1
2
3 chillers, fuel cells, air seperators, curved pipes, evaporative cooling, natural convection, solar
4

pt
5 thermal, functional and regular graded materials. According to the previous research survey
6
7 researchers did alot of work on entropy generation. Khan et al. [1] have discussed the impact of
8
9
entropy generation on viscous fluid with radiation and binary chemical reaction effects. Khan et

cri
10 al. [2] disclosed the features of entropy optimization in chemically reactive williamson nanofluid
11
12 with Joule heating. Alharbi et al. [3] considered Powell-Eyring fluid to ellaborate the effects of
13
14 Entropy generation with MHD and radiation effect. Jamshed et al. [4] have analyzed the entropy
15
in Powell-Eyring nanofluid with variable thermal conductivity and thermal radiation. Butt et al.
16

us
17 [5-7] have elaborated the entropy generation effects in a manetohydrodynamic nanofluid flow
18
19 deformed by a stretching sheet in the presence of viscous dissipation and radiation.
20
21 Several models for non-Newtonian fluids have been introduced by the scientists to explore the
22
23
24
25
26
27
28
an
features of heat transfer and fluid flow. One of the models is known as Powell-Eyring model for
non-Newtonian fluids. Scientists adopted this model because it has two main advantages over other
models (i) it is derived from kinetic theory of gases instead of empirical formula (ii) It behaves
like Newtonian fluid at high or low shear stresses. It has many applications in industrial and natural
dM
29 processes, agricultural field, thermal insulation, thermal radiation, environmental pollution and
30
31 geo thermal reservoirs. Hayat et al. [8] discussed the Powell-Eyring fluid with the effect of non-
32
33 fourier heat flux theory. Adesanyaa et al. [9] discussed the squeezing flow in Powell-Eyring fluid
34
with radiation and chemical reaction. Rehman et al. [10] analyzed the effects of stratification and
35
36 mixed convection in Powell-Eyring fluid flow. Rehman et al. [11] have analyzed the Powell-
37
38 Eyring fluid flow in the presence of stagnation point. Rahimi et al. [12] used collocation method
pte

39
40 to analyze the effects of Powell-Eyring fluid over a stretching sheet.
41 In many engineering problems internal heat generation or absorption occurs. In this manner, Hayat
42
43 et al. [13] have described the effects of heat generation/absorption in three dimensional viscoelatic
44
45 nanofluid. Hayat et al. [14] have checked the heat generation/absorption effects in Walters-B
46
nanofluid in the presence of cattaneo-christov model. Soomro et al. [15] have analyzed the effects
ce

47
48 of heat generation/absorption in the presence of stagnation point with nonlinear radiation. Qayyum
49
50 et al. [16] have ellaborated the heat generation/absorption effects over a stretching sheet in third
51
52 grade nanofluid. Anjum et al. [17] explained the heat generation/absorption effects in second grade
Ac

53 fluid deformed by a Riga plate.

54
55 Five novel aspects are there in our present research. Firstly Powell-Eyring fluid flow is applied by
56
57
58
59
60
Page 3 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3 inclined stretching sheet with convective boundary conditions. Secondly, we analyzed heat tranfer
4

pt
5 with Entropy generation along with dual stratification (nonlinear). Thirdly heat transfer is also
6
7 helped out with heat generation/absorption and radiation. Fourthly mass transfer is carried out with
8
9
and chemical reaction. Fifthly to find out solution of convergent series homotopic analysis method

cri
10 [18-24] is used and arising dominating parameters are discussed by plots. Features of skin friction,
11
12 Nusselt number and Sherwood number are also elaborated through graph.
13
14 In the abovementioned studies, no one has attempted the study of Powell-Eyring fluid on two
15
dimensional flow with nonlinear stratification over a stretchable inclined surface in collaboration
16

us
17 with entropy generation and convective heat and mass boundary conditions. Hence to fill this
18
19 breach is our basic theme.
20
21 2 Mathematical Modeling
22
23
24
25
26
27
28
an
We consider Powell-Eyring fluid flow (incompressible and steady) persuaded by linearly
stretchable sheet inclined at an angle 𝛽1 with the horizontal line. Mixed convective phenomena
dM
29
30 is applied to demonstrate the features of allied mass and and heat transport. Nonlinear stratification
31
32 is also implimented in order to analyze heat and mass flux. Heat generation/absorption and thermal
33
34 radiation assisted heat transfer phenomena. Entropy generation is also implimented. Mass transfer
35 mechanism is evaluated in the light of constructive and destructive chemical reactions. The
36
37 governing equations by implementing boundary layer approximations are as follows [25-26]:
38
pte

39
∂𝑢 ∂𝑣
40 + ∂𝑦 = 0,
41 ∂𝑥 (1)
42
43
44 ∂𝑢 ∂𝑢 1 ∂2 𝑢 1 ∂𝑢 2 ∂2 𝑢 (𝛽 (𝑇 − 𝑇∞ ) +
45 𝑢 ∂𝑥 + 𝑣 ∂𝑦 = (𝜐 + 𝜌𝛽𝐶 ) ∂𝑦 2 − 2𝜌𝛽𝐶 3 (∂𝑦) +( 𝑇 ) 𝑔 𝑠𝑖𝑛 𝛽1 ,
1 1 ∂𝑦 2 𝛽𝐶 (𝐶 − 𝐶∞ ) (2)
46
ce

47
48
∂𝑇 ∂𝑇 𝑘 ∂2 𝑇 𝑄 16𝜎∗ ∂2 𝑇 1 ∂𝑢 1 ∂𝑢
49 𝑢 ∂𝑥 + 𝑣 ∂𝑦 = 𝜌𝑐 + 𝜌𝑐0 (𝑇 − 𝑇0 ) + 𝜌𝑐 𝑇∞3 ∂𝑦 2 + (𝜇 + 𝛽𝐶 ) (∂𝑦)2 − 6𝛽𝐶 3 (∂𝑦)4 ,
∂𝑦 2 3𝑘 ∗
50 𝑝 𝑝 𝑝 1 1

51
52 (3)
Ac

53 ∂𝐶 ∂𝐶 ∂2 𝐶
54
𝑢 ∂𝑥 + 𝑣 ∂𝑦 = 𝐾1 (𝐶 − 𝐶∞ ) + 𝐷 ∂𝑦 2 , (4)
55
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 4 of 20

1
2
3 with the boundary conditions [27-28]:
4

pt
5 ∂𝑇 ∂𝐶
6 𝑢 = 𝑢𝑤 (𝑥) = 𝑐𝑥, −𝑘 ∂𝑦 = ℎ𝑓 (𝑇𝑤 − 𝑇), −𝐷 ∂𝑦 = ℎ𝑐 (𝐶𝑤 − 𝐶), 𝑣 = 0,  𝑎𝑡 𝑦 = 0,
7
8
𝑢 → 0,  𝑇 → 𝑇∞ (𝑥), 𝐶 → 𝐶∞ (𝑥) 𝑎𝑠 𝑦 → ∞,
(5)
9

cri
10 where
11 𝑇𝑤 (𝑥) = 𝑇0 + 𝑑1 𝑥 2 , 𝐶𝑤 (𝑥) = 𝐶0 + 𝑒1 𝑥 2 , 𝑇∞ (𝑥) = 𝑇0 + 𝑑2 𝑥 2 , 𝐶∞ (𝑥) = 𝐶0 + 𝑒2 𝑥 2 , (6)
12 In above expressions, 𝜐 represents fluid kinematic viscosity, 𝑢𝑤 represents stretching
13
14 velocity, 𝜌 represents density, 𝑇𝑤 the stretching temperature of fluid, thermal expansion is
15
16 represented by 𝛽𝑇 , 𝑔 stands for gravitational acceleration, 𝑇∞ the variable ambient fluid

us
17
temperature, 𝐶𝑤 the stretching concentration of heated fluid, 𝛽1 represents angle of
18
19 inclination, 𝐶1 and 𝛽 are material parameters, 𝛽𝐶 represents coefficient of mass
20
21 expansion, 𝐶∞ the variable ambient concentration, specific heat is demostrated by 𝐶𝑝 , 𝐾1
22
23
24
25
26
27
28
species coefficient,
an
is chemical reaction coefficient , ℎ𝑓 and ℎ𝑐 demonstrate heat and mass transfer coefficients,
𝑇0 and 𝐶0 represent reference temperature and concentration respectively, 𝐷 is diffusion
𝜎∗ stephen Boltzman constant and concentration,
𝑘∗ 𝑑1 , 𝑑2 ,
𝑄0 is heat
𝑒1 , and 𝑒2
dM
generation/absorption coefficient, absorption coefficient,
29
30 represents dimensional constants and 𝑘 represents thermal conductivity. It is noted that
31
32 𝑢𝑤 (𝑥) = 𝑐𝑥 at 𝑦 = 0 represents stretching velocity which is produced by applying two forces
33
34
of same magnitude but in opposite direction in order origin is kept constant, 𝑣 = 0 demonstrates
35 ∂𝑇
36 that plate is impermeable, −𝑘 ∂𝑦 = ℎ𝑓 (𝑇𝑤 − 𝑇) is a convective boundary condition due to
37
38 temperature which is derived from Newton's law of cooling and Fourier law of heat conduction,
pte

39 ∂𝐶
40 −𝐷 ∂𝑦 = ℎ𝑐 (𝐶𝑤 − 𝐶) is a convective boundary condition due to concentration and is derived
41
42 from Fick's first and second laws, 𝑢 → ∞ velocity outside the boundary layer and approaches
43
44 to zero while temperature and concentration approaches to 𝑇∞ (𝑥) and 𝐶∞ (𝑥) respectively,
45
which vary quadratically, as we move away from the origin and these temperature and
46
ce

47 concentration fields are responsible for thermal solutal stratifications.

48
49 Implementing the transformations [28-29]
50
51
52 𝑐
𝜉 = 𝑦√𝜐 , 𝑢 = 𝑐𝑥𝑓 ′ (𝜉), 𝑣 = −√𝑐𝜐𝑓(𝜉),
Ac

53
54 𝑇 − 𝑇∞ 𝐶 − 𝐶∞
55 𝜃(𝜉) = , 𝜑(𝜉) = , (7)
56
𝑇𝑤 − 𝑇0 𝐶𝑤 − 𝐶0
57
58
59
60
Page 5 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3 The continuity equation Eq.(1) vanishes, while equations (2)-(4) take the forms:
4

pt
5 (1 + 𝜀)𝑓 ‴ + 𝑓𝑓 ′′ − 𝑓 ′2 − 𝜀𝛿𝑓 ′′2 𝑓 ‴ + 𝜆𝜃 𝑠𝑖𝑛 𝛽1 + 𝜆1 𝜑 𝑠𝑖𝑛 𝛽1 = 0, (8)
6
7
8
9 1
𝜃 ′′ (1 + 𝑅) − 2𝑓 ′ 𝑆1 𝑃𝑟 − 2𝑓 ′ 𝜃𝑃𝑟 + 𝜃 ′ 𝑓𝑃𝑟 + 𝛿1 𝜃𝑃𝑟 + 𝐸𝑐 [(1 + 𝜀)𝑓 ′′2 − 3 𝜀𝛿𝑓 ′′4 ] = 0, (9)

cri
10
11
12
13
14 𝜑 ″ − 𝑆𝑐(2𝑆2 𝑓 ′ + 2𝑓 ′ 𝜑 − 𝑓𝜑 ′ ) + 𝑆𝑐𝑘𝑟𝜑 = 0, (10)
15 Dimensionless boundary conditions are:
16

us
17 𝑓 ′ (0) = 1, 𝑓(0) = 0, 𝜃 ′ (0) = −𝛾1 (1 − 𝑆1 − 𝜃(0)), 𝜑 ′ (0) = −𝛾2 (1 − 𝑆2 − 𝜑(0)),
18
19 𝑓 ′ (∞) = 0, 𝜃(∞) = 0, 𝜑(∞) = 0, (11)
20
21 where 𝑆𝑐 represents Schmidt number, 𝐸𝑐 is Eckert number, 𝑅 is radiation parameter,
22
23
24
25
26
27
28
by
an
𝛿1 is heat generation/absorption parameter, dimensionless material parameter are 𝜀 and 𝛿 ,
𝛾1 is Biot number (thermal), 𝜆1 is solutal buoyancy parameter, Prandtl number is represented
, 𝑆2 and 𝑆1 represent solutal and thermal stratified parameters, 𝛾2 is solutal Biot
number, 𝛽1 is an angle of inclination, 𝜆 is thermal buoyancy parameter, 𝑘𝑟 represents
dM
29
30
chemical reaction parameter and 𝛿1 is heat generation/absorption parameter. These
31 dimensionless parameters are expressed as given below:
32
33
𝜐 1 𝑥2𝑐 3 𝐾1 𝜐 𝑔𝛽𝑇 (𝑇𝑤 −𝑇0 )
34 𝑃𝑟 = 𝛼 , 𝜀 = 𝜇𝛽𝐶 , 𝛿 = 2𝜐𝐶 2 , 𝑘𝑟 = , 𝑆𝑐 = 𝐷 , 𝜆 = ,
35 1 1 𝑐 𝑥𝑐 2
36 𝑔𝛽𝐶 (𝐶𝑤 − 𝐶0 ) 𝑑2 𝑒2 ℎ𝑓 ℎ𝑐
37 𝜆1 = , 𝑆1 = , 𝑆2 = , 𝛾1 = , 𝛾2 = ,
38 𝑥𝑐 2 𝑑1 𝑒1 𝑐 𝑐
𝑘√𝜐 𝐷√𝜐
pte

39
40 (12)
41 16𝜎 ∗ 3 𝑄0 𝑐2
𝑅= 𝑇 ,𝛿 = , 𝐸𝑐 = ,
42 3𝑘𝑘 ∗ ∞ 1 𝜌𝐶𝑝 𝑐 𝑑1 𝐶𝑝
43
44
45
46 Surface drag force is as follows:
𝜏
𝐶𝑓𝑥 = 𝜌𝑢𝑤2 ,
ce

47 (13)
𝑤
48
49
50 where wall shear stress,
51 1 ∂𝑢 1 ∂𝑢 3
52 𝜏𝑤 = 𝜏𝑥𝑦 = (𝜇 + )( ) − ( ) ,
𝛽𝐶1 ∂𝑦 6𝛽𝐶13 ∂𝑦
Ac

53
54
55
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 6 of 20

1
2
3 1/2 𝜀
4 𝐶𝑓 𝑅𝑒𝑥 = (1 + 𝜀)𝑓 ′′ (0) − 𝛿𝑓 ′′3 (0). (14)
3

pt
5
6
7
8
Nussult number is as follows:
9 𝑥𝑞𝑤

cri
10 𝑁𝑢𝑥 = ,
11 𝑘(𝑇𝑓 − 𝑇∞ )
12
13
14 ∂𝑇
15 𝑞𝑤 = −𝑘( ),
16 ∂𝑦

us
17
18 −𝜃′ (0)
𝑁𝑢𝑥 = −1/2 .
19 𝑅𝑒𝑥 (1−𝑆1 ) (15)
20
21 Sherwood number is as follows:
22
23
24
25
26
27
28
𝑞𝑚 = −𝐷(
an
𝑆ℎ𝑥 = 𝐷(𝐶
∂𝐶
∂𝑦
),
𝑥𝑞𝑚
𝑓 −𝐶∞ )

−𝜑 ′ (0)
,
dM
29 𝑆ℎ𝑥 = −1/2
30 𝑅𝑒𝑥 ( 1 − 𝑆2 ) (16)
31
𝑢𝑤 𝑥
32 Reynolds number is represented by 𝑅𝑒𝑥 = .
33 𝜐
34
35
2.1 Entropy generation modeling
36 Here total Entropy generation rate is calculated by second law of thermodynamics. Rate of Entropy
37
38 generation is calculated by heat transfer, thermal radiation and viscous dissipation and chemical
pte

39
40 reaction. In thermal engineering most recent investigation reports that sencond law of
41
42
thermodynamics [30] is more significant than first law of thermodynamics. The main difference is
43 that first law provides only accounting or quantity of energy while second law provides variation
44
45 as well as direction of heat transfer. Bejan [31-32] is the first who studied entropy generation in a
46
convective heat transfer problems.
ce

47
48
𝐸𝑔 = (𝐸𝑔) 𝑇 + (𝐸𝑔)𝑅 + (𝐸𝑔)𝜏.𝐿 + (𝐸𝑔)𝑘𝑟 (17)
49
50
51 𝑘 ∂𝑇 2 𝑘 16𝜎 ∗ ∂𝑇 2 𝜏.𝐿 𝐷 ∂𝐶
𝐸𝑔 = (𝑇 ( ) + (𝑇 ( 𝑇 3 ) (∂𝑦) + 𝑇 +𝐶 ( ),
52 ∞ −𝑇 )2 ∂𝑦
0 ∞ −𝑇 )2 3𝑘 ∗ ∞
0 ∞ −𝑇0 ∞ −𝐶0 ∂𝑦
Ac

53
54 𝑦
(𝑇∞ −𝑇0 )2 ( )2
55 𝜂 (18)
𝐸𝑔0 = , (19)
56 𝑘(𝑇𝑤 −𝑇∞ )2
57
58
59
60
Page 7 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3
4 𝐸𝑔

pt
5 𝑁𝑔 = 𝐸𝑔
6 0 (20)
7
8 Non-dimensional form of Entropy generation is:
9

cri
10
(𝜃′ )2 (𝜃′ )2 𝑆2 1 𝜑 ′2 𝑆12 𝑏
11 𝑁𝑔 = (1−𝑆 + 𝑅 (1−𝑆 + 𝑃𝑟𝐸𝑐 (1−𝑆1 [(1 + 𝜀)𝑓 ′′2 + 3 𝜀𝛿𝑓 ′′4 ] + 𝑆 , (21)
12 1 )2 1 )2 1 )2 2 (1−𝑆1 )
2

13
14 where, first term on right hand side shows entropy generation due to heat transfer, second term
15
16 represents entropy generation due to radiation, third term shows entropy generation because of

us
17 viscous dissipation and last term shows entropy generation due to chemical reaction.
18
19
20 𝜇𝐶𝑝 𝑐2
𝑃𝑟 (= ) represents Prandtl number, 𝐸𝑐 (= (𝑇 ) is Eckert number,
21 𝑘 ∞ −𝑇0 )𝐶𝑝
22 16𝜎 ∗ 3 𝐷𝑒1
23
24
25
26
27
28
𝑅 (=
3𝑘𝑘 ∗
𝑇∞ ) is radiation parameter, 𝑏 (=

Bejan number is expressed as:

𝐵𝑒 =
an𝑘
) is diffusion variable.

Heat transfer irreversibility

(22)

(23)
dM
Total entropy
29
30
31
32 (𝜃′ )2
(1−𝑆1 )2
33 𝐵𝑒 = (𝜃′ )2 (𝜃′ )2 𝑆2 ′2 2
1 [(1+𝜀)𝑓 ′′2 +1𝜀𝛿𝑓 ′′4 ]+ 𝜑 𝑆1 𝑏 (24)
34 (1−𝑆1 )2
+𝑅
(1−𝑆1 )2
+𝑃𝑟 𝐸𝑐
(1−𝑆1 )2 3 𝑆2 (1−𝑆1 )2
35
36
37 3 Analytical method of solution
38
pte

39 To find out analytical solutions of non-linear differential equations (7)-(9), homotopic analysis
40
41 method is implimented. Non-linear (weak or strong) mathematical problems are well solved by
42
43 this technique because it is independent of large or small parameters. There is no restriction of
44 choosing the initial guesses and linear operators to obtain the analytical solution of series.
45
46 The initiatory guesses are:
ce

47
48
49 𝑓0 (𝜉) = 1 − 𝑒𝑥𝑝( − 𝜉),
50 (1 − 𝑆1 )𝛾1
51 𝜃0 (𝜉) = 𝑒𝑥𝑝( − 𝜉),
1 + 𝛾1
52
(1 − 𝑆2 )𝛾2
Ac

53 𝜑0 (𝜉) = 𝑒𝑥𝑝( − 𝜉),

54 1 + 𝛾2 (25)
55
56 Corresponding linear operators are:
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 8 of 20

1
2
3
4 𝑑3 𝑓 𝑑𝑓

pt
5 L𝑓 (𝑓) = 𝑑𝜉3 − 𝑑𝜉 ,
6
7
𝑑2 𝜃
L𝜃 (𝜃) = − 𝜃,
8 𝑑𝜉 2
9 𝑑2𝜑

cri
10 L𝜑 (𝜑) = − 𝜑,
11
𝑑𝜉 2 (26)
12 which satisfy the specified properties
13 L𝑓 [𝐴3 𝑒𝑥𝑝( − 𝜉) + 𝐴2 𝑒𝑥𝑝( 𝜉) + 𝐴1 ] = 0,
14 L𝜃 [𝐴5 𝑒𝑥𝑝( − 𝜉) + 𝐴4 𝑒𝑥𝑝( 𝜉)] = 0,
15 L𝜑 [𝐴7 𝑒𝑥𝑝( − 𝜉) + 𝐴6 𝑒𝑥𝑝( 𝜉)] = 0,
16

us
(27)
17
18 where A i (𝑖 = 1 − 7) are the optional constants.
19
20 3.1 Zeroth-order problems
21
22
23
24
25
26
27
28
∼

an ∼
(1 − 𝑞)L𝑓 [𝑓 (𝜉; 𝑞) − 𝑓0 (𝜉)] = 𝑞ℏ𝑓 N𝑓 [𝑓 (𝜉; 𝑞), 𝜃(𝜉; 𝑞), 𝜑(𝜉; 𝑞)] ,
∼ ∼
(1 − 𝑞)L𝜃 [𝜃(𝜉; 𝑞) − 𝜃0 (𝜉)] = 𝑞ℏ𝜃 N𝜃 [𝜃(𝜉; 𝑞), 𝑓(𝜉; 𝑞)] ,
∼ ∼
(1 − 𝑞)L𝜑 [𝜑(𝜉; 𝑞) − 𝜑0 (𝜉)] = 𝑞ℏ𝜑 N𝜑 [𝜑(𝜉; 𝑞), 𝑓(𝜉; 𝑞), ],
∼

∼
∼

∼
∼
(28)

(29)
dM
29 (30)
30
31 ∼ ∼ ∼ ∼
32 𝑓 ′ (0; 𝑞) = 1, 𝑓 (0; 𝑞) = 0, 𝜃 ′ (0; 𝑞) = −𝛾1 [1 − 𝜃(0) − 𝑆1 ] ,
∼ ∼ ∼ ∼ ∼
33
34
𝜑 ′ (0; 𝑞) = −𝛾2 [1 − 𝜑(0) − 𝑆2 ], 𝑓 ′ (∞; 𝑞) = 0, 𝜃(∞; 𝑞) = 0, 𝜑(∞; 𝑞) = 0,
(31)
35
36
∼ ∼ ∼
37 ∼ ∼ ∼ ∂3 𝑓 (𝜉;𝑞) ∼ ∂2 𝑓 (𝜉;𝑞) ∂𝑓 (𝜉;𝑞)
38 N𝑓 [𝑓 (𝜉, 𝑞), 𝜃(𝜉; 𝑞), 𝜑(𝜉; 𝑞)] = (1 + 𝜀) + 𝑓(𝜉, 𝑞) + ( ∂𝜉 )2
∂𝜉 3 ∂𝜉 2
∼ ∼
pte

39
40 ∂2𝑓 (𝜉; 𝑞) 2 ∂3 𝑓(𝜉; 𝑞)
41
−𝜀𝛿( )
∂𝜉 2 ∂𝜉 3
42 ∼ ∼
43 +𝜆𝜃(𝜉, 𝑞) 𝑠𝑖𝑛 𝛽1 + 𝜆1 𝜑(𝜉, 𝑞) 𝑠𝑖𝑛 𝛽1 ,
44 (32)
45 ∼ ∼ ∼ ∼
∼ ∼
∂2 𝜃(𝜉;𝑞) ∂𝑓 (𝜉;𝑞) ∂𝑓 (𝜉;𝑞)
46 N𝜃 [𝑓 (𝜉, 𝑞), 𝜃(𝜉; 𝑞), ] = (1 + 𝑅) − 2𝑆1 𝑃𝑟 − 2𝜃(𝜉, 𝑞)𝑃𝑟 +
∂𝜉 2 ∂𝜉 ∂𝜉
ce

47 ∼
48 ∼ ∂𝜃(𝜉; 𝑞) ∼
49 𝑃𝑟𝑓(𝜉, 𝑞) + 𝛿1 𝑃𝑟𝜃(𝜉, 𝑞)
50 ∂𝜉
51
52
Ac

53
54
55 (33)
56
57
58
59
60
Page 9 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3 ∼ 2
4 ∂2 𝑓(𝜉; 𝑞)

pt
5 (1 + 𝜀) ( )
∂𝜉 2
6 +𝐸𝑐 ,
7 ∼ 4
2
8 1 ∂ 𝑓(𝜉; 𝑞)
− 𝜀𝛿 ( )
9
[ 3 ∂𝜉 2 ]

cri
10
11 ∼ ∼
12 ∂𝑓 (𝜉;𝑞) ∂𝑓 (𝜉;𝑞) ∼
∼ ∼ 2∼ 2𝑆2 +2 𝜑(𝜉; 𝑞)
13 ∂ 𝜑(𝜉;𝑞) ∂𝜉 ∂𝜉
14
N𝜑 [𝑓(𝜉, 𝑞), 𝜑(𝜉; 𝑞), ] = − 𝑆𝑐 [ ∼ ∼ ]
∂𝜉 2 ∂𝑓 (𝜉;𝑞) ∂𝜑 (𝜉;𝑞)
15 − ∂𝜉 ∂𝜉
16 ∼

us
17 +𝑆𝑐𝑘𝑟𝜑(𝜉, 𝑞),
18
(34)
19 [0,1] represents the value of embedding parameter 𝑞 while ℏ𝑓 , ℏ𝜃 , and ℏ𝜑 are known as
20
21 auxiliary parameters containing non-zero values.
22
23
24
25
26
27
28
3.2 m-th-order problems

𝑓
an
ℏ𝑓 R 𝑚 (𝜉) = L𝑓 [𝑓𝑚 (𝜉) − 𝜒𝑚 𝑓𝑚−1 (𝜉)],
ℏ𝜃 R𝜃𝑚 (𝜉) = L𝜃 [𝜃𝑚 (𝜉) − 𝜒𝑚 𝜃𝑚−1 (𝜉)],
𝜑
dM
29 ℏ𝜑 R 𝑚 (𝜉) = L𝜑 [𝜑𝑚 (𝜉) − 𝜒𝑚 𝜑𝑚−1 (𝜉)],
30 (35)
31
32 𝑓𝑚′ (0) = 0, 𝑓(0) = 0, 𝜃𝑚
′ (0) ′ (0)
= 𝛾1 𝜃𝑚 (0), 𝜑𝑚 = −𝛾2 𝜑𝑚 (0),
33 ′
34
𝑓𝑚 (∞) = 0, 𝜃𝑚 (∞) = 0, 𝜑𝑚 (∞) = 0,
(36)
35
36
𝑓 ″
37 ′′′
R 𝑚 (𝜉) = (1 + 𝛼)𝑓𝑚−1 + ∑𝑚−1 ′
(𝑓𝑚−1−𝑘 ′2
𝑓𝑘 ) − 𝑓𝑚−1 − 𝜀𝛿 ∑𝑚−1 ′′′
𝑓𝑚−1−𝑘 ∑𝑘𝑙=0 ′′
(𝑓𝑘−𝑙 𝑓𝑙′′ ) +
𝑘=0 𝑘=0
38
𝜆 𝑠𝑖𝑛 𝛽1 𝜃𝑚−1 +
pte

39
40 𝜆1 𝑠𝑖𝑛 𝛽1 𝜑𝑚−1 ,
𝑚−1
41 (37)
42 R𝜃𝑚 (𝜉) = (1 + ′′
𝑅)𝜃𝑚−1 − ′
2𝑆1 𝑃𝑟𝑓𝑚−1 − 2𝑃𝑟 ∑ ′
(𝑓𝑚−1−𝑘 𝜃𝑘 ) + 𝛿1 𝑃𝑟𝜃𝑚−1 +
43 𝑘=0
44 𝑘 𝑘
45 1
𝐸𝑐 [(1 + 𝜀) ∑ ′′
(𝑓𝑘−𝑙 𝑓𝑙′′ )2 − 𝜀𝛿 ∑ ′′
(𝑓𝑘−𝑙 𝑓𝑙′′ )4 ]
46 3
𝑙=0 𝑙=0
ce

47 𝑚−1
48 ′
49 +𝑃𝑟 ∑ (𝜃𝑚−1−𝑘 𝑓𝑘 ),
50 𝑘=0
𝑚−1 𝑚−1 (38)
51
𝜑 ′′ ′ ′ ′
52 R 𝑚 (𝜉) = 𝜑𝑚−1 − 𝑆𝑐 [2𝑆2 𝑓𝑚−1 +2 ∑ 𝑓𝑚−1−𝑘 𝜑𝑘 −∑ 𝜑𝑚−1−𝑘 𝑓𝑘 ] + 𝑆𝑐𝑘𝑟𝜑𝑚−1
Ac

53 𝑘=0 𝑘=0
54
55
(39)
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 10 of 20

1
2
3 0, 𝑚 ≤ 1
4 𝜒𝑚 = { .
1, 𝑚 > 1

pt
5 (40)
6
For 𝑞 = 0 and 𝑞 = 1 , one can write
7 𝑓(𝜉; 0) = 𝑓0 (𝜉), 𝑓(𝜉; 1) = 𝑓(𝜉),
8
9 𝜃(𝜉; 0) = 𝜃0 (𝜉), 𝜃(𝜉; 1) = 𝜃(𝜉),

cri
10
11 𝜑(𝜉; 0) = 𝜑0 (𝜉), 𝜑(𝜉; 1) = 𝜑(𝜉), (41)
12
13
and with diversity of 𝑞 from 𝑧𝑒𝑟𝑜 to 𝑜𝑛𝑒 solution begins from first approximations
14 𝑓0 (𝜉) , 𝜃0 (𝜉) and 𝜑0 (𝜉) to final solutions. By using 𝑞 = 1 and Taylor's series, we get
15 𝑓(𝜉) = 𝑓0 (𝜉) + ∑∞ 𝑚=1 𝑓𝑚 (𝜉),
∞
16

us
17 𝜃(𝜉) = 𝜃0 (𝜉) + ∑ 𝜃𝑚 (𝜉),
18 𝑚=1
19 ∞
20 𝜑(𝜉) = 𝜑0 (𝜉) + ∑ 𝜑𝑚 (𝜉),
21
𝑚=1
22
23
24
25
26
27
28
an
The appropriate solutions 𝑓𝑚 , 𝜃𝑚 and 𝜑𝑚 corresponding to (𝑓𝑚∗ , 𝜃𝑚
∗

𝑓𝑚 (𝜉) = 𝑓𝑚? (𝜉) + 𝐴1 + 𝐴2 𝑒 𝜉 + 𝐴3 𝑒 −𝜉 ,

𝜃𝑚 (𝜉) = 𝜃𝑚
𝜑𝑚 (𝜉) = 𝜑𝑚
? (𝜉)
? (𝜉)
+ 𝐴4 𝑒 𝜉 + 𝐴5 𝑒 −𝜉 ,
+ 𝐴6 𝑒 𝜉 + 𝐴7 𝑒 −𝜉 ,
∗ )
and 𝜑𝑚 are
(42)
dM
29 (43)
30
31
3.3 Convergence analysis
32
33
It is clear from the figure (see Fig. 1) −1.5 ≤ ℏ𝑓 ≤ −0.1 , −1.4 ≤ ℏ𝜃 ≤ −0.1 and
34 −1.4 ≤ ℏ𝜑 ≤ −0.2 are attained range values of auxiliary parameters ℏ𝑓 , ℏ𝜃 ,and ℏ𝜑
35
36 which represent the convergence region of the series solution. Well known Homotopic analysis
37
38 technique is applied to get these ranges corresponding to velocity, temperature and concentration,
pte

39
40 the range, where plotted ℎ − 𝑐𝑢𝑟𝑣𝑒𝑠 are horizontal indicated as convergence region.
41
42
43
44
45
46
ce

47
48
49
50
51
52
Ac

53
54
55
56
57
58
59
60
Page 11 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3
4

pt
5
6
7
8
9

cri
10
11
12
13
14
15
16

us
17
18
19
20
21
22 Fig. 1 h-curves for 𝑓(𝜉), 𝜃(𝜉), 𝜙(𝜉)
23
24
25
26
27
28
4 Results and discussion
an
dM
29 Temperature, velocity and concentration distributions are illustrated in this section. Fig. 2
30 portrays the increasing behavior of thermal buoyancy parameter 𝜆 on velocity profile 𝑓 ′ (𝜉) .
31
32 It shows that velocity decays when 𝜆 increases. Physically, fluid convection due to gravity
33
34 increses with the increment of 𝜆 which slows down the velocity profile. Fig. 3 illustrates that
35
36 larger 𝜀 (material fluid parameter) leads to larger horizontal velocity component. Physically,
37 viscosity of fluid decreases against dominant 𝜀 and consequently velocity raises. Thus enhanced
38
pte

39 velocity field appears. Fig. 4 exhibits role of 𝛽1 (angle of inclination) on velocity field.
40
41 Physically as the angle of inclination increases, velocity field also increases because buoyancy
42
43 forces increase in the presence of gravitational force. Hence velocity field enhances. Fig. 5
44 shows the conduct of temperature field due to the impact of Prandtl number . It describes that
45
46 when Pr grows , temperature field decays. According to physical justification low thermal
ce

47
48 diffusivity is responsible for higher Prandtl number 𝑃𝑟 which consequently results in less heat
49
50 transfer from heated wall to cold fluid. Thus, decrement occurs in the fluid's temperature Fig. 6
51 depicts the character of 𝑅 (radiation parameter) on temperature field 𝜃(𝜉) . Infact larger 𝑅
52
Ac

53 produces larger radiative heat flux on the surface which leads to higher temperature 𝜃(𝜉).
54
55 Impact of 𝛿1 (heat generation/absorption parameter) on temperature enclosure 𝜃(𝜉) is
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 12 of 20

1
2
3 displayed in Fig. 7 . It has been found that with the increment of heat generation parameter more
4

pt
5 heat produces due to which temperature profile increases. Though opposite behavior of
6
7 temperature field is found for heat absorption parameter. Character of temperature field due to
8
9 thermal Biot number 𝛾1 is expressed in Fig. 8 . As thermal Biot number increases, heat

cri
10 transfer rate increases and consequently temperature of the fluid grows. Behavior of solutal Biot
11
12 number 𝛾2 on concentration profile is depicted in Fig. 9 . Larger 𝛾2 corresponds to higher
13
14 concentration field. Physically due to the increment of 𝛾2 , mass transfer coefficient enhances
15
16 and as a result concentration distribution enhances. Fig. 10 elucidates characteristics of thermal

us
17 stratification parameter 𝑆1 on temperature field. It shows decreasing trend of temperature
18
19 enclosure and related boundary layer. Physically, higher stratified parameter results in low
20
21 convective flow between heated wall and ambient fluid. Thus, temperature field decays. Fig. 11.
22
23
24
25
26
27
28
an
reflects the impact of chemical reaction parameter 𝑘𝑟 on concentration field. Concentration
field enhances for dominant values of costructive chemical reaction. For dominant constructive
chemical reaction parameter, more fluid particles are produced as a product. Hence concentration
field enhances. Fig. 12 demonstrates impact of solutal stratification parameter 𝑆2 on
dM
29
30
concentration field. Dominant 𝑆2 is responsible for lower concentration. Infact larger 𝑆2
31 reduces the concentration difference between concentration of the sheet and ambient fluid which
32
33 behaves as a resistance for mass transfer. Hence, concentration field decays. Diversity in
34
35 concentration distribution due to 𝑆𝑐 is scrutinized in Fig. 13. Concentration field reflects
36
37
decreasing behavior for larger 𝑆𝑐 . It describes that because of low mass diffusion 𝑆𝑐 raises,
38 which results reduction in concentration enclosure. Fig. 14 displays the impact of Prandtl
pte

39
40 number 𝑃𝑟 on entropy generation. Entropy generation rate increases for dominant 𝑃𝑟 . Fluids
41
42 with higher Pr corresponding to lower thermal conductivity. Thus fluid has small temperature field
43
44
and in more probable or equilibrium state. Hence entropy generation rate increases. Fig. 15
45 demonstrates that there is an obvious enhancement in entropy generation rate corresponding to
46
ce

47 Eckert number Ec, Physically, difference between boundary layer enthalpy and kinetic energy of
48
49 particles in flow increase which results enhancement in irreversibility rate, thus entropy generation
50
rate increases. Fig. 16 investigates the behavior of 𝜀 and solutal buoyancy parameter 𝜆1
51
52 1/2
on skin friction 𝑅𝑒𝑥 𝐶𝑓 . It is noticed that with the increment in 𝜀 and 𝜆1 . Skin friction
Ac

53
54 coefficient decays. Fig . 17 displays the behavior of thermal stratification parameter 𝑆1 and
55
56 Prandtl number 𝑃𝑟 on Nusselt number enclosure. It shows as 𝑆1 and 𝑃𝑟 grows Nusselt
57
58
59
60
Page 13 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3 number rises. Fig. 18 reveals the response of Schmidt number and chemical reaction parameter
4

pt
5 on Sherwood number. It is noticed that with the increment of 𝑆𝑐 and 𝑘𝑟 Sherwood number
6
7 gets boost.
8
9 Table 1 represents the comparison of skin friction with Wahab et al. [33] and Javed et al. [34]. It

cri
10 is concluded that all the results are in good agreement.
11
12
13
14
15
16

us
17
18
19
20
21
22
23
24
25
26
27
28
Fig. 2 Analysis of 𝜆 on 𝑓'(𝜉)
an Fig. 3 Analysis of 𝜀 on 𝑓'(𝜉)
dM
29
30
31
32
33
34
35
36
37
38
pte

39
40
41
42
43
44
45 Fig. 4 Analysis of 𝛽1 on 𝜃(𝜉) Fig. 5 Analysis of 𝑃𝑟 on 𝜃(𝜉)
46
ce

47
48
49
50
51
52
Ac

53
54
55
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 14 of 20

1
2
3
4

pt
5
6
7
8
9

cri
10
11
12
13
14
15
16
Fig. 6 Analysis of 𝑅 on 𝜃(𝜉) Fig. 7 Analysis of 𝛿1 on 𝜃(𝜉)

us
17
18
19
20
21
22
23
24
25
26
27
28
an
dM
29
30
31
32
33
F

34
35
Fig. 8 Analysis of 𝛾1on 𝜃(𝜉) Fig. 9 Analysis of 𝛾2 on 𝜑(𝜉)
36
37
38
pte

39
40
41
42
43
44
45
46
ce

47
48
49
50
51
52
Ac

53
54 Fig. 10 Analysis of 𝑆1on 𝜃(𝜉) Fig. 11 Analysis of 𝑘𝑟 on 𝜑(𝜉)
55
56
57
58
59
60
Page 15 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3
4

pt
5
6
7
8
9

cri
10
11
12
13
14
15
16

us
17
18 Fig. 12 Analysis of 𝑆2 on 𝜑(𝜉) Fig. 13 Analysis of 𝑆𝑐 on 𝜑(𝜉)
19
20
21
22
23
24
25
26
27
28
an
dM
29
30
31
32
33
34 Fig. 14 Analysis of 𝑃𝑟on 𝑁𝑔 Fig. 15 Analysis of 𝐸𝑐 on 𝑁𝑔
35
36
37
38
pte

39
40
41
42
43
44
45
46
ce

47
48
49
50
51
52
Ac

53 Fig. 16 Analysis of 𝜆1 and 𝜀 on 𝐶𝑓 Fig. 17 Analysis of 𝑆1 and 𝑃𝑟on 𝑁𝑢

54
55
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 16 of 20

1
2
3
4

pt
5
6
7
8
9

cri
10
11
12
13
14
15
16

us
17 Fig. 18 Analysis of 𝑘𝑟 and 𝑆𝑐on 𝑆ℎ
18
19
20
21
1/2
22 Table 1. Comparison of skin friction coefficient 𝐶𝑓 𝑅𝑒𝑥 of the present values [in brackets]
23
24
25
26
27
28
𝜀/𝛿
0.0
0.0
-1
(-1)
0.2
-1.0954
(-1.095445)
0.4
an
with the previous published results Javed et al. and (Wahab et al.) for 𝜆 = 0 and 𝜆1 = 0

-1.1832
(-1.264911)
0.6
-1.2649
(-1.264911)
0.8
-1.3416
(-1.341641)
1.0
-1.4142
(-1.414214)
dM
29 [-1] [-1.09545] [-1.26492] [-1.264910] [-1.341639] [-1.414213]
30
0.1 -1 -1.0940 -1.1808 -1.2620 -1.3384 -1.4107
31
32 (-1) (-1.093953) (-1.18842) (-1.261993) (-1.338379) (-1.410732)
33 [-1] [-1.093951] [-1.18842] [-1.261991] [-1.338377] [-1.410730]
34 0.2 -1 -1.0924 -1.1784 -1.2590 -1.3351 -1.4072
35 (-1) (-1.092445) (-1.178431) (-1.259022) (-1.335054) (-1.407183)
36 [-1] [-1.092443] [-1.178429] [-1.259020] [-1.335054] [-1.407183]
37
38
0.3 -1 -1.0909 -1.1776 -1.2560 -1.3317 -1.4036
(-1) (-1.090921) (-1.175981) (-1.255996) (-1.331665) (-1.403562)
pte

39
40 [-1] [-1.090920] [-1.175979] [-1.255994] [-1.331663] [-1.403561]
41 0.4 -1 -1.0894 -1.1735 -1.2529 -1.3282 -1.3999
42 (-1) (-1.089381) (-1.173490) (-1.252912) (-1.328205) (-1.399867)
43
44
[-1] [-1.089380] [-1.173490] [-1.252910] [-1.328203] [-1.399865]
45 0.5 -1 -1.0878 -1.1710 -1.2498 -1.3247 -1.3961
46 (-1) (-1.087823) (-1.170957) (-1.249765) (-1.324671) (-1.396090)
ce

47 [-1] [-1.087821] [-1.170955] [-1.249763] [-1.324669] [-1.396089]

48 0.6 -1 -1.0862 -1.1684 -1.2466 -1.3211 -1.3922
49
(-1) (-1.086247) (-1.168379) (-1.246551) (-1.321057) (-1.39228)
50
51 [-1] [-1.086247] [-1.168377] [-1.246550] [-1.246551] [-1.39228]
52 0.7 -1 -1.0847 -1.1658 -1.2433 -1.3174 -1.3883
Ac

53 (-1) (-1.084653) (-1.165752) (-1.24367) (-1.317359) (-1.388272)

54 [-1] [-1.084653] [-1.165752] [-1.24366] [-1.317357] [-1.388270]
55
0.8 -1 -1.0830 -1.1631 -1.2399 -1.3136 -1.3842
56
57
58
59
60
Page 17 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3 (-1) (-1.083040) (-1.163075) (-1.239906) (-1.313568) (-1.384217)
4

pt
5 [-1] [-1.083039] [-1.163073] [-1.239902] [-1.313565] [-1.384217]
6 0.9 -1 -1.0814 -1.1603 -1.2365 -1.3097 -1.3801
7 (-1) (-1.081407) (-1.160345) (-1.236464) (-1.309677) (-1.3880053)
8 [-1] [-1.081405] [-1.160343] [-1.236463] [-1.309677] [-1.3880053]
9

cri
10 1.0 -1 -1.0798 -1.1576 -1.2329 -1.3057 -1.3758
11 (-1) (-1.079753) (-1.157556) (-1.232933) (-1.305679) (-1.375771)
12
13 [-1] [-1.079750] [-1.157555] [-1.232931] [-1.305677] [-1.375769]
14
15
16

us
17 5 Summary
18
19 In present investigation phenomena of radiation and heat generation/absorption with nonlinear
20
21
stratification in Powell-Eyring fluid flow in assistance with Entropy generation condition are
22
23
24
25
26
27
28
•
•
•
an
analysed and declared. The important points are summarised as:
Angle of inclination executes gain in velocity field because of higher rate of transfer of heat.
Radiation parameter concludes raise in temperature field due to higher heat flux on surface.
Heat generation/absorption results increased temperature profile beacuse of heat generation.
dM
29
• Solutal and thermal Biot number are liable for higher concentration and temperature fields
30
31 respectively.
32
33 • Thermal and solutal stratified parameter accordingly lessen the temperature and concentration
34
35 fields respectively.
36
37
• Chemical reaction parameter boost up the concentration field.
38 • Entropy generation increases with the increment of Prandtl number and Eckert number.
pte

39
40 Nomenclature
41
42
43 𝑁𝑔 Entropy generation (non- 𝑆1 Thermal stratification
44 dimensional) parameter
45 𝑢, 𝑣 Velocity component 𝑆2 Solutal stratification parameter
46
𝐶 Fluid concentration 𝑔 Gravitational acceleration
ce

47
48 𝐶0 Reference concentration 𝑘 Thermal conductivity
49 𝐶∞ Ambient concentration 𝑘𝑟 Chemical reaction parameter
50 𝐶𝑓 Variable concentration 𝐾1 Chemical reaction coefficient
51
52
𝑇 Fluid temperature Greek Symbols
𝑇0 𝜐
Ac

53 Reference temperature Kinematic viscosity

54 𝑇∞ Ambient temperature 𝛿, 𝜀 Dimensionless parameters
55 𝑇𝑓 Heated fluid temperature 𝜉 Similarity variable
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 18 of 20

1
2
3
𝐶𝑓𝑥 Skin friction coefficient 𝜑 Dimensionless concentration
4

pt
5 𝑄0 Heat generation/Absorption 𝜆 Thermal buoyancy parameter
6 coefficient
7 𝑢𝑤 Stretching velocity 𝜆1 Solutal buoyancy parameter
8 𝛽, 𝐶1 Material parameters 𝛾1 Thermal Biot number
9
𝑆ℎ𝑥 Sherwood number 𝜃

cri
10
Dimensionless temperature
11 𝑅 Radiation parameter 𝜌 Fluid density
12 𝐸𝑐 Eckert number 𝑘* Absorption coefficient
13 𝐵𝑒 Bejan number 𝑏 Diffusion variable
14
𝐷 Diffusion species coefficient 𝛿1 Heat generation/absorption
15
16
parameter

us
17 𝑁𝑢𝑥 Nusselt number 𝛾2 Solutal Biot number
18 𝑅𝑒 Reynolds number 𝛽1 Angle of inclination
19 𝑆𝑐 Schmidt number 𝛽𝑡 Coefficient of thermal
20 expansion
21
22
𝑃𝑟 Prandtl number 𝛽𝑐 Coefficient of mass expansion
23
24
25
26
27
28
Cp
𝑑1 , 𝑑2 , 𝑒1 , 𝑒2
ℎ𝑓
ℎ𝑐
Dimensional constant
an
Specific heat capacity

Heat transfer coefficient

Mass transfer coefficient
dM
29
30
31 Reference
32
33 [1] M. I. Khan, S. Ahmad, T. Hayat and A. Alsaedi, Entropy Generation and Activation
34
Energy Impact on Radiative Flow of Viscous Fluid in Presence of Binary Chemical
35
36 Reaction, International Journal of Chemical Reactor Engineering; 2018; 20180045
37 [2] M. I. Khan, S. Qayyum, T. Hayat, M. Imran Khan and A. Alsaedi, Entropy optimization
38 in flow of Williamson nanofluid in the presence of chemical reaction and Joule heating,
pte

39 International Journal of Heat and Mass Transfer, 133 (2019) 959--967

40 [3] S. O. Alharbi, A. Dawar , Z. Shah , W. Khan, M. Idrees , S. Islam and I. Khan, Flow over
41 an Unsteady Oscillatory Porous Stretching Surface under the Impact of Thermal
42
Radiation and Heat Source/Sink,Entropy Generation in MHD Eyring--Powell Fluid,
43
44 2018, Applied Sciences
45 [4] W. Jamshed and A. Aziz, A comparative entropy based analysis of Cu and
46 Fe3O4/methanol Powell-Eyring nanofluid in solar thermal collectors subjected to thermal
ce

47 radiation, variable thermal conductivity and impact of different nanoparticles shape,

48 Results in Physics, 9 (2018) 195-205
49 [5] A. S. Butt, A. Ali and A. Mehmood, Entropy analysis in MHD nanofluid flow near a
50
51
convectively heated stretching surface, Int. J. Exergy, 20 (2016) 318-342
52 [6] A. S. Butt, A. Ali, T. Nazim; and A. Mehmood, Entropy Production in Mixed Convective
Ac

53 Magnetohydrodynamic Flow of Nanofluid Over a Linearly Stretching Sheet, Journal of

54 Nanofluids, 6 (2017) 379-389
55 [7] A. S. Butt, A. Ali, R. Masood and Z. Hussain, Parametric Study of Entropy Generation
56
57
58
59
60
Page 19 of 20 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1

1
2
3 Effects in Magnetohydrodynamic Radiative Flow of Second Grade Nanofluid Past a
4

pt
5
Linearly Convective Stretching Surface Embedded in a Porous Medium, Journal of
6 Nanofluids, 7 (2018) 1004-1023
7 [8] T. Hayat, M. Zubair, M. Waqas, A. Alsaedi and M. Ayub, On doubly stratified
8 chemically reactive flow of Powell--Eyring liquid subject to non-Fourier heat flux theory,
9 Results in Physics 7, (2017) 99--106

cri
10 [9] S. O. Adesanyaa, H. A. Ogunseyeb, and S. Jangilic, Unsteady squeezing flow of a
11
radiative Eyring-Powell fluid channel flow with chemical reactions, International Journal
12
13
of Thermal Sciences 125 (2018) 440--447
14 [10] K. Rehman, M. Y. Malik, T. Salahuddin and M. Naseer, Dual stratified mixed convection
15 flow of Eyring-Powell fluid over an inclined stretching cylinder with heat
16 generation/absorption effect, AIP ADVANCES 6, (2016)

us
17 [11] K. Rehman, M.Y. Malik and O.D. Makinde, Parabolic curve fitting study subject to Joule
18 heating in MHD thermally stratified mixed convection stagnation point flow of Eyring-
19
Powell fluid induced by an inclined cylindrical surface, Journal of King Saud University -
20
21 - Science 30 (2018) 440--449
22 [12] J. Rahimi, D.D. Ganji, M. Khaki and Kh. Hosseinzadeh, Solution of the boundary layer
23
24
25
26
27
28
[13] an
flow of an Eyring-Powell non-Newtonian fluid over a linear stretching sheet by
collocation method, Alexandria Engineering Journal, 56 (2017) 621--627
T. Hayat, S. Qayyum, S. A. Shehzad and A. Alsaedi, Cattaneo-Christov double-diffusion
theory for three-dimensional flow of viscoelastic nanofluid with the effect of heat
generation/absorption, Results in Physics 8 (2018) 489--495
dM
29 [14] T. Hayat, S. Qayyum, S. Shehzad and A. Alsaedi, Chemical reaction and heat
30 generation/absorption aspects in flow of
31 [15] Walters-B nanofluid with Cattaneo-Christov double-diffusion, Results in Physics 7
32 (2017) 4145--4152
33 [16] F. A. Soomro, R. Haq, Qasem M, Al-Mdallal and Q. Zhang, Heat generation/absorption
34
and nonlinear radiation effects on stagnation point flow of nanofluid along a moving
35
36
surface, Results in Physics 8 (2018) 404--414
37 [17] S. Qayyum , T. Hayat, and A. Alsaedi, Thermal radiation and heat generation/absorption
38 aspects in third grade magneto-nanofluid over a slendering stretching sheet with
pte

39 Newtonian conditions, Physica B: Condensed Matter 537 (2018) 139--149

40 [18] A. Anjum, N.A. Mir, M. Farooq, M. Javed, S. Ahmad, M.Y. Malik and A.S. Alshomrani,
41 Physical aspects of heat generation/absorption in the second grade fluid flow due to Riga
42
plate: Application of Cattaneo-Christov approach, Results in Physics 9 (2018) 955--960
43
44 [19] Liao. S. J, Homotopy analysis method in non-linear differential equation 2012, Springer
45 and Higher Education Press, Heidelberg.
46 [20] Liao. S. J, Advances in the Homotopy Analysis Method 2014, World Scientific Amazon.
ce

47 [21] Liao SJ. Beyond Perturbation: Introduction to Homotopy analysis method. Boca Raton:
48 chapman and Hall, CRC Press; 2003.
49 [22] S. Ahmad, M. Farooq, M. Javed and Aisha Anjum, Slip analysis of squeezing flow using
50
51
doubly stratified fluid, Results in Physics 9 (2018) 527--533
52 [23] M. Farooq, S. Ahmad, M. Javed and Aisha Anjum. Analysis of Cattaneo-Christov heat
Ac

53 and mass fluxes in the squeezed flow embedded in porous medium with variable mass
54 diffusivity, Results in Physics 7 (2017) 3788--3796
55 [24] S. Ahmad, M. Farooq, M. Javed and Aisha Anjum, Double stratification effects in
56
57
58
59
60
 AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-108963.R1 Page 20 of 20

1
2
3 chemically reactive squeezed Sutterby fluid flow with thermal radiation and mixed
4

pt
5
convection, Results in Physics 8 (2018) 1250--1259
6 [25] I. Jabeen, M. Farooq and N. A. Mir,Variable mass and thermal properties in three-
7 dimensional viscous flow: Application of Darcy law, Journal of Central South University,
8 26 (2019) 1271--1282 |
9 [26] B. J. Gireesha, R. S. R. Gorla1 and B. Mahanthesh, Effect of Suspended Nanoparticles on

cri
10 Three-Dimensional MHD Flow, Heat and Mass Transfer of Radiating Eyring-Powell
11
Fluid Over a Stretching Sheet, Journal of Nanofluids, 4 (2015) 1--11
12
13
[27] T. Hayat and M. Farooq, Melting Heat Transfer in the Stagnation Point Flow of Powell-
14 Eyring Fluid, Journal of Thermophysics and Heat Transfer, 27 (2013) 761-766
15 [28] M. Ramzan, M. Farooq, T Hayat and Jae Dong Chung, Radiative and Joule heating e ects
16 in the MHD ßow of a micropolar ßuid with partial slip and convective boundary

us
17 condition, Journal of Molecular Liquids, 221 (2016) 394-400
18 [29] I. Jabeen, M. Farooq and N. A. Mir, Description of stratification phenomena in the fluid
19
reservoirs with first-order chemical reaction, Advances in Mechanical Engineering 11
20
21 (2019) 1--9
22 [30] I. Khan, S. Fatima, M.Y. Malik and T. Salahudd, Exponentially varying viscosity of
23
24
25
26
27
28
[31]
[32]
an
magnetohydrodynamic mixed convection Eyring-Powell nanofluid flow over an inclined
surface, Results in Physics 8 (2018) 1194--1203
Bejan. A, Thermodynamics today, Energy, 160 (2018) 1208-1219
Bejan. A, The thermodynamic design of heat and mass transfer process and devices, Heat
Fluid Flow, 8 (1987) 258-276
dM
29 [33] Bejan. A, A study of entropy generation in fundamental convective heat transfer. ASME
30 J. Heat Transf, 101 (1979) 718-725
31 [34] H. A. Wahab, S. Hussain1, S. Bhatti and M. Naeem, Mixed convection flow of Powell-
32 Eyring fluid over a stretching cylinder with Newtonian heating, Kuwait J. Sci, 43 (2016)
33 1-13
34
[35] T. Javed, N. Ali, Z. Abbas & M. Sajid. Flow of an Eyring-Powell Non-Newtonian Fluid
35
36
over a Stretching Sheet. Chemical Engineering Communications, 200 (2013) 327-336.
37
38
pte

39
40
41
42
43
44
45
46
ce

47
48
49
50
51
52
Ac

53
54
55
56
57
58
59
60