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Darcy–Forchheimer flow over an exponentially stretching curved surface with Cattaneo–Christov double diffusion
Journal Pre-proof Darcy-Forchheimer flow over an exponentially stretching curved surface with Cattaneo-Christov double diffusion Taseer Muhammad, Kiran Rafique, Mir Asma, Metib Alghamdi
PII: DOI: Reference:
S0378-4371(19)32198-3 https://doi.org/10.1016/j.physa.2019.123968 PHYSA 123968
To appear in:
Physica A
Received date : 5 May 2019 Revised date : 16 October 2019 Please cite this article as: T. Muhammad, K. Rafique, M. Asma et al., Darcy-Forchheimer flow over an exponentially stretching curved surface with Cattaneo-Christov double diffusion, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.123968. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.
© 2019 Published by Elsevier B.V.
Journal Pre-proof
Darcy-Forchheimer ‡ow over an exponentially stretching curved surface with Cattaneo-Christov double di¤usion Taseer Muhammad1¤ , Kiran Ra…que2 , Mir Asma3 and Metib Alghamdi4 Department of Mathematics, Government College Women University, Sialkot 51310,
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1
Pakistan 2
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala
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3
Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan
Lumpur 50603, Malaysia 4
Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia
Correspondence: [email protected] (Taseer Muhammad)
Pr e-
¤
Abstract: This article deals with Darcy-Forchheimer viscous liquid ‡ow by an exponentially stretching curved surface. Flow in permeable space is speci…ed via Darcy-Forchheimer relation. Cattaneo-Christov mass and heat di¤usion relations are considered in the mathematical formulation. Appropriate variables lead to highly non-linear ordinary di¤erential equations. The obtained problem is solved numerically through NDSolve technique. The outcomes of di¤erent sundry vari-
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ables on velocity, temperature and concentration are sketched and discussed. The physical quantities like skin friction and heat and mass transfer rates are examined graphically. Our results indicate
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that the heat and mass transfer rates are enhanced for larger values of thermal and concentration relaxation parameters respectively.
Keywords: Exponentially stretching curved surface; Darcy-Forchheimer ‡ow; CattaneoChristov double di¤usion; Numerical solution.
Introduction
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1
Flow analysis over an stretchable surface has several applications in many industrial and technological processes. Such applications include extrusion processes, glass …ber, crystal growing, metal mining, production of plastic and rubber sheet, paper production, crystal growing, glass blowing and continuous casting, liquid layers in condensation process and several others [1 ¡ 10] Thus Crane [11] discussed ‡ow past a stretching plate. Gupta and Gupta [12] studied stretching ‡ow subject to suction and injection. Mgyari and Keller [13] presented 1
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‡ow behavior and heat transfer due to an exponentially stretching surface in presence of exponential temperature distribution. Khan and Sanjayanand [14] analyzed viscoelastic ‡uid ‡ow by exponential extendable sheet. It is remarkable to indicate that numerous attempts are discussed for linear or non-linear extending surfaces. In such attempts, the curved ex-
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tending surface is not addressed. Thus Sajid et al. [15] investigated viscous liquid ‡ow over a curved extendable surface. Flow due to porous curved shrinkable/stretchable surface is examined by Rosca and Pop [16]. Magnetohydrodynamic micropolar liquid ‡ow by curved
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stretchable surface with thermal radiation is studied by Naveed et al. [17]. Abbas et al. [18] reported nanoliquid ‡ow by curved stretchable surface subject to slip e¤ects. Hayat et al. [19] investigated MHD ‡ow of by curved sheet with thermal radiation and chemical reaction. Okechi et al. [20] explained boundary-layer ‡ow by exponential stretchable curved surface.
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Hayat et al. [21] studied MHD micropolar ‡uid ‡ow by curved stretchable surface in the presence of homogeneous-heterogeneous reactions. Hayat et al. [22] also examined nano‡uid ‡ow by non-linear stretchable curved surface with convective mass and heat conditions. The ‡uid ‡ows and transport process in permeable space have wide scope of signi…cance and applications in concoction, modern, pharmaceutical and ecological frameworks. Such applications incorporate displaying of oil supply in protecting procedures, geothermal warmth exchanger formats, ground water frameworks, unre…ned petroleum creation, atomic
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waste transfer, units of the vitality stockpiling, water development in stores and numerous others. Adjusted type of traditional Darcy model is non-Darcian permeable media which
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fuses idleness and limit highlights. Much business related to permeable space issues are contemplated by utilizing Darcy’s hypothesis [23] The conventional Darcian law is reasonable just under the restricted scope of lower velocity and little porosity. This law is unable for high velocities. Along these lines Forchheimer [24] accounted the squared velocity factor in momentum articulation to investigate such highlights. Muskat [25] named this factor as
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"Forchheimer term". Seddeek [26] researched blended convective Darcy-Forchheimer stream with viscous dissemination and thermophoresis highlights. Systematic answer for BrinkmanForchheimer-expanded Darcy ‡ow model is considered by Jha and Kaurangini [27] Hydromagnetic Darcy-Forchheimer stream with variable thickness is broke down by Pal and Mondal [28] Flow of Maxwell material prompted by the convectively warmed sheet in a DarcyForchheimer permeable medium is accounted for by Sadiq and Hayat [29] Constrained convection stagnation-point ‡ow towards a contracting sheet in a Darcy-Forchheimer permeable 2
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space is examined by Bakar et al. [30] Muhammad et al. [31] talks about Darcy-Forchheimer 3D nano‡uid ‡ow subject to convective surface condition. Umavathi et al. [32] analyzed convective nano‡uid ‡ow in a vertical rectangular channel by using Darcy-Forchheimer permeable medium. Cattaneo-Christov warmth motion and homogeneous-heterogeneous responses
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in Darcy-Forchheimer ‡ow is analyzed by Hayat et al. [33] Hayat et al. [34] likewise considered Darcy-Forchheimer ‡ow of viscous liquid because of a bended stretchable surface with homogeneous-heterogeneous responses and Cattaneo-Christov heat ‡ux.
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The heat transfer as a wave instead of di¤usion is a concern of impressive enthusiasm for late analysts and architects. It is a result of huge applications in modern, assembling and metallurgical procedures. Such applications incorporate power age, cooling of electronic gadgets, vitality creation, atomic reactors cooling, nano‡uid mechanics and various others.
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Accessible writing observers that traditional Fourier conduction law [35] is for the most part used for warmth move component. Cattaneo [36] included relaxation time for heat transport along engendering of heat waves with limited speed. Christov [37] further adjusted Cattaneo [36] hypothesis by utilizing time subsidiary with Oldroyd upper convective subordinate. Straughan [38] examined warm convection with Cattaneo-Christov theory. Soundness for Cattaneo-Christov condition is analyzed by Ciarletta and Straughan [39] Haddad [40] detailed warm soundness in Brinkman permeable space with Cattaneo-Christov warmth theory.
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Hayat et al. [41] inspected Cattaneo-Christov warmth theory in ‡ow past an extending sheet of variable thickness. Hayat et al. [42] likewise researched viscoelastic liquid ‡ow by consider-
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ing Cattaneo-Christov warmth theory. Sui et al. [43] considered Cattaneo-Christov twofold di¤usion theory in Maxwell nano‡uid ‡ow. Hayat et al. [44] analyzed three-dimensional nano‡uid ‡ow with Cattaneo-Christov double di¤usion. Hayat et al. [45] additionally considered Darcy-Forchheimer ‡ow because of unsteady bended stretchable surface with CattaneoChristov double di¤usion theory. Further latest studies on Cattaneo-Christov double di¤usion
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are seen via the attempts [46 ¡ 50]
Having above discussion in mind, our objective here is to study Darcy-Forchheimer ‡ow of viscous ‡uid with Cattaneo-Christov double di¤usion. Flow is generated by an exponentially stretching curved surface. Curvilinear coordinates are applied as best …t for surface boundary. The governing mathematical problems are computed numerically by NDSolve technique. The obtained numerical results are sketched and discussed. Furthermore, the skin friction coe¢cient and local Nusselt and Sherwood numbers are analyzed graphically. 3
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2
Modeling
Let us consider steady two dimensional (2D) ‡ow of viscous ‡uid bounded by an exponentially stretching curved surface. Flow is generated due to an exponentially stretching curved surface coiled in a circle of radius . An incompressible ‡uid saturates the porous space
of
characterizing Darcy-Forchheimer model. Let () = denotes the exponential velocity with 0. Heat and mass transfer mechanisms are portrayed through Cattaneo-Christov
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heat theory. Resulting boundary layer emphasized for viscous ‡uid ‡ow are [20 46] : f( + ) g + = 0
(1)
2 1 = +
(2)
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1 + + =¡ ( + ) ( + ) ( + ) µ 2 ¶ 1 + + ¡ ¡ ¤ ¡ 2 2 2 ( + ) ( + )
(3)
Here and denotes the ‡ow velocities in - and -directions respectively, (= ) stands for kinematic viscosity, ¤ for permeability of porous space, =
12 ¤
for non-uniform inertia
coe¢cient of porous medium, for pressure and for drag coe¢cient. Cattaneo-Christov
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double di¤usion theory is established in specifying thermal and concentration di¤usions with heat and mass ‡uxes relaxations. The frame indi¤erent generalization regarding Fourier’s
derived by [46] :
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and Fick’s law (which are named as Cattaneo-Christov anomalous di¤usion expressions) are ¶ q q+ + Vrq ¡ qrV+ (rV) q = ¡r µ ¶ J ¤ J + + VrJ ¡ JrV+ (rV) J = ¡r
µ
(4) (5)
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¤
Here q and J stands for heat and mass ‡uxes respectively, for mass di¤usion coe¢cient, for thermal conductivity and ¤ and ¤ for relaxation time of heat and mass ‡uxes. The classical Fourier’s and Fick’s laws are deduced by putting ¤ = ¤ = 0 in Eqs. (4) and (5) ¡ ¢ By considering steady ‡ow q = J = 0 and incompressibility condition (rV = 0) Eqs. (4) and (5) yield
q + ¤ (Vrq ¡ qrV) = ¡r
(6)
J + ¤ (VrJ ¡ JrV) = ¡r
(7)
4
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Then the energy and concentration equations in the absence of thermal radiation are given by + + ¤ © = ( + )
µ
2 1 + 2 ( + )
¶
with the boundary conditions
! 0
! 0
! 1 ! 1 as ! 1
(11)
µ ¶2 2 µ ¶ 2 = + + + 2 + 2 + Ã ! µ ¶2 2 + + + 2 + + + 2
Pr e-
2
¡
2 ( + )
and
(12)
µ ¶2 2 µ ¶ 2 = + + + 2 + 2 + Ã ! µ ¶2 2 + + + 2 + + + 2
2
2 ( + )
(13)
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¡
al
©
= = 1 + 0 2 = = 1 + 0 2 at = 0 (10)
Here ©
(9)
= 0
p ro
= =
(8)
of
µ 2 ¶ 1 ¤ + + © = + ( + ) 2 ( + )
in which = ( ) stands for thermal di¤usivity, for temperature, for concentration, and for surface temperature and concentration respectively, 1 and 1 for ambient
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‡uid temperature and concentration respectively. Selecting [20] : q ¡ ¢12 0 = = () = ¡ + 2 ( () + 0 ()) = 2
= 1 + 0 2 () = 1 + 0 2 () = 2 2 ()
(14)
Continuity equation (1) is identically satis…ed while Eqs. (2), (3) and (8) ¡ (13) yield 0
=
1 2 0 +
1 1 + 2 0 0 2 00 ¡ 00 2 ¡ 2 ( ) + + + ( + ) ( + ) 0 0 02 + (4 + 0 ) 2 ¡ 2 ¡ 2 = + ( + )
(15)
000 +
5
(16)
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¢ 1 ¡ ( + )3 00 + ( + )2 0 + ( + )2 0 ¡ ( + )2 0 Pr 0 2 2
( + ) 2 00 ¡ 2 2 ( + ) 02 00 ¡
3 02 0 2
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B B 2 ¡ 1 B ¡ 32 2 ( + ) 02 0 ¡ 2 ( + ) 00 + 2 ( + ) 02 @ 2 2 2 ¡ 2 ( + ) 0 0 + 2 0 + 2 0 0 + 2 02 ¢ 1 ¡ ( + )3 00 + ( + )2 0 + ( + )2 0 ¡ ( + )2 0 0 2 2
2 00
2 2
02 00
3 02 0 2
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( + ) ¡ ( + ) ¡ B B 3 2 2 ¡ 2 B ¡ 2 ( + ) 02 0 ¡ 2 ( + ) 00 + 2 ( + ) 02 @ 2 2 2 ¡ 2 ( + ) 0 0 + 2 0 + 2 0 0 + 2 02 = 0 0 = 1 = 1 = 1 at = 0
1
C C C = 0 (17) A 1
C C C = 0 (18) A
Pr e-
0 ! 0 00 ! 0 ! 0 ! 0 as ! 1
(19) (20)
Eliminating pressure from Eqs. (15) and (16), we get
2 1 1 00 0 00 000 ¡ 000 2 + 3 + 2 + + + ( + ) ( + ) ( + ) µ ¶ 3 3 0 00 1 0 02 00 0 ¡ ¡ ¡ ¡ 2 + + + ( + )3 ( + )2 µ ¶ 1 0 00 02 ¡2 2 + =0 +
(21)
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+
where stands for local porosity parameter, for curvature parameter, for Forchheimer
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number, Pr for Prandtl number, for Schmidt number, A for temperature exponent, B for concentration exponent, 1 for thermal relaxation parameter and 2 for concentration relaxation parameter. These variables are 2
¶12
=
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=
µ
= ¤12 Pr = 1 = ¤ 2 = ¤ ¤ (22)
Skin friction and heat and mass transfer rates are given by
in which Re =
¡ Re ¢12
00
1 0
9 > (0) > > =
= (0) ¡ ¡ ¢ Re ¡12 = ¡0 (0) 2 ¡ ¢ Re ¡12 = ¡0 (0) 2
2
stands for local Reynolds number.
6
> > > ;
(23)
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3
Numerical results and discussion
The local similar solutions of Eqs. (17) (18) and (21) through the boundary conditions (19) and (20) are obtained numerically by NDSolve technique. Our purpose here is to examine the contributions of di¤erent physical parameters like Forchheimer number ( ), curvature
of
parameter () thermal relaxation parameter ( 1 ) local porosity parameter () Schmidt number () temperature exponent () Prandtl number (Pr) concentration exponent ()
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and concentration relaxation parameter ( 2 ) on the velocity 0 (), temperature () and concentration () …elds. Fig. 1 presents the impacts of curvature parameter on velocity …eld 0 (). An increment in curvature parameter lead to higher velocity …eld 0 (). For larger curvature parameter the surface radius increases which produces an enhancement in ‡uid velocity. Fig. 2 is plotted to study the e¤ect of local porosity parameter on ve-
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locity pro…le 0 (). Velocity …eld 0 () is a decreasing function of local porosity parameter . Fig. 3 depicts the variation in velocity …eld 0 () for varying Forchheimer number . An increment in Forchheimer number correspond to lower velocity …eld 0 (). Fig. 4 present the curves of temperature …eld () for varying values of curvature parameter For larger curvature parameter , the temperature …eld () enhances. E¤ect of local porosity parameter on temperature pro…le () is shown in Fig. 5. Higher causes an enhancement
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in the temperature …eld (). Fig. 6 presents that increasing value of Forchheimer number shows an increment in temperature …eld (). Fig. 7 presents the curves of temperature
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…eld () for various values of Prandtl number . Temperature …eld () show decreasing trend for increasing values of Prandtl number . Fig. 8 illustrates the behavior of temperature exponent on temperature …eld (). Both temperature …eld () and thermal layer thickness are reduced for larger temperature exponent Fig. 9 shows that larger thermal relaxation parameter 1 lead to lower temperature () and less thickness of thermal layer.
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Fig. 10 plots the concentration …eld () for distinct values of values of curvature parameter . Larger curvature parameter lead to higher concentration …eld () Fig. 11 is sketched to examine the behavior of local porosity parameter on concentration …eld () Increasing value of local porosity parameter depicts increasing trend for concentration …eld (). Fig. 12 presents that concentration …eld () is an increasing function of Forchheimer number Fig. 13 plots the concentration …eld () for varying Schmidt number Larger Schmidt number shows lower concentration …eld () Fig. 14 illustrates the impact of concen-
7
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tration exponent on concentration …eld () An increment in concentration exponent lead to lower concentration …eld () Fig. 15 is plotted to analyze that how concentration …eld () is a¤ected with the variation of concentration relaxation parameter 2 It has been
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noticed that concentration …eld () is reduced for varying concentration relaxation para¡ ¢12 meter 2 Figs. 16 and 17 present the plots of skin friction coe¢cient Re2 for various values of local porosity parameter Forchheimer number and curvature parameter ¡ ¢12 The magnitude of skin friction coe¢cient Re2 is enhanced for increasing values of local
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porosity parameter and Forchheimer number Figs. 18 is sketched to examine the behavior of thermal relaxation parameter 1 and curvature parameter on local Nusselt number ¡ ¢ ¡ ¢¡12 Re ¡12 Clearly the magnitude of local Nusselt number Re2 is enhanced 2
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for higher values of thermal relaxation parameter 1 Figs. 19 presents change in local Sher¡ ¢¡12 wood number Re2 for varying values of concentration relaxation parameter 2 ¡ ¢¡12 against curvature parameter . The magnitude of local Sherwood number Re2
is higher for larger concentration relaxation parameter 2 . Table 1 is developed to validate present data with previous published data by Okechi et al. [20] From this Table, we examined that present NDSolve solution is in excellent agreement with previous solution by Okechi et al. [20] in a limiting case. 1.0
al
0.3, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1
0.8
f '
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0.6
K 0.2, 0.4, 0.6, 0.8
0.4
0.2
0.0
Jo
0
1
2
3
4
5
Fig. 1. Curves of 0 () for .
8
6
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1.0
K 0.9, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1 0.8
f '
0.6
0.0, 0.2, 0.4, 0.5
of
0.4
0.0 0
1
2
3
p ro
0.2
4
5
6
Fig. 2. Curves of 0 () for . 1.0
Pr e-
0.3, A B 0.1, K 0.9, Pr Sc 1.0, 1 2 0.1
0.8
f '
0.6
Fr 0.0, 0.3, 0.5, 0.7 0.4
0.2
0.0 1
2
3
4
5
6
al
0
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Fig. 3. Curves of 0 () for .
1.0
0.3, A B 0.1, F r 0.2, Pr Sc 1.0, 1 2 0.1
0.8
0.6
K 0.3, 0.5, 0.7, 0.9
Jo
0.4
0.2
0.0
0
1
2
3
Fig. 4. Curves of () for .
9
4
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1.0
K 0.9, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1 0.8
0.6
0.2, 0.3, 0.4, 0.5
of
0.4
0.0 0
2
4
p ro
0.2
6
8
10
Fig. 5. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
Fr 0.0, 0.3, 0.5, 0.7 0.4
0.2
0.0 2
4
6
8
10
al
0
urn
Fig. 6. Curves of () for .
1.0
0.3, A B 0.1, F r 0.2, K 0.9, Sc 1.0, 1 2 0.1
0.8
0.6
Pr 0.3, 0.7, 1.0, 1.5
Jo
0.4
0.2
0.0
0
2
4
6
8
Fig. 7. Curves of () for Pr.
10
10
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1.0
0.3, B 0.1, F r 0.2, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
A 0.0, 0.1, 0.2, 0.3
of
0.4
0.0 0
2
4
p ro
0.2
6
8
10
Fig. 8. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, Fr 0.2, K 0.9, Pr Sc 1.0, 2 0.1
0.8
0.6
1 0.0, 0.7, 1.4, 2.1 0.4
0.2
0.0 2
4
al
0
6
8
urn
Fig. 9. Curves of () for 1 .
1.0
0.3, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1
0.8
K 0.3, 0.5, 0.7, 0.9
0.6
Jo
0.4
0.2
0.0 0
1
2
3
Fig. 10. Curves of () for .
11
4
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1.0
K 0.9, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1 0.8
0.6
0.2, 0.3, 0.4, 0.5
of
0.4
0.0 0
2
4
p ro
0.2
6
8
10
Fig. 11. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
Fr 0.0, 0.1, 0.2, 0.3 0.4
0.2
0.0 2
4
6
8
10
al
0
urn
Fig. 12. Curves of () for .
1.0
0.3, A B 0.1, F r 0.2, K 0.9, Pr 1.0, 1 2 0.1
0.8
Sc 0.0, 0.5, 1.0, 1.5
0.6
Jo
0.4
0.2
0.0
0
2
4
6
8
Fig. 13. Curves of () for .
12
10
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1.0
0.3, A 0.1, Fr 0.2, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
B 0.0, 0.3, 0.5, 0.7
of
0.4
0.2
0
2
p ro
0.0 4
6
8
10
Fig. 14. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, Fr 0.2, K 0.9, Pr Sc 1.0, 1 0.1
0.8
0.6
2 0.0, 1.3, 2.3, 3.3 0.4
0.2
0.0 2
4
6
8
al
0
Fig. 15. Curves of () for 2 . Fr 0.2, A B 0.1, Pr Sc 1.0, 1 2 0.1
urn
2.0
2
2.4
2.6
Jo
Re s
12C f
2.2
0.1, 0.2, 0.3, 0.4
2.8
3.0 1.0
Fig. 16 Curves of
1.2
1.4
1.6
1.8
2.0
K
¡ Re ¢12 2
for and when = 0
13
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0.3, A B 0.1, Pr Sc 1.0, 1 2 0.1
2.2
2
2.6
of
Re s
12C f
2.4
Fr 0.1, 0.2, 0.3, 0.4
2.8
1.0
1.2
1.4
1.6
K
Fig. 17 Curves of
2
for and when = 0
12Nus
1 0.1, 0.2, 0.3, 0.4
0.80
s
2
0.78 0.76 0.74 0.72
1.2
1.4
al
1.0
¡ Re ¢¡12 2
1.6
1.8
2.0
K
for 1 and when = 0
urn
Fig. 18 Curves of
0.3, F r 0.2, A B 0.1, Pr Sc 1.0, 1 0.1
0.85
2 0.1, 0.3, 0.5, 0.7
0.80
Jo
s
2
12Shs
2.0
Pr e-
0.82
1 Re s
1.8
0.3, F r 0.2, A B 0.1, Pr Sc 1.0, 2 0.1
0.84
1 Res
¡ Re ¢12
p ro
3.0
0.75
1.0
Fig. 19 Curves of
1.2
1.4
1.6
1.8
2.0
K
¡ Re ¢¡12 2
for 2 and when = 0
14
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Table 1. Comparative study of ¡
¡ Re ¢12
¡
2
for di¤erent values of when = = 0
¡ Re ¢12 2
5
14196
10 13467
13467
20 13135
13135
Conclusions
p ro
4
14196
of
NDSolve Okechi et al. [20]
Darcy-Forchheimer ‡ow of viscous ‡uid over an exponentially stretching curved surface with Cattaneo-Christov double di¤usion is analyzed. Main points are listed below.
Pr e-
² Larger curvature parameter show increasing trend for velocity 0 (), temperature () and concentration () …elds.
² An increment in inertia coe¢cient and local porosity parameter lead to lower velocity …eld 0 () while opposite trend is noticed for temperature () and concentration () …elds.
al
² Temperature …eld () and associated thermal layer thickness show decreasing behavior for higher Prandtl number Pr
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² Concentration …eld () and related concentration layer thickness are reduced for higher Schmidt number
² Skin friction coe¢cient is enhanced for higher local porosity parameter and Forchheimer number
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² Local Nusselt number is higher for larger thermal relaxation parameter 1 ² Local Sherwood number is an increasing function of concentration relaxation parameter 2.
Acknowledgment The authors extend their appreciation to the Deanship of Scienti…c Research at King Khalid University for funding this work through Research Groups Program under grant number (R.G.P2./19/40). 15
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References [1] A. Alsaedi, T. Hayat, T. Muhammad and S.A. Shehzad, MHD three-dimensional ‡ow of viscoelastic ‡uid over an exponentially stretching surface with variable thermal con-
of
ductivity, Comp. Math. Math. Phys., 56 (2016) 1665-1678. [2] K.L. Hsiao, Stagnation electrical MHD nano‡uid mixed convection with slip boundary
p ro
on a stretching sheet, Appl. Thermal Eng., 98 (2016) 850-861.
[3] K.L. Hsiao, Combined electrical MHD heat transfer thermal extrusion system using Maxwell ‡uid with radiative and viscous dissipation e¤ects, Appl. Thermal Eng., 112 (2017) 1281-1288.
Pr e-
[4] K.L. Hsiao, To promote radiation electrical MHD activation energy thermal extrusion manufacturing system e¢ciency by using Carreau-Nano‡uid with parameters control method, Energy, 130 (2017) 486-499.
[5] K.L. Hsiao, Micropolar nano‡uid ‡ow with MHD and viscous dissipation e¤ects towards a stretching sheet with multimedia feature, Int. J. Heat Mass Transfer, 112 (2017) 983990.
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[6] T. Hayat, T. Muhammad and A. Alsaedi, Impact of Cattaneo-Christov heat ‡ux in three-dimensional ‡ow of second grade ‡uid over a stretching surface, Chinese J. Phys.,
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55 (2017) 1242-1251.
[7] H. Wang, Z. Hu, W. Lu and M.D. Thouless, The e¤ect of coupled wear and creep during grid-to-rod fretting, Nuclear Eng. Design, 318 (2017) 163-173. [8] Z. Hu, Developments of analyses on grid-to-rod fretting problems in pressurized water
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reactors, Progress Nuclear Energy, 106 (2018) 293-299. [9] Z. Hu, H. Wang, M.D. Thouless and W. Lu, An approach of adaptive e¤ective cycles to couple fretting wear and creep in …nite-element modeling, Int. J. Solids Structures, 139 (2018) 302-311.
[10] B.J. Gireesha, B. Mahanthesh, O.D. Makinde and T. Muhammad, E¤ects of Hall current on transient ‡ow of dusty ‡uid with nonlinear radiation past a convectively heated stretching plate, Defect Di¤usion Forum, 387 (2018) 352-363. 16
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[11] L. J. Crane, Flow past a stretching plate, J. Appl. Math. Phys. (ZAMP), 21 (1970) 645-647. [12] P. Gupta, A. Gupta, Heat and mass transfer on a stretching sheet with suction or
of
blowing, Can J Chem Eng, 55(1977) 744-746. [13] E. Magyari and B. Keller, Heat and mass transfer in the boundary layers on an expo-
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nentially stretching continuous surface, J. Phy. D: Appl. Phy., 32 (1999) 577-585. [14] S.K. Khan and E. Sanjayanand, Viscoelastic boundary layer ‡ow and heat transfer over an exponential stretching sheet, Int. J. Heat Mass Transfer, 48 (2005) 1534-1542. [15] M. Sajid, N. Ali, T. Javed and Z. Abbas, Stretching a curved surface in a viscous ‡uid,
Pr e-
Chin. Phys. Lett., 27 (2010) 024703.
[16] N.C. Rosca and I. Pop, Unsteady boundary layer ‡ow over a permeable curved stretching/shrinking surface, Eur. J. Mech. B. Fluids, 51 (2015) 61-67. [17] M. Naveed, Z. Abbas and M. Sajid, MHD ‡ow of a microploar ‡uid due to curved stretching surface with thermal radiation, J. Appl. Fluid Mech., 9 (2016) 131-138.
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[18] Z. Abbas, M. Naveed and M. Sajid, Hydromagnetic slip ‡ow of nano‡uid over a curved stretching surface with heat generation and thermal radiation, J. Mol. Liq., 215 (2016)
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756-762.
[19] T. Hayat, M. Rashid, M. Imtiaz and A. Alsaedi, MHD convective ‡ow due to a curved surface with thermal radiation and chemical reaction, J. Mol. Liq., 225 (2017) 482-489. [20] N.F. Okechi, M. Jalil and S. Asghar, Flow of viscous ‡uid along an exponentially stretch-
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ing curved surface, Results Phys., 7 (2017) 2851-2854. [21] T. Hayat, R. Sajjad, R. Ellahi, A. Alsaedi and T. Muhammad, Homogeneousheterogeneous reactions in MHD ‡ow of micropolar ‡uid by a curved stretching surface, J. Mol. Liq., 240 (2017) 209-220. [22] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, Numerical study for nano‡uid ‡ow due to a nonlinear curved stretching surface with convective heat and mass conditions, Results Phys., 7 (2017) 3100-3106. 17
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[23] H. Darcy, Les Fontaines Publiques De La Ville De Dijon, Victor Dalmont, Paris (1856). [24] P. Forchheimer, Wasserbewegung durch boden, Zeitschrift Ver. D. Ing., 45 (1901) 17821788.
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[25] M. Muskat, The ‡ow of homogeneous ‡uids through porous media, Edwards, MI. (1946). [26] M.A. Seddeek, In‡uence of viscous dissipation and thermophoresis on Darcy-
p ro
Forchheimer mixed convection in a ‡uid saturated porous media, J. Colloid Interface Sci., 293 (2006) 137-142.
[27] B.K. Jha and M.L. Kaurangini, Approximate analytical solutions for the nonlinear Brinkman-Forchheimer-extended Darcy ‡ow model, Appl. Math., 2 (2011) 1432-1436.
Pr e-
[28] D. Pal and H. Mondal, Hydromagnetic convective di¤usion of species in DarcyForchheimer porous medium with non-uniform heat source/sink and variable viscosity, Int. Commun. Heat Mass Transfer, 39 (2012) 913-917.
[29] M.A. Sadiq and T. Hayat, Darcy-Forchheimer ‡ow of magneto Maxwell liquid bounded by convectively heated sheet, Results Phys., 6 (2016) 884-890.
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[30] S.A. Bakar, N.M. Ari…n, R. Nazar, F.M. Ali and I. Pop, Forced convection boundary layer stagnation-point ‡ow in Darcy-Forchheimer porous medium past a shrinking sheet,
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Frontiers Heat Mass Transfer, 7 (2016) 38. [31] T. Muhammad, A. Alsaedi, T. Hayat and S.A. Shehzad, A revised model for DarcyForchheimer three-dimensional ‡ow of nano‡uid subject to convective boundary condition, Results Phys., 7 (2017) 2791-2797.
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[32] J.C. Umavathi, O. Ojjela and K. Vajravelu, Numerical analysis of natural convective ‡ow and heat transfer of nano‡uids in a vertical rectangular duct using Darcy-ForchheimerBrinkman model, Int. J. Thermal Sci., 111 (2017) 511-524. [33] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, Darcy-Forchheimer ‡ow with Cattaneo-Christov heat ‡ux and homogeneous-heterogeneous reactions, Plos One, 12 (2017) e0174938.
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[34] T. Hayat, R. S. Saif, R. Ellahi, T. Muhammad and B. Ahmad, Numerical study for Darcy-Forchheimer ‡ow due to a curved stretching surface with Cataneo-Christov heat ‡ux and homogeneous-heterogeneous reactions, Results Phys., 7 (2017) 2886-2892.
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[35] J.B.J. Fourier, Théorie Analytique De La Chaleur, Paris (1822). [36] C. Cattaneo, Sulla conduzione del calore, Atti Semin Mat Fis Univ Modena Reggio
p ro
Emilia, 3 (1948) 83-101.
[37] C.I. Christov, On frame indi¤erent formulation of the Maxwell-Cattaneo model of …nitespeed heat conduction, Mech Res Commun. 36 (2009) 481-486.
[38] B. Straughan, Thermal convection with the Cattaneo-Christov model, Int J Heat Mass
Pr e-
Transfer, 53 (2010) 95-98.
[39] M. Ciarletta and B. Straughan, Uniqueness and structural stability for the CattaneoChristov equations, Mech Res Commun. 37 (2010) 445-447. [40] S.A.M. Haddad, Thermal instability in Brinkman porous media with Cattaneo-Christov heat ‡ux, Int J Heat Mass Transfer, 68 (2014) 659-658.
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[41] T. Hayat, M. Farooq, A. Alsaedi and F. Al-Solamy. Impact of Cattaneo-Christov heat ‡ux in the ‡ow over a stretching sheet with variable thickness, AIP Adv. 5 (2015) 087159.
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[42] T. Hayat, T. Muhammad, A. Alsaedi and M. Mustafa, A comparative study for ‡ow of viscoelastic ‡uids with Cattaneo-Christov heat ‡ux, Plos One, 11 (2016) e0155185. [43] J. Sui, L. Zheng, X. Zhang, Boundary layer heat and mass transfer with CattaneoChristov double-di¤usion in upper-convected Maxwell nano‡uid past a stretching sheet
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with slip velocity, Int J Therm Sci, 104 (2016) 461-468. [44] T. Hayat, T. Muhammad, A. Alsaedi and B. Ahmad, Three-dimensional ‡ow of nano‡uid with Cattaneo-Christov double di¤usion, Results Phys, 6 (2016) 897-903. [45] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, Darcy-Forchheimer ‡ow due to a curved stretching surface with Cattaneo-Christov double di¤usion: A numerical study, Results Phys, 7 (2017) 2663-2670.
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[46] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, An optimal study for DarcyForchheimer ‡ow with generalized Fourier’s and Fick’s laws, Results Phys., 7 (2017) 2878-2885. [47] T. Hayat, T. Ayub, T. Muhammad and A. Alsaedi, Flow of variable thermal conductivity
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Oldroyd-B ‡uid with generalized Fourier’s and Fick’s laws, J. Mol. Liq., 234 (2017) 9-17. [48] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, On model for ‡ow of Burgers nano‡uid
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with Cattaneo-Christov double di¤usion, Chinese J. Phys., 55 (2017) 916-929. [49] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, Three-dimensional ‡ow of Prandtl ‡uid with Cattaneo-Christov double di¤usion, Results Phys., 9 (2018) 290-296.
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[50] T. Hayat, T. Ayub, T. Muhammad and B. Ahmad, Nonlinear computational treatment for couple stress ‡uid ‡ow with Cattaneo-Christov double di¤usion and homogeneous-
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heterogeneous reactions, Int. J. Chem. Reactor Eng., 17 (2019) 20180056.
20
Journal Pre-proof Darcy-Forchheimer flow of viscous fluid is modeled.
Flow is induced by an exponentially stretching curved surface.
Flow in porous medium is described by Darcy-Forchheimer model.
Cattaneo-Christov double diffusion theory is utilized.
Numerical solutions are obtained through NDSolve technique.
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Dear Editor, Hope you are fine and doing well. We have no conflict of interest for this submission.
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Best regards!
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Taseer Muhammad
PII: DOI: Reference:
S0378-4371(19)32198-3 https://doi.org/10.1016/j.physa.2019.123968 PHYSA 123968
To appear in:
Physica A
Received date : 5 May 2019 Revised date : 16 October 2019 Please cite this article as: T. Muhammad, K. Rafique, M. Asma et al., Darcy-Forchheimer flow over an exponentially stretching curved surface with Cattaneo-Christov double diffusion, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.123968. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.
© 2019 Published by Elsevier B.V.
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Darcy-Forchheimer ‡ow over an exponentially stretching curved surface with Cattaneo-Christov double di¤usion Taseer Muhammad1¤ , Kiran Ra…que2 , Mir Asma3 and Metib Alghamdi4 Department of Mathematics, Government College Women University, Sialkot 51310,
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1
Pakistan 2
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala
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3
Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan
Lumpur 50603, Malaysia 4
Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia
Correspondence: [email protected] (Taseer Muhammad)
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¤
Abstract: This article deals with Darcy-Forchheimer viscous liquid ‡ow by an exponentially stretching curved surface. Flow in permeable space is speci…ed via Darcy-Forchheimer relation. Cattaneo-Christov mass and heat di¤usion relations are considered in the mathematical formulation. Appropriate variables lead to highly non-linear ordinary di¤erential equations. The obtained problem is solved numerically through NDSolve technique. The outcomes of di¤erent sundry vari-
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ables on velocity, temperature and concentration are sketched and discussed. The physical quantities like skin friction and heat and mass transfer rates are examined graphically. Our results indicate
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that the heat and mass transfer rates are enhanced for larger values of thermal and concentration relaxation parameters respectively.
Keywords: Exponentially stretching curved surface; Darcy-Forchheimer ‡ow; CattaneoChristov double di¤usion; Numerical solution.
Introduction
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1
Flow analysis over an stretchable surface has several applications in many industrial and technological processes. Such applications include extrusion processes, glass …ber, crystal growing, metal mining, production of plastic and rubber sheet, paper production, crystal growing, glass blowing and continuous casting, liquid layers in condensation process and several others [1 ¡ 10] Thus Crane [11] discussed ‡ow past a stretching plate. Gupta and Gupta [12] studied stretching ‡ow subject to suction and injection. Mgyari and Keller [13] presented 1
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‡ow behavior and heat transfer due to an exponentially stretching surface in presence of exponential temperature distribution. Khan and Sanjayanand [14] analyzed viscoelastic ‡uid ‡ow by exponential extendable sheet. It is remarkable to indicate that numerous attempts are discussed for linear or non-linear extending surfaces. In such attempts, the curved ex-
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tending surface is not addressed. Thus Sajid et al. [15] investigated viscous liquid ‡ow over a curved extendable surface. Flow due to porous curved shrinkable/stretchable surface is examined by Rosca and Pop [16]. Magnetohydrodynamic micropolar liquid ‡ow by curved
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stretchable surface with thermal radiation is studied by Naveed et al. [17]. Abbas et al. [18] reported nanoliquid ‡ow by curved stretchable surface subject to slip e¤ects. Hayat et al. [19] investigated MHD ‡ow of by curved sheet with thermal radiation and chemical reaction. Okechi et al. [20] explained boundary-layer ‡ow by exponential stretchable curved surface.
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Hayat et al. [21] studied MHD micropolar ‡uid ‡ow by curved stretchable surface in the presence of homogeneous-heterogeneous reactions. Hayat et al. [22] also examined nano‡uid ‡ow by non-linear stretchable curved surface with convective mass and heat conditions. The ‡uid ‡ows and transport process in permeable space have wide scope of signi…cance and applications in concoction, modern, pharmaceutical and ecological frameworks. Such applications incorporate displaying of oil supply in protecting procedures, geothermal warmth exchanger formats, ground water frameworks, unre…ned petroleum creation, atomic
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waste transfer, units of the vitality stockpiling, water development in stores and numerous others. Adjusted type of traditional Darcy model is non-Darcian permeable media which
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fuses idleness and limit highlights. Much business related to permeable space issues are contemplated by utilizing Darcy’s hypothesis [23] The conventional Darcian law is reasonable just under the restricted scope of lower velocity and little porosity. This law is unable for high velocities. Along these lines Forchheimer [24] accounted the squared velocity factor in momentum articulation to investigate such highlights. Muskat [25] named this factor as
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"Forchheimer term". Seddeek [26] researched blended convective Darcy-Forchheimer stream with viscous dissemination and thermophoresis highlights. Systematic answer for BrinkmanForchheimer-expanded Darcy ‡ow model is considered by Jha and Kaurangini [27] Hydromagnetic Darcy-Forchheimer stream with variable thickness is broke down by Pal and Mondal [28] Flow of Maxwell material prompted by the convectively warmed sheet in a DarcyForchheimer permeable medium is accounted for by Sadiq and Hayat [29] Constrained convection stagnation-point ‡ow towards a contracting sheet in a Darcy-Forchheimer permeable 2
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space is examined by Bakar et al. [30] Muhammad et al. [31] talks about Darcy-Forchheimer 3D nano‡uid ‡ow subject to convective surface condition. Umavathi et al. [32] analyzed convective nano‡uid ‡ow in a vertical rectangular channel by using Darcy-Forchheimer permeable medium. Cattaneo-Christov warmth motion and homogeneous-heterogeneous responses
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in Darcy-Forchheimer ‡ow is analyzed by Hayat et al. [33] Hayat et al. [34] likewise considered Darcy-Forchheimer ‡ow of viscous liquid because of a bended stretchable surface with homogeneous-heterogeneous responses and Cattaneo-Christov heat ‡ux.
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The heat transfer as a wave instead of di¤usion is a concern of impressive enthusiasm for late analysts and architects. It is a result of huge applications in modern, assembling and metallurgical procedures. Such applications incorporate power age, cooling of electronic gadgets, vitality creation, atomic reactors cooling, nano‡uid mechanics and various others.
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Accessible writing observers that traditional Fourier conduction law [35] is for the most part used for warmth move component. Cattaneo [36] included relaxation time for heat transport along engendering of heat waves with limited speed. Christov [37] further adjusted Cattaneo [36] hypothesis by utilizing time subsidiary with Oldroyd upper convective subordinate. Straughan [38] examined warm convection with Cattaneo-Christov theory. Soundness for Cattaneo-Christov condition is analyzed by Ciarletta and Straughan [39] Haddad [40] detailed warm soundness in Brinkman permeable space with Cattaneo-Christov warmth theory.
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Hayat et al. [41] inspected Cattaneo-Christov warmth theory in ‡ow past an extending sheet of variable thickness. Hayat et al. [42] likewise researched viscoelastic liquid ‡ow by consider-
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ing Cattaneo-Christov warmth theory. Sui et al. [43] considered Cattaneo-Christov twofold di¤usion theory in Maxwell nano‡uid ‡ow. Hayat et al. [44] analyzed three-dimensional nano‡uid ‡ow with Cattaneo-Christov double di¤usion. Hayat et al. [45] additionally considered Darcy-Forchheimer ‡ow because of unsteady bended stretchable surface with CattaneoChristov double di¤usion theory. Further latest studies on Cattaneo-Christov double di¤usion
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are seen via the attempts [46 ¡ 50]
Having above discussion in mind, our objective here is to study Darcy-Forchheimer ‡ow of viscous ‡uid with Cattaneo-Christov double di¤usion. Flow is generated by an exponentially stretching curved surface. Curvilinear coordinates are applied as best …t for surface boundary. The governing mathematical problems are computed numerically by NDSolve technique. The obtained numerical results are sketched and discussed. Furthermore, the skin friction coe¢cient and local Nusselt and Sherwood numbers are analyzed graphically. 3
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2
Modeling
Let us consider steady two dimensional (2D) ‡ow of viscous ‡uid bounded by an exponentially stretching curved surface. Flow is generated due to an exponentially stretching curved surface coiled in a circle of radius . An incompressible ‡uid saturates the porous space
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characterizing Darcy-Forchheimer model. Let () = denotes the exponential velocity with 0. Heat and mass transfer mechanisms are portrayed through Cattaneo-Christov
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heat theory. Resulting boundary layer emphasized for viscous ‡uid ‡ow are [20 46] : f( + ) g + = 0
(1)
2 1 = +
(2)
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1 + + =¡ ( + ) ( + ) ( + ) µ 2 ¶ 1 + + ¡ ¡ ¤ ¡ 2 2 2 ( + ) ( + )
(3)
Here and denotes the ‡ow velocities in - and -directions respectively, (= ) stands for kinematic viscosity, ¤ for permeability of porous space, =
12 ¤
for non-uniform inertia
coe¢cient of porous medium, for pressure and for drag coe¢cient. Cattaneo-Christov
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double di¤usion theory is established in specifying thermal and concentration di¤usions with heat and mass ‡uxes relaxations. The frame indi¤erent generalization regarding Fourier’s
derived by [46] :
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and Fick’s law (which are named as Cattaneo-Christov anomalous di¤usion expressions) are ¶ q q+ + Vrq ¡ qrV+ (rV) q = ¡r µ ¶ J ¤ J + + VrJ ¡ JrV+ (rV) J = ¡r
µ
(4) (5)
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¤
Here q and J stands for heat and mass ‡uxes respectively, for mass di¤usion coe¢cient, for thermal conductivity and ¤ and ¤ for relaxation time of heat and mass ‡uxes. The classical Fourier’s and Fick’s laws are deduced by putting ¤ = ¤ = 0 in Eqs. (4) and (5) ¡ ¢ By considering steady ‡ow q = J = 0 and incompressibility condition (rV = 0) Eqs. (4) and (5) yield
q + ¤ (Vrq ¡ qrV) = ¡r
(6)
J + ¤ (VrJ ¡ JrV) = ¡r
(7)
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Then the energy and concentration equations in the absence of thermal radiation are given by + + ¤ © = ( + )
µ
2 1 + 2 ( + )
¶
with the boundary conditions
! 0
! 0
! 1 ! 1 as ! 1
(11)
µ ¶2 2 µ ¶ 2 = + + + 2 + 2 + Ã ! µ ¶2 2 + + + 2 + + + 2
Pr e-
2
¡
2 ( + )
and
(12)
µ ¶2 2 µ ¶ 2 = + + + 2 + 2 + Ã ! µ ¶2 2 + + + 2 + + + 2
2
2 ( + )
(13)
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¡
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©
= = 1 + 0 2 = = 1 + 0 2 at = 0 (10)
Here ©
(9)
= 0
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= =
(8)
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µ 2 ¶ 1 ¤ + + © = + ( + ) 2 ( + )
in which = ( ) stands for thermal di¤usivity, for temperature, for concentration, and for surface temperature and concentration respectively, 1 and 1 for ambient
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‡uid temperature and concentration respectively. Selecting [20] : q ¡ ¢12 0 = = () = ¡ + 2 ( () + 0 ()) = 2
= 1 + 0 2 () = 1 + 0 2 () = 2 2 ()
(14)
Continuity equation (1) is identically satis…ed while Eqs. (2), (3) and (8) ¡ (13) yield 0
=
1 2 0 +
1 1 + 2 0 0 2 00 ¡ 00 2 ¡ 2 ( ) + + + ( + ) ( + ) 0 0 02 + (4 + 0 ) 2 ¡ 2 ¡ 2 = + ( + )
(15)
000 +
5
(16)
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¢ 1 ¡ ( + )3 00 + ( + )2 0 + ( + )2 0 ¡ ( + )2 0 Pr 0 2 2
( + ) 2 00 ¡ 2 2 ( + ) 02 00 ¡
3 02 0 2
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B B 2 ¡ 1 B ¡ 32 2 ( + ) 02 0 ¡ 2 ( + ) 00 + 2 ( + ) 02 @ 2 2 2 ¡ 2 ( + ) 0 0 + 2 0 + 2 0 0 + 2 02 ¢ 1 ¡ ( + )3 00 + ( + )2 0 + ( + )2 0 ¡ ( + )2 0 0 2 2
2 00
2 2
02 00
3 02 0 2
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( + ) ¡ ( + ) ¡ B B 3 2 2 ¡ 2 B ¡ 2 ( + ) 02 0 ¡ 2 ( + ) 00 + 2 ( + ) 02 @ 2 2 2 ¡ 2 ( + ) 0 0 + 2 0 + 2 0 0 + 2 02 = 0 0 = 1 = 1 = 1 at = 0
1
C C C = 0 (17) A 1
C C C = 0 (18) A
Pr e-
0 ! 0 00 ! 0 ! 0 ! 0 as ! 1
(19) (20)
Eliminating pressure from Eqs. (15) and (16), we get
2 1 1 00 0 00 000 ¡ 000 2 + 3 + 2 + + + ( + ) ( + ) ( + ) µ ¶ 3 3 0 00 1 0 02 00 0 ¡ ¡ ¡ ¡ 2 + + + ( + )3 ( + )2 µ ¶ 1 0 00 02 ¡2 2 + =0 +
(21)
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+
where stands for local porosity parameter, for curvature parameter, for Forchheimer
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number, Pr for Prandtl number, for Schmidt number, A for temperature exponent, B for concentration exponent, 1 for thermal relaxation parameter and 2 for concentration relaxation parameter. These variables are 2
¶12
=
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=
µ
= ¤12 Pr = 1 = ¤ 2 = ¤ ¤ (22)
Skin friction and heat and mass transfer rates are given by
in which Re =
¡ Re ¢12
00
1 0
9 > (0) > > =
= (0) ¡ ¡ ¢ Re ¡12 = ¡0 (0) 2 ¡ ¢ Re ¡12 = ¡0 (0) 2
2
stands for local Reynolds number.
6
> > > ;
(23)
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3
Numerical results and discussion
The local similar solutions of Eqs. (17) (18) and (21) through the boundary conditions (19) and (20) are obtained numerically by NDSolve technique. Our purpose here is to examine the contributions of di¤erent physical parameters like Forchheimer number ( ), curvature
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parameter () thermal relaxation parameter ( 1 ) local porosity parameter () Schmidt number () temperature exponent () Prandtl number (Pr) concentration exponent ()
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and concentration relaxation parameter ( 2 ) on the velocity 0 (), temperature () and concentration () …elds. Fig. 1 presents the impacts of curvature parameter on velocity …eld 0 (). An increment in curvature parameter lead to higher velocity …eld 0 (). For larger curvature parameter the surface radius increases which produces an enhancement in ‡uid velocity. Fig. 2 is plotted to study the e¤ect of local porosity parameter on ve-
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locity pro…le 0 (). Velocity …eld 0 () is a decreasing function of local porosity parameter . Fig. 3 depicts the variation in velocity …eld 0 () for varying Forchheimer number . An increment in Forchheimer number correspond to lower velocity …eld 0 (). Fig. 4 present the curves of temperature …eld () for varying values of curvature parameter For larger curvature parameter , the temperature …eld () enhances. E¤ect of local porosity parameter on temperature pro…le () is shown in Fig. 5. Higher causes an enhancement
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in the temperature …eld (). Fig. 6 presents that increasing value of Forchheimer number shows an increment in temperature …eld (). Fig. 7 presents the curves of temperature
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…eld () for various values of Prandtl number . Temperature …eld () show decreasing trend for increasing values of Prandtl number . Fig. 8 illustrates the behavior of temperature exponent on temperature …eld (). Both temperature …eld () and thermal layer thickness are reduced for larger temperature exponent Fig. 9 shows that larger thermal relaxation parameter 1 lead to lower temperature () and less thickness of thermal layer.
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Fig. 10 plots the concentration …eld () for distinct values of values of curvature parameter . Larger curvature parameter lead to higher concentration …eld () Fig. 11 is sketched to examine the behavior of local porosity parameter on concentration …eld () Increasing value of local porosity parameter depicts increasing trend for concentration …eld (). Fig. 12 presents that concentration …eld () is an increasing function of Forchheimer number Fig. 13 plots the concentration …eld () for varying Schmidt number Larger Schmidt number shows lower concentration …eld () Fig. 14 illustrates the impact of concen-
7
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tration exponent on concentration …eld () An increment in concentration exponent lead to lower concentration …eld () Fig. 15 is plotted to analyze that how concentration …eld () is a¤ected with the variation of concentration relaxation parameter 2 It has been
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noticed that concentration …eld () is reduced for varying concentration relaxation para¡ ¢12 meter 2 Figs. 16 and 17 present the plots of skin friction coe¢cient Re2 for various values of local porosity parameter Forchheimer number and curvature parameter ¡ ¢12 The magnitude of skin friction coe¢cient Re2 is enhanced for increasing values of local
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porosity parameter and Forchheimer number Figs. 18 is sketched to examine the behavior of thermal relaxation parameter 1 and curvature parameter on local Nusselt number ¡ ¢ ¡ ¢¡12 Re ¡12 Clearly the magnitude of local Nusselt number Re2 is enhanced 2
Pr e-
for higher values of thermal relaxation parameter 1 Figs. 19 presents change in local Sher¡ ¢¡12 wood number Re2 for varying values of concentration relaxation parameter 2 ¡ ¢¡12 against curvature parameter . The magnitude of local Sherwood number Re2
is higher for larger concentration relaxation parameter 2 . Table 1 is developed to validate present data with previous published data by Okechi et al. [20] From this Table, we examined that present NDSolve solution is in excellent agreement with previous solution by Okechi et al. [20] in a limiting case. 1.0
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0.3, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1
0.8
f '
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0.6
K 0.2, 0.4, 0.6, 0.8
0.4
0.2
0.0
Jo
0
1
2
3
4
5
Fig. 1. Curves of 0 () for .
8
6
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1.0
K 0.9, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1 0.8
f '
0.6
0.0, 0.2, 0.4, 0.5
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0.4
0.0 0
1
2
3
p ro
0.2
4
5
6
Fig. 2. Curves of 0 () for . 1.0
Pr e-
0.3, A B 0.1, K 0.9, Pr Sc 1.0, 1 2 0.1
0.8
f '
0.6
Fr 0.0, 0.3, 0.5, 0.7 0.4
0.2
0.0 1
2
3
4
5
6
al
0
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Fig. 3. Curves of 0 () for .
1.0
0.3, A B 0.1, F r 0.2, Pr Sc 1.0, 1 2 0.1
0.8
0.6
K 0.3, 0.5, 0.7, 0.9
Jo
0.4
0.2
0.0
0
1
2
3
Fig. 4. Curves of () for .
9
4
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1.0
K 0.9, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1 0.8
0.6
0.2, 0.3, 0.4, 0.5
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0.4
0.0 0
2
4
p ro
0.2
6
8
10
Fig. 5. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
Fr 0.0, 0.3, 0.5, 0.7 0.4
0.2
0.0 2
4
6
8
10
al
0
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Fig. 6. Curves of () for .
1.0
0.3, A B 0.1, F r 0.2, K 0.9, Sc 1.0, 1 2 0.1
0.8
0.6
Pr 0.3, 0.7, 1.0, 1.5
Jo
0.4
0.2
0.0
0
2
4
6
8
Fig. 7. Curves of () for Pr.
10
10
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1.0
0.3, B 0.1, F r 0.2, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
A 0.0, 0.1, 0.2, 0.3
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0.4
0.0 0
2
4
p ro
0.2
6
8
10
Fig. 8. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, Fr 0.2, K 0.9, Pr Sc 1.0, 2 0.1
0.8
0.6
1 0.0, 0.7, 1.4, 2.1 0.4
0.2
0.0 2
4
al
0
6
8
urn
Fig. 9. Curves of () for 1 .
1.0
0.3, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1
0.8
K 0.3, 0.5, 0.7, 0.9
0.6
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0.4
0.2
0.0 0
1
2
3
Fig. 10. Curves of () for .
11
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1.0
K 0.9, A B 0.1, Fr 0.2, Pr Sc 1.0, 1 2 0.1 0.8
0.6
0.2, 0.3, 0.4, 0.5
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0.4
0.0 0
2
4
p ro
0.2
6
8
10
Fig. 11. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
Fr 0.0, 0.1, 0.2, 0.3 0.4
0.2
0.0 2
4
6
8
10
al
0
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Fig. 12. Curves of () for .
1.0
0.3, A B 0.1, F r 0.2, K 0.9, Pr 1.0, 1 2 0.1
0.8
Sc 0.0, 0.5, 1.0, 1.5
0.6
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0.4
0.2
0.0
0
2
4
6
8
Fig. 13. Curves of () for .
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10
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1.0
0.3, A 0.1, Fr 0.2, K 0.9, Pr Sc 1.0, 1 2 0.1 0.8
0.6
B 0.0, 0.3, 0.5, 0.7
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0.4
0.2
0
2
p ro
0.0 4
6
8
10
Fig. 14. Curves of () for . 1.0
Pr e-
0.3, A B 0.1, Fr 0.2, K 0.9, Pr Sc 1.0, 1 0.1
0.8
0.6
2 0.0, 1.3, 2.3, 3.3 0.4
0.2
0.0 2
4
6
8
al
0
Fig. 15. Curves of () for 2 . Fr 0.2, A B 0.1, Pr Sc 1.0, 1 2 0.1
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2.0
2
2.4
2.6
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Re s
12C f
2.2
0.1, 0.2, 0.3, 0.4
2.8
3.0 1.0
Fig. 16 Curves of
1.2
1.4
1.6
1.8
2.0
K
¡ Re ¢12 2
for and when = 0
13
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0.3, A B 0.1, Pr Sc 1.0, 1 2 0.1
2.2
2
2.6
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Re s
12C f
2.4
Fr 0.1, 0.2, 0.3, 0.4
2.8
1.0
1.2
1.4
1.6
K
Fig. 17 Curves of
2
for and when = 0
12Nus
1 0.1, 0.2, 0.3, 0.4
0.80
s
2
0.78 0.76 0.74 0.72
1.2
1.4
al
1.0
¡ Re ¢¡12 2
1.6
1.8
2.0
K
for 1 and when = 0
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Fig. 18 Curves of
0.3, F r 0.2, A B 0.1, Pr Sc 1.0, 1 0.1
0.85
2 0.1, 0.3, 0.5, 0.7
0.80
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s
2
12Shs
2.0
Pr e-
0.82
1 Re s
1.8
0.3, F r 0.2, A B 0.1, Pr Sc 1.0, 2 0.1
0.84
1 Res
¡ Re ¢12
p ro
3.0
0.75
1.0
Fig. 19 Curves of
1.2
1.4
1.6
1.8
2.0
K
¡ Re ¢¡12 2
for 2 and when = 0
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Table 1. Comparative study of ¡
¡ Re ¢12
¡
2
for di¤erent values of when = = 0
¡ Re ¢12 2
5
14196
10 13467
13467
20 13135
13135
Conclusions
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4
14196
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NDSolve Okechi et al. [20]
Darcy-Forchheimer ‡ow of viscous ‡uid over an exponentially stretching curved surface with Cattaneo-Christov double di¤usion is analyzed. Main points are listed below.
Pr e-
² Larger curvature parameter show increasing trend for velocity 0 (), temperature () and concentration () …elds.
² An increment in inertia coe¢cient and local porosity parameter lead to lower velocity …eld 0 () while opposite trend is noticed for temperature () and concentration () …elds.
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² Temperature …eld () and associated thermal layer thickness show decreasing behavior for higher Prandtl number Pr
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² Concentration …eld () and related concentration layer thickness are reduced for higher Schmidt number
² Skin friction coe¢cient is enhanced for higher local porosity parameter and Forchheimer number
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² Local Nusselt number is higher for larger thermal relaxation parameter 1 ² Local Sherwood number is an increasing function of concentration relaxation parameter 2.
Acknowledgment The authors extend their appreciation to the Deanship of Scienti…c Research at King Khalid University for funding this work through Research Groups Program under grant number (R.G.P2./19/40). 15
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References [1] A. Alsaedi, T. Hayat, T. Muhammad and S.A. Shehzad, MHD three-dimensional ‡ow of viscoelastic ‡uid over an exponentially stretching surface with variable thermal con-
of
ductivity, Comp. Math. Math. Phys., 56 (2016) 1665-1678. [2] K.L. Hsiao, Stagnation electrical MHD nano‡uid mixed convection with slip boundary
p ro
on a stretching sheet, Appl. Thermal Eng., 98 (2016) 850-861.
[3] K.L. Hsiao, Combined electrical MHD heat transfer thermal extrusion system using Maxwell ‡uid with radiative and viscous dissipation e¤ects, Appl. Thermal Eng., 112 (2017) 1281-1288.
Pr e-
[4] K.L. Hsiao, To promote radiation electrical MHD activation energy thermal extrusion manufacturing system e¢ciency by using Carreau-Nano‡uid with parameters control method, Energy, 130 (2017) 486-499.
[5] K.L. Hsiao, Micropolar nano‡uid ‡ow with MHD and viscous dissipation e¤ects towards a stretching sheet with multimedia feature, Int. J. Heat Mass Transfer, 112 (2017) 983990.
al
[6] T. Hayat, T. Muhammad and A. Alsaedi, Impact of Cattaneo-Christov heat ‡ux in three-dimensional ‡ow of second grade ‡uid over a stretching surface, Chinese J. Phys.,
urn
55 (2017) 1242-1251.
[7] H. Wang, Z. Hu, W. Lu and M.D. Thouless, The e¤ect of coupled wear and creep during grid-to-rod fretting, Nuclear Eng. Design, 318 (2017) 163-173. [8] Z. Hu, Developments of analyses on grid-to-rod fretting problems in pressurized water
Jo
reactors, Progress Nuclear Energy, 106 (2018) 293-299. [9] Z. Hu, H. Wang, M.D. Thouless and W. Lu, An approach of adaptive e¤ective cycles to couple fretting wear and creep in …nite-element modeling, Int. J. Solids Structures, 139 (2018) 302-311.
[10] B.J. Gireesha, B. Mahanthesh, O.D. Makinde and T. Muhammad, E¤ects of Hall current on transient ‡ow of dusty ‡uid with nonlinear radiation past a convectively heated stretching plate, Defect Di¤usion Forum, 387 (2018) 352-363. 16
Journal Pre-proof
[11] L. J. Crane, Flow past a stretching plate, J. Appl. Math. Phys. (ZAMP), 21 (1970) 645-647. [12] P. Gupta, A. Gupta, Heat and mass transfer on a stretching sheet with suction or
of
blowing, Can J Chem Eng, 55(1977) 744-746. [13] E. Magyari and B. Keller, Heat and mass transfer in the boundary layers on an expo-
p ro
nentially stretching continuous surface, J. Phy. D: Appl. Phy., 32 (1999) 577-585. [14] S.K. Khan and E. Sanjayanand, Viscoelastic boundary layer ‡ow and heat transfer over an exponential stretching sheet, Int. J. Heat Mass Transfer, 48 (2005) 1534-1542. [15] M. Sajid, N. Ali, T. Javed and Z. Abbas, Stretching a curved surface in a viscous ‡uid,
Pr e-
Chin. Phys. Lett., 27 (2010) 024703.
[16] N.C. Rosca and I. Pop, Unsteady boundary layer ‡ow over a permeable curved stretching/shrinking surface, Eur. J. Mech. B. Fluids, 51 (2015) 61-67. [17] M. Naveed, Z. Abbas and M. Sajid, MHD ‡ow of a microploar ‡uid due to curved stretching surface with thermal radiation, J. Appl. Fluid Mech., 9 (2016) 131-138.
al
[18] Z. Abbas, M. Naveed and M. Sajid, Hydromagnetic slip ‡ow of nano‡uid over a curved stretching surface with heat generation and thermal radiation, J. Mol. Liq., 215 (2016)
urn
756-762.
[19] T. Hayat, M. Rashid, M. Imtiaz and A. Alsaedi, MHD convective ‡ow due to a curved surface with thermal radiation and chemical reaction, J. Mol. Liq., 225 (2017) 482-489. [20] N.F. Okechi, M. Jalil and S. Asghar, Flow of viscous ‡uid along an exponentially stretch-
Jo
ing curved surface, Results Phys., 7 (2017) 2851-2854. [21] T. Hayat, R. Sajjad, R. Ellahi, A. Alsaedi and T. Muhammad, Homogeneousheterogeneous reactions in MHD ‡ow of micropolar ‡uid by a curved stretching surface, J. Mol. Liq., 240 (2017) 209-220. [22] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, Numerical study for nano‡uid ‡ow due to a nonlinear curved stretching surface with convective heat and mass conditions, Results Phys., 7 (2017) 3100-3106. 17
Journal Pre-proof
[23] H. Darcy, Les Fontaines Publiques De La Ville De Dijon, Victor Dalmont, Paris (1856). [24] P. Forchheimer, Wasserbewegung durch boden, Zeitschrift Ver. D. Ing., 45 (1901) 17821788.
of
[25] M. Muskat, The ‡ow of homogeneous ‡uids through porous media, Edwards, MI. (1946). [26] M.A. Seddeek, In‡uence of viscous dissipation and thermophoresis on Darcy-
p ro
Forchheimer mixed convection in a ‡uid saturated porous media, J. Colloid Interface Sci., 293 (2006) 137-142.
[27] B.K. Jha and M.L. Kaurangini, Approximate analytical solutions for the nonlinear Brinkman-Forchheimer-extended Darcy ‡ow model, Appl. Math., 2 (2011) 1432-1436.
Pr e-
[28] D. Pal and H. Mondal, Hydromagnetic convective di¤usion of species in DarcyForchheimer porous medium with non-uniform heat source/sink and variable viscosity, Int. Commun. Heat Mass Transfer, 39 (2012) 913-917.
[29] M.A. Sadiq and T. Hayat, Darcy-Forchheimer ‡ow of magneto Maxwell liquid bounded by convectively heated sheet, Results Phys., 6 (2016) 884-890.
al
[30] S.A. Bakar, N.M. Ari…n, R. Nazar, F.M. Ali and I. Pop, Forced convection boundary layer stagnation-point ‡ow in Darcy-Forchheimer porous medium past a shrinking sheet,
urn
Frontiers Heat Mass Transfer, 7 (2016) 38. [31] T. Muhammad, A. Alsaedi, T. Hayat and S.A. Shehzad, A revised model for DarcyForchheimer three-dimensional ‡ow of nano‡uid subject to convective boundary condition, Results Phys., 7 (2017) 2791-2797.
Jo
[32] J.C. Umavathi, O. Ojjela and K. Vajravelu, Numerical analysis of natural convective ‡ow and heat transfer of nano‡uids in a vertical rectangular duct using Darcy-ForchheimerBrinkman model, Int. J. Thermal Sci., 111 (2017) 511-524. [33] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, Darcy-Forchheimer ‡ow with Cattaneo-Christov heat ‡ux and homogeneous-heterogeneous reactions, Plos One, 12 (2017) e0174938.
18
Journal Pre-proof
[34] T. Hayat, R. S. Saif, R. Ellahi, T. Muhammad and B. Ahmad, Numerical study for Darcy-Forchheimer ‡ow due to a curved stretching surface with Cataneo-Christov heat ‡ux and homogeneous-heterogeneous reactions, Results Phys., 7 (2017) 2886-2892.
of
[35] J.B.J. Fourier, Théorie Analytique De La Chaleur, Paris (1822). [36] C. Cattaneo, Sulla conduzione del calore, Atti Semin Mat Fis Univ Modena Reggio
p ro
Emilia, 3 (1948) 83-101.
[37] C.I. Christov, On frame indi¤erent formulation of the Maxwell-Cattaneo model of …nitespeed heat conduction, Mech Res Commun. 36 (2009) 481-486.
[38] B. Straughan, Thermal convection with the Cattaneo-Christov model, Int J Heat Mass
Pr e-
Transfer, 53 (2010) 95-98.
[39] M. Ciarletta and B. Straughan, Uniqueness and structural stability for the CattaneoChristov equations, Mech Res Commun. 37 (2010) 445-447. [40] S.A.M. Haddad, Thermal instability in Brinkman porous media with Cattaneo-Christov heat ‡ux, Int J Heat Mass Transfer, 68 (2014) 659-658.
al
[41] T. Hayat, M. Farooq, A. Alsaedi and F. Al-Solamy. Impact of Cattaneo-Christov heat ‡ux in the ‡ow over a stretching sheet with variable thickness, AIP Adv. 5 (2015) 087159.
urn
[42] T. Hayat, T. Muhammad, A. Alsaedi and M. Mustafa, A comparative study for ‡ow of viscoelastic ‡uids with Cattaneo-Christov heat ‡ux, Plos One, 11 (2016) e0155185. [43] J. Sui, L. Zheng, X. Zhang, Boundary layer heat and mass transfer with CattaneoChristov double-di¤usion in upper-convected Maxwell nano‡uid past a stretching sheet
Jo
with slip velocity, Int J Therm Sci, 104 (2016) 461-468. [44] T. Hayat, T. Muhammad, A. Alsaedi and B. Ahmad, Three-dimensional ‡ow of nano‡uid with Cattaneo-Christov double di¤usion, Results Phys, 6 (2016) 897-903. [45] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, Darcy-Forchheimer ‡ow due to a curved stretching surface with Cattaneo-Christov double di¤usion: A numerical study, Results Phys, 7 (2017) 2663-2670.
19
Journal Pre-proof
[46] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, An optimal study for DarcyForchheimer ‡ow with generalized Fourier’s and Fick’s laws, Results Phys., 7 (2017) 2878-2885. [47] T. Hayat, T. Ayub, T. Muhammad and A. Alsaedi, Flow of variable thermal conductivity
of
Oldroyd-B ‡uid with generalized Fourier’s and Fick’s laws, J. Mol. Liq., 234 (2017) 9-17. [48] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, On model for ‡ow of Burgers nano‡uid
p ro
with Cattaneo-Christov double di¤usion, Chinese J. Phys., 55 (2017) 916-929. [49] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, Three-dimensional ‡ow of Prandtl ‡uid with Cattaneo-Christov double di¤usion, Results Phys., 9 (2018) 290-296.
Pr e-
[50] T. Hayat, T. Ayub, T. Muhammad and B. Ahmad, Nonlinear computational treatment for couple stress ‡uid ‡ow with Cattaneo-Christov double di¤usion and homogeneous-
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heterogeneous reactions, Int. J. Chem. Reactor Eng., 17 (2019) 20180056.
20
Journal Pre-proof Darcy-Forchheimer flow of viscous fluid is modeled.
Flow is induced by an exponentially stretching curved surface.
Flow in porous medium is described by Darcy-Forchheimer model.
Cattaneo-Christov double diffusion theory is utilized.
Numerical solutions are obtained through NDSolve technique.
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Dear Editor, Hope you are fine and doing well. We have no conflict of interest for this submission.
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Best regards!
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Pr e-
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Taseer Muhammad