- Email: [email protected]
Theoretical and numerical investigation of Carreau–Yasuda fluid flow subject to Soret and Dufour effects
Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects
Journal Pre-proof
Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects M. Ijaz Khan, T. Hayat, Sidra Afzal, M. Imran Khan, A. Alsaedi PII: DOI: Reference:
S0169-2607(19)31731-6 https://doi.org/10.1016/j.cmpb.2019.105145 COMM 105145
To appear in:
Computer Methods and Programs in Biomedicine
Received date: Revised date: Accepted date:
7 October 2019 17 October 2019 18 October 2019
Please cite this article as: M. Ijaz Khan, T. Hayat, Sidra Afzal, M. Imran Khan, A. Alsaedi, Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105145
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.
Highlights • Here flow of Carreau-Yasuda fluid is addressed over a porous surface. • Energy equation is modeled subject to Soret and Dofour effects. • Mixed convection is considered. • Numerical results are calculated via bvp4c.
1
Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects M. Ijaz Khan1 , T. Hayat , Sidra Afzal , M. Imran Khan2 and A. Alsaedi
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80257, Jeddah 21589, Saudi Arabia
Heriot Watt University, Edinburgh Campus, Edinburgh EH14 4AS, United Kingdom
Abstract: Background: Newtonian fluids can be categorized by a single coefficient of viscosity for specific temperature. This viscosity will change with temperature; it doesn’t change with strain rate. Just a small group of liquids show such steady consistency. A fluid whose viscosity changes subject to relative flow velocity is called non-Newtonian liquids. Here we have summarized a result for the flow of Carreau-Yasuda fluid over a porous stretchable surface. Mixed convection is considered. Modeling of energy expression is performed subject to Soret and Dufour effects.
Method: The nonlinear PDE’s are changed to ODE’s through suitable transformations and then solved for numerical solutions via Built-in shooting method (bvp4c).
Results: Variation of important variables is studied on the concentration, temperature and velocity fields. Tabular representation for study of skin friction and heat transfer rate is presented for important variables. Our results show that velocity decreases versus higher estimations of Weissenberg number, porosity parameter, buoyancy ratio and mixed convection parameter. Temperature decays via Weissenberg number and porosity parameter. Increase in concentration is noticed through higher Soret number and porosity parameter. Skin friction and heat transfer rate (Nusselt number) boosts versus larger porosity parameter and Prandtl number respectively while it decays against Weissenberg number and Dufour and Eckert number.
Keywords: Carreau Yasuda Fluid model; Porous medium; Mixed convection; Soret and Dufour effect; Viscous dissipation. 1 2
Corresponding author Email: [email protected] (M. Ijaz Khan) Corresponding author Email: [email protected] (M. Imran Khan)
1
1
Introduction
Non-Newtonian fluids are much important in industrial and physiological fields. A wide range of researches in this regard on non-Newtonian fluids have been carried out [1 − 5]. Non-linear relation between stress and deformation rate exists in such materials. Some general examples of these fluids are blood, toothpaste, shampoo etc. With shear thickening/thinning variables the Carreau-Yasuda model is a model to understand rheological behavior of these materials. Hayat et al. [6] examined Hall and Ohmic effects in asymmetric channel on Carreau-Yasuda model. Flow behavior of Carreau-Yasuda model with Hall effects subject to curved channel is analyzed by Abbasi et al. [7]. Peralta et al. [8] analyzed flow of Carreau-Yasuda fluid with free draining flow on vertical plate. Soret and Dufour effects are important in heat transfer and flow phenomena during double diffusive convective process. Thermal and solutal buoyancies coexist in such process. In most research works Soret and Dufour effects are neglected due to their smaller magnitude order as compared to Fourier’s and Fick’s laws. Alao et al. [9] scrutinized chemically reacting fluid along with thermal radiation under Soret and Dufour effects. Analysis of Soret and Dufour effect in horizontal cavity is done by Wang et al. [10]. Omowaye et al. [11] scrutinized convective flow of fluid with Soret and Dufour effects. Viscoelastic flow over a porous medium with Soret and Dufour effects is highlighted by Hayat et al. [12]. Investigations on porous space have increasing trend because of its increasing uses in numerous applied fields having industrial and geological importance. Porous medium concepts are used extensively in filtration, geophysics, petroleum geology and material sciences. Chaelek et al. [13] pursued work on porous medium gas burner. Yang et al. [14] studied gas permeability measurement in porous medium. Noreena et al. [15] discussed heat transport in electroosmotic flow through Darcy porous medium with peristaltic pumping. Hongsheng et al. [16] deliberated fluid combustion in a porous medium burner. Khan et al. [17] examined second grade liquid flow with magnetohydrodynamic. The basic aims of this research communication is to analyze flow of non-Newtonian material subject to Soret and Dufour effects over a porous space. Effects of dissipation and mixed convection along with Soret Dufour are also taken into account. The nonlinear ODE’s are solved numerically via bvp4c (Shooting method). Some studies on different fluid models are listed in [18 − 25] Results are graphically plotted. 2
2
Modelling
Fig. 1 is illustrated for the physical interpretation of fluid flow (Carreau-Yasuda fluid) over a stretched surface where the surface is porous. Flow is generated with rate Magnetic effects is not considered. Let ∞ and ∞ respectively the wall and ambient temperature and concentration.
Fig. 1. Flow geometry For Carreau-Yasuda fluid we have stress tensor as ´ −1 ³ ˜ τ = ∞ + (0 − ∞ ) 1 + (Γ) A ˙
(1)
h i ˜ = 1 (grad ) + (grad ) A 2
(2)
where
˙ =
r
³ ´2 ˜ 2 A
(3)
˜ , and Γ respectively indicate viscosity at zero shear rate, viscosity where 0 ∞ A at infinite rate, first Rivlin Erickson tensor, shear rate and Carreau Yasuda fluid variables. Assume ∞ = 0, stress tensor reduces to ³ ´ −1 ˜ τ = 0 1 + (Γ) A ˙
(4)
+ = 0
(5)
Governing equations of the problem are addressed as:
3
2
+ = 2 + Γ
¡ −1 ¢
2
( + 1) 2
³
− ∗ + [ ( − ∞ ) + ( − ∞ )]
³ ´2 0 2 + = 2 + ( ) ¡ −1 ¢ ³ ´2 ³ ´ 2 0 + ( ) + 2 Γ
2 + = 2 +
µ
¶
2 2
´ ⎫ ⎬
(6)
⎫ ⎬
(7)
⎭
⎭
(8)
Conditions on boundary are taken as
⎫ = 0 ⎬ → ∞ ⎭
= = 0 = = → 0 → ∞ → ∞
(9)
where denotes kinematic viscosity, ∗ porosity rate, thermal conductivity, density, concentration susceptibility, specific heat, mass diffusivity„ thermal mean temperature and fluid mean temperature. Letting following transformations √ = () = − () () = p − ∞ () = − = ∞ 0
we get dimensionless form of governing equations as 000 + ( )
00
−∞ −∞
⎫ ⎬ ⎭
( − 1) ( + 1) 000 00 ( ) − ∗ 0 + + − 02 + 00 = 0
+ Pr
002
µ ¶ −1 00 1+ ( ) ( ) + Pr 0 + Pr 00 = 0 00 + 00 + 0 = 0
⎫ (0) = = 0 (0) = 1 (∞) → 0 ⎬ (0) = 1 (∞) → 0 (0) = 1 (∞) → 0 ⎭ −0 √
0
0
where the dimensionless variables are defined as
4
(10)
(11)
(12)
(13)
(14)
q 3 = Γ =
=
Re2
=
2 ( −∞ ) 3 −∞ ) ∗ = ∗ = ( Re = = 2 ( −∞ ) 2 ( −∞ ) ( −∞ ) Pr = ( ) = (() = ( = ( −∞ ) −∞ ) −∞ )
⎫ ⎬ ⎭
(15)
where ∗ Re Pr and respectively denote the Weissenberg number, Grashof number, porosity parameter, buoyancy ratio, local Reynold number, mixed convection, Dufour number, Prandtl number, Eckert number, Soret number and Lewis number .
3
Physical quantities
Mathematically
=
, 2 = ( − ∞ ) ( )
(16)
where and represent shear stress and heat flux, given by ∙µ ¶ ¸ ¡ ¢ ³ ´ ³ ´ −1 = ( )=0 = 0 1 + Γ =0 h ³ ´i p 0 = − = − ( − ∞ ) (0) =0
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(17)
The dimensionless form of skin drag force and Nusselt number are expressed as h (Re ) = 00 (0) + 1 2
− 12
− (Re )
4
−1
00 +1
( ) ( )
= 0 (0)
Numerical solution
i ⎫ ⎬ ⎭
(18)
Eqs. (11 − 14) giving velocity, temperature ,concentration equations and the boundary conditions are highly non-linear ODEs. Shooting method is used to simplify these highly non-linear ODEs. For this purpose we let = 1
⎫ 0 ⎪ = 3 = 1 ⎪ ⎪ ⎬ 0 2 = 5 2 = 5 = 2 ⎪ ⎪ ⎪ 0 2 ⎭ = 7 2 = 7 = 3
= 2
= 4
= 6
2 2
= 3
3 3
Eqs. (11 − 13) are converted into following first order ODEs 5
(19)
´¡ ³ ¢ 0 ∗ 2 − 4 − 6 + (2 )2 − 1 3 3 = 1 − ( ) (3 ) (−1)(+1) ³ ´ 0 ( ) + 1 − Pr 3 − Pr 1 5 5 = − Pr (3 )2 ( ) (−1) 3 0
7 = −2 − 1 5
subject to boundary conditions ⎫ ⎪ 1 (0) − 0 = 0 2 (0) − 1 = 0 2 (∞) → 0 ⎪ ⎪ ⎬ 4 (0) − 1 = 0 4 (∞) → 0
6 (0) − 1 = 0 6 (∞) → 0
⎪ ⎪ ⎪ ⎭
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(20)
(21)
These simplified equations are then used in bvp4c MATLAB software to analyze variation of variables on subsequent profiles.
5
Results and discussion
With increase in suction parameter 0 , decline in velocity of the fluid is observed in Fig. 2. With increased suction more fluid particles move towards the wall and the velocity boundary layer declines as a result. Fig. 3 shows decreasing velocity with higher Weissenberg number . Weissenberg number gives comparison of the relaxation time ratio with time of specific process. With enhanced increase in relaxation time causes impedance in fluid flow resulting in decreased velocity. Higher porosity parameter ∗ declines velocity of the fluid due to increased resistance in fluid flow (see Fig. 4). The mixed convection parameter when increased shows enhancement in fluid velocity as highlighted in Fig. 5. This happens because of more effective buoyance force as compared to viscous forces. With increase in buoyancy ratio , velocity becomes higher (see Fig. 6) because concentration expansion effects are dominated over thermal expansion. The power law index increases velocity of the fluid as depicted in Fig.7.
6
Fig. 2. 0 () against 0 .
Fig. 3. 0 () against .
Fig. 4. 0 () against ∗ .
Fig. 5. 0 () against .
Fig. 6. 0 () against .
Fig. 7. 0 () against .
Fig. 8 shows that temperature decreases with increasing Pr. With increase in Pr, decrease in thermal diffusivity is observed which lowers temperature. Increasing Eckert number increases temperature profile (see Fig. 9). Kinetic energy of the system is enhanced with higher values of Eckert number and this increases the temperature. Dufour number when increased, elevates fluid temperature as observed through Fig. 10. In energy expression Dufour number gets involved due to concentration gradient. Hence temperature increases for larger concentration gradient. Fig. 11 shows decreasing temperature with increasing suction parameter 0 .
7
Fig. 8. () against Pr.
Fig. 9. () against .
Fig. 10. () against .
Fig. 11. () against 0 .
Higher Soret number increases fluid concentration in Fig. 12. Temperature gradient enhances with increasing Soret number that gives more convective flow. Fig. 13 shows decreasing concentration with higher Lewis number . Lewis number being directly related to viscosity elevates fluid viscosity which increases concentration of the fluid. Fig. 14 and 15 illustrate increasing concentration with higher values of suction parameter 0 and Prandtl number Pr.
Fig. 12. () against .
Fig. 13. () against .
Fig. 14. () against 0 .
Fig. 15. () against Pr.
8
6
Physical Quantities
Now we have elaborated the behavior of (skin friction) and (Nusselt number) through various parameters. Table 1 depicts that skin friction is higher with increasing ∗ whereas decrease in is noticed for and . Table 2 represents that decreases with and while it enhances with Pr. Table 1: Skin friction values under variation of relevant variables
∗
− Re12
1
0.2
0.9428
0.1
2
0.9465
3
0.9463
1
0.2
0.9387
0.3
0.9342
0.4
0.9293
0.1
0.4
1.0060
0.6
1.0650
0.8
1.1210
Table 2: Nusselt number values under variation of relevant variables Pr
Re12
1.0
0.3
0.5
1.1200
1.1
1.2070
1.2
1.2910
1.0
0.4
1.0130
0.5
0.9076
0.6
0.8047
0.3
7
0.6
1.0210
0.7
0.9220
0.8
0.8219
Conclusion
Major findings of this research are: 9
• Velocity decreases with larger 0 , and ∗ . • Velocity increases with higher , and . • Temperature declines with increase in Pr and 0 while opposite behavior is observed with increase in and . • Increase in concentration is examined for rising , 0 and Pr whereas opposite trend is observed with increasing . • decreases for higher and ∗ while it enhances against . • increases versus larger Pr and declines against and .
References [1] GH. R. Kefayati and H. Tang, Three-dimensional Lattice Boltzmann simulation on thermosolutal convection and entropy generation of Carreau-Yasuda fluids, Int. J. Heat Mass Transf., 131 (2019) 346-364. [2] Z. Alloui and P. Vasseur, Natural convection of Carreau—Yasuda non-Newtonian fluids in a vertical cavity heated from the sides, Int. J. Heat Mass Transf., 84 (2015) 912-924. [3] M. M. Bhatti, T. Abbas, M. M. Rashidi, M. El-Sayed and Z. Yang , Entropy generation on MHD Eyring—Powell nanofluid through a permeable stretching surface, Entropy, 18 (2016) 224. [4] T. Hayat, M.I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, Int. J. Heat Mass Transf., 99 (2016) 702-710. [5] M. I. Khan, M. Waqas, T. Hayat and A. Alsaedi, A comparative study of Casson fluid with homogeneous-heterogeneous reactions, J. Colloid Interface Sci., 498 (2017) 85-90. [6] T. Hayat, F.M. Abbasi, A. Alsaedi and F. Alsaadi, Hall and Ohmic heating effects on the peristaltic transport of Carreau—Yasuda fluid in an asymmetric channel, Z. Naturforschung, 69, 43—51 (2014)
10
[7] F.M. Abbasi, T. Hayat and A. Alsaedi, Numerical analysis for MHD peristaltic transport of Carreau-Yasuda fluid in a curved channel with Hall effects. J Magn Magn Mater, 382, 104—110 (2015) [8] J. M. Peralta, B. E. Meza and S. E. Zorrilla, Analytical solutions for the free-draining flow of a Carreau-Yasuda fluid on a vertical plate, Chemical Engineering Science, 168, 391—402 (2017) [9] F.I. Alao, A.I. Fagbade and B.O. Falodun, Effects of thermal radiation, Soret and Dufour on an unsteady heat and mass transfer flow of a chemically reacting fluid past a semiinfinite vertical plate with viscous dissipation, Journal of the Nigerian Mathematical Society, 35, 142—158 (2016) [10] J.Wang, M. Yang and Y. Zhang, Onset of double-diffusive convection in horizontal cavity with Soret and Dufour effects, International Journal of Heat and Mass Transfer, 78, 1023—1031 (2014) [11] A.J. Omowaye, A.I. Fagbade and A.O. Ajayi, Dufour and soret effects on steady MHD convective flow of a fluid in a porous medium with temperature dependent viscosity: Homotopy analysis approach, Journal of the Nigerian Mathematical Society, 34, 343— 360 (2015) [12] T. Hayat, M. Mustafa and I. Pop, Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid, Commun Nonlinear Sci Numer Simulat, 15, 1183— 1196 (2010) [13] A. Chaelek, U. M. Grareb and S. Jugjai, Self-aspirating/air-preheating porous medium gas burner, Applied Thermal Engineering , 153, 181—189 (2019) [14] D. Yang, W. Wang, W. Chen, X. Tan and L. Wang, Revisiting the methods for gas permeability measurement in tight porous medium, Journal of Rock Mechanics and Geotechnical Engineering, 11, 263-276 (2019) [15] S. Noreena, Quratulain and D. Tripathi, Heat transfer analysis on electroosmotic flow via peristaltic pumping in nonDarcy porous medium, Thermal Science and Engineering Progress, 11, 254—262 (2019) 11
[16] L. Hongsheng, W. Dan, X. Maozhao, W. Songxiang and L. Lin, Experimental study on the pre-evaporation pulse combustion of liquid fuelwithin a porous medium burner, Experimental Thermal and Fluid Science, 103, 286-294 (2019) [17] M. Khan, S. H. Ali, T. Hayat and C. Fetecau, MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium, International Journal of Non-Linear Mechanics, 43, 302 — 319 (2008) [18] K. L. Hsiao, To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method, Energy, 130, 486-499 (2017) [19] K. L. Hsiao, Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects, Applied Thermal Engineering, 112, 1281-1288 (2017) [20] M. Waqas, A mathematical and computational framework for heat transfer analysis of ferromagnetic non-newtonian liquid subjected to heterogeneous and homogeneous reactions, Journal of Magnetism and Magnetic Materials, 493, 165646 (2020) [21] K. L. Hsiao, Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature, International Journal of Heat and Mass Transfer, 112, 983-990 (2017) [22] M. Waqas, Simulation of revised nanofluid model in the stagnation region of cross fluid by expanding-contracting cylinder, International Journal of Numerical Methods for Heat & Fluid Flow, Doi: 10.1108/HFF-12-2018-0797, (2019) [23] K. L. Hsiao, Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet, Applied Thermal Engineering„ 98, 850-861 (2016) [24] M.I. Khan, M. Tamoor, T. Hayat and A. Alsaedi, MHD Boundary layer thermal slip flow by nonlinearly stretching cylinder with suction/blowing and radiation, Results in Physics, 7, 1207—1211 (2017) [25] M. I. Khan, T. Hayat, A. Alsaedi, S. Qayyum and M. Tamoor, Entropy optimization and quartic autocatalysis in MHD chemically reactive stagnation point flow of Sisko nanomaterial, International Journal of Heat and Mass Transfer, 127, 829-837 (2018) 12
Declaration of Competing Interest The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere.
14
Journal Pre-proof
Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects M. Ijaz Khan, T. Hayat, Sidra Afzal, M. Imran Khan, A. Alsaedi PII: DOI: Reference:
S0169-2607(19)31731-6 https://doi.org/10.1016/j.cmpb.2019.105145 COMM 105145
To appear in:
Computer Methods and Programs in Biomedicine
Received date: Revised date: Accepted date:
7 October 2019 17 October 2019 18 October 2019
Please cite this article as: M. Ijaz Khan, T. Hayat, Sidra Afzal, M. Imran Khan, A. Alsaedi, Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105145
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.
Highlights • Here flow of Carreau-Yasuda fluid is addressed over a porous surface. • Energy equation is modeled subject to Soret and Dofour effects. • Mixed convection is considered. • Numerical results are calculated via bvp4c.
1
Theoretical and numerical investigation of Carreau-Yasuda fluid flow subject to Soret and Dufour effects M. Ijaz Khan1 , T. Hayat , Sidra Afzal , M. Imran Khan2 and A. Alsaedi
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80257, Jeddah 21589, Saudi Arabia
Heriot Watt University, Edinburgh Campus, Edinburgh EH14 4AS, United Kingdom
Abstract: Background: Newtonian fluids can be categorized by a single coefficient of viscosity for specific temperature. This viscosity will change with temperature; it doesn’t change with strain rate. Just a small group of liquids show such steady consistency. A fluid whose viscosity changes subject to relative flow velocity is called non-Newtonian liquids. Here we have summarized a result for the flow of Carreau-Yasuda fluid over a porous stretchable surface. Mixed convection is considered. Modeling of energy expression is performed subject to Soret and Dufour effects.
Method: The nonlinear PDE’s are changed to ODE’s through suitable transformations and then solved for numerical solutions via Built-in shooting method (bvp4c).
Results: Variation of important variables is studied on the concentration, temperature and velocity fields. Tabular representation for study of skin friction and heat transfer rate is presented for important variables. Our results show that velocity decreases versus higher estimations of Weissenberg number, porosity parameter, buoyancy ratio and mixed convection parameter. Temperature decays via Weissenberg number and porosity parameter. Increase in concentration is noticed through higher Soret number and porosity parameter. Skin friction and heat transfer rate (Nusselt number) boosts versus larger porosity parameter and Prandtl number respectively while it decays against Weissenberg number and Dufour and Eckert number.
Keywords: Carreau Yasuda Fluid model; Porous medium; Mixed convection; Soret and Dufour effect; Viscous dissipation. 1 2
Corresponding author Email: [email protected] (M. Ijaz Khan) Corresponding author Email: [email protected] (M. Imran Khan)
1
1
Introduction
Non-Newtonian fluids are much important in industrial and physiological fields. A wide range of researches in this regard on non-Newtonian fluids have been carried out [1 − 5]. Non-linear relation between stress and deformation rate exists in such materials. Some general examples of these fluids are blood, toothpaste, shampoo etc. With shear thickening/thinning variables the Carreau-Yasuda model is a model to understand rheological behavior of these materials. Hayat et al. [6] examined Hall and Ohmic effects in asymmetric channel on Carreau-Yasuda model. Flow behavior of Carreau-Yasuda model with Hall effects subject to curved channel is analyzed by Abbasi et al. [7]. Peralta et al. [8] analyzed flow of Carreau-Yasuda fluid with free draining flow on vertical plate. Soret and Dufour effects are important in heat transfer and flow phenomena during double diffusive convective process. Thermal and solutal buoyancies coexist in such process. In most research works Soret and Dufour effects are neglected due to their smaller magnitude order as compared to Fourier’s and Fick’s laws. Alao et al. [9] scrutinized chemically reacting fluid along with thermal radiation under Soret and Dufour effects. Analysis of Soret and Dufour effect in horizontal cavity is done by Wang et al. [10]. Omowaye et al. [11] scrutinized convective flow of fluid with Soret and Dufour effects. Viscoelastic flow over a porous medium with Soret and Dufour effects is highlighted by Hayat et al. [12]. Investigations on porous space have increasing trend because of its increasing uses in numerous applied fields having industrial and geological importance. Porous medium concepts are used extensively in filtration, geophysics, petroleum geology and material sciences. Chaelek et al. [13] pursued work on porous medium gas burner. Yang et al. [14] studied gas permeability measurement in porous medium. Noreena et al. [15] discussed heat transport in electroosmotic flow through Darcy porous medium with peristaltic pumping. Hongsheng et al. [16] deliberated fluid combustion in a porous medium burner. Khan et al. [17] examined second grade liquid flow with magnetohydrodynamic. The basic aims of this research communication is to analyze flow of non-Newtonian material subject to Soret and Dufour effects over a porous space. Effects of dissipation and mixed convection along with Soret Dufour are also taken into account. The nonlinear ODE’s are solved numerically via bvp4c (Shooting method). Some studies on different fluid models are listed in [18 − 25] Results are graphically plotted. 2
2
Modelling
Fig. 1 is illustrated for the physical interpretation of fluid flow (Carreau-Yasuda fluid) over a stretched surface where the surface is porous. Flow is generated with rate Magnetic effects is not considered. Let ∞ and ∞ respectively the wall and ambient temperature and concentration.
Fig. 1. Flow geometry For Carreau-Yasuda fluid we have stress tensor as ´ −1 ³ ˜ τ = ∞ + (0 − ∞ ) 1 + (Γ) A ˙
(1)
h i ˜ = 1 (grad ) + (grad ) A 2
(2)
where
˙ =
r
³ ´2 ˜ 2 A
(3)
˜ , and Γ respectively indicate viscosity at zero shear rate, viscosity where 0 ∞ A at infinite rate, first Rivlin Erickson tensor, shear rate and Carreau Yasuda fluid variables. Assume ∞ = 0, stress tensor reduces to ³ ´ −1 ˜ τ = 0 1 + (Γ) A ˙
(4)
+ = 0
(5)
Governing equations of the problem are addressed as:
3
2
+ = 2 + Γ
¡ −1 ¢
2
( + 1) 2
³
− ∗ + [ ( − ∞ ) + ( − ∞ )]
³ ´2 0 2 + = 2 + ( ) ¡ −1 ¢ ³ ´2 ³ ´ 2 0 + ( ) + 2 Γ
2 + = 2 +
µ
¶
2 2
´ ⎫ ⎬
(6)
⎫ ⎬
(7)
⎭
⎭
(8)
Conditions on boundary are taken as
⎫ = 0 ⎬ → ∞ ⎭
= = 0 = = → 0 → ∞ → ∞
(9)
where denotes kinematic viscosity, ∗ porosity rate, thermal conductivity, density, concentration susceptibility, specific heat, mass diffusivity„ thermal mean temperature and fluid mean temperature. Letting following transformations √ = () = − () () = p − ∞ () = − = ∞ 0
we get dimensionless form of governing equations as 000 + ( )
00
−∞ −∞
⎫ ⎬ ⎭
( − 1) ( + 1) 000 00 ( ) − ∗ 0 + + − 02 + 00 = 0
+ Pr
002
µ ¶ −1 00 1+ ( ) ( ) + Pr 0 + Pr 00 = 0 00 + 00 + 0 = 0
⎫ (0) = = 0 (0) = 1 (∞) → 0 ⎬ (0) = 1 (∞) → 0 (0) = 1 (∞) → 0 ⎭ −0 √
0
0
where the dimensionless variables are defined as
4
(10)
(11)
(12)
(13)
(14)
q 3 = Γ =
=
Re2
=
2 ( −∞ ) 3 −∞ ) ∗ = ∗ = ( Re = = 2 ( −∞ ) 2 ( −∞ ) ( −∞ ) Pr = ( ) = (() = ( = ( −∞ ) −∞ ) −∞ )
⎫ ⎬ ⎭
(15)
where ∗ Re Pr and respectively denote the Weissenberg number, Grashof number, porosity parameter, buoyancy ratio, local Reynold number, mixed convection, Dufour number, Prandtl number, Eckert number, Soret number and Lewis number .
3
Physical quantities
Mathematically
=
, 2 = ( − ∞ ) ( )
(16)
where and represent shear stress and heat flux, given by ∙µ ¶ ¸ ¡ ¢ ³ ´ ³ ´ −1 = ( )=0 = 0 1 + Γ =0 h ³ ´i p 0 = − = − ( − ∞ ) (0) =0
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(17)
The dimensionless form of skin drag force and Nusselt number are expressed as h (Re ) = 00 (0) + 1 2
− 12
− (Re )
4
−1
00 +1
( ) ( )
= 0 (0)
Numerical solution
i ⎫ ⎬ ⎭
(18)
Eqs. (11 − 14) giving velocity, temperature ,concentration equations and the boundary conditions are highly non-linear ODEs. Shooting method is used to simplify these highly non-linear ODEs. For this purpose we let = 1
⎫ 0 ⎪ = 3 = 1 ⎪ ⎪ ⎬ 0 2 = 5 2 = 5 = 2 ⎪ ⎪ ⎪ 0 2 ⎭ = 7 2 = 7 = 3
= 2
= 4
= 6
2 2
= 3
3 3
Eqs. (11 − 13) are converted into following first order ODEs 5
(19)
´¡ ³ ¢ 0 ∗ 2 − 4 − 6 + (2 )2 − 1 3 3 = 1 − ( ) (3 ) (−1)(+1) ³ ´ 0 ( ) + 1 − Pr 3 − Pr 1 5 5 = − Pr (3 )2 ( ) (−1) 3 0
7 = −2 − 1 5
subject to boundary conditions ⎫ ⎪ 1 (0) − 0 = 0 2 (0) − 1 = 0 2 (∞) → 0 ⎪ ⎪ ⎬ 4 (0) − 1 = 0 4 (∞) → 0
6 (0) − 1 = 0 6 (∞) → 0
⎪ ⎪ ⎪ ⎭
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(20)
(21)
These simplified equations are then used in bvp4c MATLAB software to analyze variation of variables on subsequent profiles.
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Results and discussion
With increase in suction parameter 0 , decline in velocity of the fluid is observed in Fig. 2. With increased suction more fluid particles move towards the wall and the velocity boundary layer declines as a result. Fig. 3 shows decreasing velocity with higher Weissenberg number . Weissenberg number gives comparison of the relaxation time ratio with time of specific process. With enhanced increase in relaxation time causes impedance in fluid flow resulting in decreased velocity. Higher porosity parameter ∗ declines velocity of the fluid due to increased resistance in fluid flow (see Fig. 4). The mixed convection parameter when increased shows enhancement in fluid velocity as highlighted in Fig. 5. This happens because of more effective buoyance force as compared to viscous forces. With increase in buoyancy ratio , velocity becomes higher (see Fig. 6) because concentration expansion effects are dominated over thermal expansion. The power law index increases velocity of the fluid as depicted in Fig.7.
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Fig. 2. 0 () against 0 .
Fig. 3. 0 () against .
Fig. 4. 0 () against ∗ .
Fig. 5. 0 () against .
Fig. 6. 0 () against .
Fig. 7. 0 () against .
Fig. 8 shows that temperature decreases with increasing Pr. With increase in Pr, decrease in thermal diffusivity is observed which lowers temperature. Increasing Eckert number increases temperature profile (see Fig. 9). Kinetic energy of the system is enhanced with higher values of Eckert number and this increases the temperature. Dufour number when increased, elevates fluid temperature as observed through Fig. 10. In energy expression Dufour number gets involved due to concentration gradient. Hence temperature increases for larger concentration gradient. Fig. 11 shows decreasing temperature with increasing suction parameter 0 .
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Fig. 8. () against Pr.
Fig. 9. () against .
Fig. 10. () against .
Fig. 11. () against 0 .
Higher Soret number increases fluid concentration in Fig. 12. Temperature gradient enhances with increasing Soret number that gives more convective flow. Fig. 13 shows decreasing concentration with higher Lewis number . Lewis number being directly related to viscosity elevates fluid viscosity which increases concentration of the fluid. Fig. 14 and 15 illustrate increasing concentration with higher values of suction parameter 0 and Prandtl number Pr.
Fig. 12. () against .
Fig. 13. () against .
Fig. 14. () against 0 .
Fig. 15. () against Pr.
8
6
Physical Quantities
Now we have elaborated the behavior of (skin friction) and (Nusselt number) through various parameters. Table 1 depicts that skin friction is higher with increasing ∗ whereas decrease in is noticed for and . Table 2 represents that decreases with and while it enhances with Pr. Table 1: Skin friction values under variation of relevant variables
∗
− Re12
1
0.2
0.9428
0.1
2
0.9465
3
0.9463
1
0.2
0.9387
0.3
0.9342
0.4
0.9293
0.1
0.4
1.0060
0.6
1.0650
0.8
1.1210
Table 2: Nusselt number values under variation of relevant variables Pr
Re12
1.0
0.3
0.5
1.1200
1.1
1.2070
1.2
1.2910
1.0
0.4
1.0130
0.5
0.9076
0.6
0.8047
0.3
7
0.6
1.0210
0.7
0.9220
0.8
0.8219
Conclusion
Major findings of this research are: 9
• Velocity decreases with larger 0 , and ∗ . • Velocity increases with higher , and . • Temperature declines with increase in Pr and 0 while opposite behavior is observed with increase in and . • Increase in concentration is examined for rising , 0 and Pr whereas opposite trend is observed with increasing . • decreases for higher and ∗ while it enhances against . • increases versus larger Pr and declines against and .
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Declaration of Competing Interest The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere.
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