Abstract
Two-dimensional forced convective steady boundary layer flow of non-Newtonian Eyring–Powell nanofluid over a moving plate in a porous medium in the presence of a parallel free-stream is investigated. The governing coupled non-linear partial differential equations (PDEs) along with boundary conditions are transformed into a set of non-linear coupled ordinary differential equations (ODEs) by using appropriate transformations. The obtained non-linear ODEs with modified boundary conditions are converted into a system of first-order ODEs which are solved using the classical and efficient shooting method. Dual solutions for velocity, temperature and nanoparticle concentration distributions for Eying–Powell fluids similar to Newtonian fluid in some special flow situations are obtained, when the plate and free-stream are moving along mutually opposite directions. The stability analysis of the obtained solutions is performed and it is found that the upper branch solutions are physically stable, while lower branch solutions are unstable. The impacts of different dimensionless physical parameters on velocity, temperature and nanoparticle concentration are reported in the form of graphs and tables. An important result is obtained and it reveals that the ‘dual solutions’ character has been destroyed if resistance due to the porous medium is raised up to a definite level (i.e., permeability parameter \(K > 0.07979\)), though the range of existence of unique solution becomes larger with further increase of resistance due to porous medium. It is also observed that heat transfer rate diminishes with increasing thermophoresis parameter, Brownian diffusion parameter and Lewis number in all the cases, whereas mass transfer rate enhances with thermophoresis parameter (for dual solutions), Brownian diffusion parameter (for unique solutions) and Lewis number (for unique solutions). Further, skin-friction coefficient, i.e., the surface drag force, increases with permeability parameter, suction/injection parameter and decreases with Eyring–Powell fluid parameter. Also, increments in permeability parameter and the suction/injection parameter lead to the delay in the boundary layer separation. The critical values of velocity ratio parameter beyond which the boundary layer separation appears are − 0.5476432, − 0.5987132, − 0.704862, − 0.816944, − 0.9365732, − 0.96179102, − 1.057104, − 1.062004, − 1.09222, − 1.115824, − 1.193413, − 1.591023 and − 1.898366 for \(K = 0\), 0.01, 0.03, 0.05, 0.07, 0.074, 0.08, 0.082, 0.085, 0.09, 0.1, 0.15 and 0.2, respectively.
Similar content being viewed by others
References
S U S Choi, ASME 66, 99 (1995)
S U S Choi, Z G Zhang, W Yu, Lockwood and E A Grulke, Appl. Phys. Lett. 79, 2252 (2001)
K V Wong and O D Leon, Adv. Mech. Eng. 2010, 519659 (2010)
Y M Xuan and Q Li, Int. J. Heat Fluid Flow 21, 58 (2000)
S K Das, S U S Choi, W Yu and T Pradet, Nanofluids: Science and technology (Wiley, New Jersey, 2007)
W Daungthongsuk and S Wongwises, Renew. Sust. Energ. Rev. 11, 797 (2007)
V Trisaksri and S Wongwises, Renew. Sust. Energ. Rev. 11, 512 (2007)
X-Q Wang and A S Mujumdar, Int. J. Therm. Sci. 46, 1 (2007)
X-Q Wang and A S Mujumdar, Braz. J. Chem. Eng. 25, 613 (2008)
X-Q Wang and A S Mujumdar, Braz. J. Chem. Eng. 25, 631 (2008)
S Kakac and A Pramuanjaroenkij, Int. J. Heat Mass Transf. 52, 3187 (2009)
J Buongiorno, ASME J. Heat Transf. 128, 240 (2006)
D A Nield and A V Kuznetsov, Int. J. Heat Mass Transf. 52, 5792 (2009)
D A Nield and A V Kuznetsov, Int. J. Heat Mass Transf. 54, 374 (2011)
A V Kuznetsov and D A Nield, Int. J. Therm. Sci. 49, 243 (2010)
A V Kuznetsov and D A Nield, Int. J. Therm. Sci. 50, 712 (2011)
W A Khan and I Pop, Int. J. Heat Mass Transf. 53, 2477 (2010)
N Bachok, A Ishak and I Pop, Int. J. Therm. Sci. 49, 1663 (2010)
M I Khan, S Qayyum, S Farooq, T Hayat and A Alsaedi, Pramana – J. Phys. 93: 62 (2019)
S Ghosh and S Mukhopadhyay, Pramana – J. Phys. 94: 61 (2020)
R E Powell and H Eyring, Nature 154, 427 (1944)
M Jalil, S Asghar and S M Imran, Int. J. Heat Mass Transf. 65, 73 (2013)
A Mushtaq, M Mustafa, T Hayat, M Rahi and A Alsaedi, Z. Naturforsch. A 68a, 791 (2013)
M Poonia and R Bhargava, J. Thermophys. Heat Transf. 28, 499 (2014)
T Hayat, I Ullah, T Muhammad, A Alsaedi and S A Shehzad, Chin. Phys. B 25, 074701 (2016)
T Hayat, R Sajjad, T Muhammad, A Alsaedi and R Ellahi, Results Phys. 7, 535 (2017)
W Ibrahim and B Hindebu, Nonlinear Eng. 8, 303 (2019)
M Y Malik, I Khan, A Hussain and T Salahuddin, AIP Adv. 5, 117118 (2015)
G Sowmya, B J Gireesha, S Sindhu and B C Prasannakumara, Commun. Theor. Phys. 72(2), 025004 (2020)
B J Gireesha, M Umeshaiah, B C Prasannakumara, N S Shashikumar and M Archana, Physica A 549, 124051 (2020)
R J Punith Gowda, R N Kumar, A Aldalbahi, A Issakhov, B C Prasannakumara, M Rahimi-Gorji and M Rahaman, Surf. Interfaces 22, 100364 (2021)
M G Reddy, P Vijayakumari, L Krishna, K G Kumar and B C Prasannakumara, Multidiscip. Model. Mater. Struct. 16(6), 1669 (2020)
A Roja, B J Gireesha and B C Prasannakumara, Multidiscip. Model. Mater. Struct. 16(6), 1475 (2020)
H Blasius, Zeits. f. Math. u Phys. 56, 1 (1908)
E Pohlhausen, ZAMM 1, 121 (1921)
L Howarth, Proc. R. Soc. Lond. A 164, 547 (1938)
A M M Abu-Sitta, Appl. Math. Comput. 64, 73 (1994)
L Wang, Appl. Math. Comput. 157, 1 (2004)
R Cortell, Chin. Phys. Lett. 25, 1340 (2008)
B C Sakiadis, AIChE J. 7, 26 (1961)
T A Abdelhafez, Int. J. Heat Mass Transf. 28, 1234 (1985)
N Afzal, A Badaruddin and A A Elgarvi, Int. J. Heat Mass Transf. 36, 3399 (1993)
R C Bataller, Appl. Math. Comput. 198, 333 (2008)
M Y Hussaini, W D Lakin and A Nachman, SIAM J. Appl. Math. 47, 699 (1987)
H T Lin, K Y Wu and H L Hoh, Int. J. Heat Mass Transf. 36, 3547 (1993)
A Ishak, R Nazar and I Pop, Chin. Phys. Lett. 24, 2274 (2007)
A Ishak, R Nazar and I Pop, Int. J. Heat Mass Transf. 50, 4743 (2007)
A Ishak, R Nazar and I Pop, Can. J. Phys. 85, 869 (2007)
A Ishak, R Nazar and I Pop, Heat Mass Transf. 45, 563 (2009)
A Ishak, R Nazar and I Pop, Chem. Eng. J. 148, 63 (2009)
P D Weidman, D G Kubitschek and A M J Davis, Int. J. Eng. Sci. 44, 730 (2006)
A Ishak, Chin. Phys. Lett. 26, 034701 (2009)
S Mukhopadhyay, K Bhattacharyya and G C Layek, Int. J. Heat Mass Transf. 54, 2751 (2011)
C Y Wang, Int. J. Nonlinear Mech. 43, 377 (2008)
A Ishak, R Nazar and I Pop, Int. J. Heat Mass Transf. 51, 1150 (2008)
A Ishak, Y Y Lok and I Pop, Chem. Eng. Commun. 197, 1417 (2010)
K Bhattacharyya, Int. J. Heat Mass Transf. 55, 3482 (2012)
H Rosali, A Ishak and I Pop, Int. Commun. Heat Mass Transf. 38, 1029 (2011).
K Bhattacharyya and K Vajravelu, Commun. Nonlinear Sci. Numer. Simulat. 17, 2728 (2012)
N Bachok, A Ishak and I Pop, Int. J. Heat Mass Transf. 55, 2102 (2012)
K Vajravelu, G Sarojamma, K Sreelakshmi and C Kalyani, Int. J. Mech. Sci. 130, 119 (2017)
K Naganthran, R Nazar and I Pop, Int. J. Mech. Sci. 131–132, 663 (2017)
G S Seth, A K Singha, M S Mandal, A Banerjee and K Bhattacharyya, Int. J. Mech. Sci. 134, 98 (2017)
S A Bakar, N MArifin, R Nazar, F M Ali, N Bachok and I Pop, J. Porous Media 21(7), 623 (2018)
I Mustafa, T Javed, A Ghaffari and H Khalil, Pramana – J. Phys. 93: 53 (2019)
T Hayat, Z Iqbal, M Qasim and S Obaidat, Int. J. Heat Mass Transf. 55, 1817 (2012)
P Kundu, V Kumar and I M Mishra, Powder Technol. 303, 278 (2016)
J H Merkin, J. Eng. Math. 20, 171 (1985)
N A L Aladdin, N Bachok and I Pop, Alex. Eng. J. 59, 657 (2020)
Acknowledgements
The research of A K Verma is supported by the Council of Scientific and Industrial Research, New Delhi, Ministry of Human Resources Development of India Grant (09/013(0724)/2017-EMR-I) and the work of A K Gautam is funded by the University Grants Commission, New Delhi, Ministry of Human Resources Development, Government of India Grant (1220/(CSIR-UGC NET DEC. 2016)). The authors are also thankful to the anonymous reviewers for their constructive comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Verma, A.K., Gautam, A.K., Bhattacharyya, K. et al. Boundary layer flow of non-Newtonian Eyring–Powell nanofluid over a moving flat plate in Darcy porous medium with a parallel free-stream: Multiple solutions and stability analysis. Pramana - J Phys 95, 173 (2021). https://doi.org/10.1007/s12043-021-02215-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-021-02215-9
Keywords
- Eyring–Powell nanofluids
- multiple solutions
- moving flat plate
- parallel free-stream
- porous medium
- stability analysis