Stability Solution of Unsteady Stagnation-Point Flow and Heat Transfer over a Stretching/Shrinking Sheet in Nanofluid with Slip Velocity Effect
DOI:
https://doi.org/10.37934/cfdl.14.1.6686Keywords:
Nanofluid, Slip velocity, Stability analysis, Stagnation-point flow, Stretching/ shrinking, Unsteady boundary layerAbstract
Computational of unsteady flow with slip condition is essential since physically the heat transfer process is time-dependent and there may exist slip between fluid and surface. Therefore, this study aims to investigate the unsteady stagnation-point flow and heat transfer over a stretching/shrinking sheet immersed in nanofluid in the presence of slip velocity. By applying boundary layer theory and Tiwari-Das model, the governing equations are developed and transformed into a system of ordinary differential equations using similarity transformation, which are then solved numerically using bvp4c solver in MATLAB. The influence of slip velocity, stretching/shrinking parameters, nanoparticle volume fraction and unsteadiness parameter on the local skin friction coefficient, local Nusselt number, as well as velocity and temperature profiles are analysed. There are three types of nanoparticles considered, namely Copper (Cu), Alumina (Al203), Titania (TiO2) and water (H20) is the base fluid. It is found that dual solutions occur for certain parameters and the stability analysis is performed. The analysis shows that the first solutions are found to be stable than the second solution. The local skin friction coefficient and local Nusselt number are increasing with slip velocity, nanoparticle volume fraction for shrinking case; however, the opposite trend is observed for stretching case. By raising 20% of nanoparticle volume fraction for the shrinking sheet ( the friction at the surface increases 24% and 15.5% for heat transfer rate for the first solution. Moreover, for 10% of nanoparticle volume fraction for the shrinking sheet (and the first solution, varying slip parameters from 0 to 0.2, give rise to approximately 21% of the friction at the surface and 68% of the heat transfer rate.
References
Schlichting, H. Boundary Layer Theory. 7th edition. McGraw-Hill, New York, 1979.
Thipse, S. S. Advanced Thermodynamics. Alpha Science International Limited, 2013.
Yang, Junlin, Rolf Bos, Gerald F. Belder, Jan Engel, and Henk J. Busscher. "Deposition of oral bacteria and polystyrene particles to quartz and dental enamel in a parallel plate and stagnation point flow chamber." Journal of Colloid and Interface Science 220, no. 2 (1999): 410-418. https://doi.org/10.1006/jcis.1999.6539
Montecchio, Francesco, Henry Persson, Klas Engvall, Jack Delin, and Roberto Lanza. "Development of a stagnation point flow system to screen and test TiO2-based photocatalysts in air purification applications." Chemical Engineering Journal 306 (2016): 734-744. https://doi.org/10.1016/j.cej.2016.07.117
Bachok, Norfifah, Anuar Ishak, and Ioan Pop. "The boundary layers of an unsteady stagnation-point flow in a nanofluid." International Journal of Heat and Mass Transfer 55, no. 23-24 (2012): 6499-6505. https://doi.org/10.1016/j.ijheatmasstransfer.2012.06.050
Bhattacharyya, Krishnendu. "Heat transfer analysis in unsteady boundary layer stagnation-point flow towards a shrinking/stretching sheet." Ain Shams Engineering Journal 4, no. 2 (2013): 259-264. https://doi.org/10.1016/j.asej.2012.07.002
Aman, Fazlina, Anuar Ishak, and Ioan Pop. "Magnetohydrodynamic stagnation-point flow towards a stretching/shrinking sheet with slip effects." International Communications in Heat and Mass Transfer 47 (2013): 68-72. https://doi.org/10.1016/j.icheatmasstransfer.2013.06.005
Sharma, Rajesh, Anuar Ishak, and Ioan Pop. "Stability analysis of magnetohydrodynamic stagnation-point flow toward a stretching/shrinking sheet." Computers & Fluids 102 (2014): 94-98. https://doi.org/10.1016/j.compfluid.2014.06.022
Ishak, Anuar, Roslinda Nazar, Norihan M. Arifin, and Ioan Pop. "Dual solutions in mixed convection flow near a stagnation point on a vertical porous plate." International Journal of Thermal Sciences 47, no. 4 (2008): 417-422. https://doi.org/10.1016/j.ijthermalsci.2007.03.005
Hayat, T., and M. Nawaz. "Unsteady stagnation point flow of viscous fluid caused by an impulsively rotating disk." Journal of the Taiwan Institute of Chemical Engineers 42, no. 1 (2011): 41-49. https://doi.org/10.1016/j.jtice.2010.04.006
Lok, Y. Y., J. H. Merkin, and I. Pop. "Mixed convection flow near the axisymmetric stagnation point on a stretching or shrinking cylinder." International Journal of Thermal Sciences 59 (2012): 186-194. https://doi.org/10.1016/j.ijthermalsci.2012.04.008
Roşca, Alin V., Natalia C. Roşca, and Ioan Pop. "Axisymmetric stagnation point flow and heat transfer towards a permeable moving flat plate with surface slip condition." Applied Mathematics and Computation 233 (2014): 139-151. https://doi.org/10.1016/j.amc.2014.01.130
Mahabaleshwar, U. S., K. R. Nagaraju, M. N. Nadagoud, R. Bennacer, and Dumitru Baleanu. "An MHD viscous liquid stagnation point flow and heat transfer with thermal radiation and transpiration." Thermal Science and Engineering Progress 16 (2020): 100379. https://doi.org/10.1016/j.tsep.2019.100379
Ghasemi, S. E., and Mohammad Hatami. "Solar radiation effects on MHD stagnation point flow and heat transfer of a nanofluid over a stretching sheet." Case Studies in Thermal Engineering 25 (2021): 100898. https://doi.org/10.1016/j.csite.2021.100898
Mathew, Alphonsa, Sujesh Areekara, A. S. Sabu, and S. Saleem. "Significance of multiple slip and nanoparticle shape on stagnation point flow of silver-blood nanofluid in the presence of induced magnetic field." Surfaces and Interfaces (2021): 101267. https://doi.org/10.1016/j.surfin.2021.101267
Aly, Emad H., and I. Pop. "MHD flow and heat transfer near stagnation point over a stretching/shrinking surface with partial slip and viscous dissipation: Hybrid nanofluid versus nanofluid." Powder Technology 367 (2020): 192-205. https://doi.org/10.1016/j.powtec.2020.03.030
Jamaludin, Anuar, Kohilavani Naganthran, Roslinda Nazar, and Ioan Pop. "MHD mixed convection stagnation-point flow of Cu-Al2O3/water hybrid nanofluid over a permeable stretching/shrinking surface with heat source/sink." European Journal of Mechanics-B/Fluids 84 (2020): 71-80. https://doi.org/10.1016/j.euromechflu.2020.05.017
Zainal, Nurul Amira, Roslinda Nazar, Kohilavani Naganthran, and Ioan Pop. "MHD mixed convection stagnation point flow of a hybrid nanofluid past a vertical flat plate with convective boundary condition." Chinese Journal of Physics 66 (2020): 630-644. https://doi.org/10.1016/j.cjph.2020.03.022
Uddin, Ziya, Korimerla Sai Vishwak, and Souad Harmand. "Numerical duality of MHD stagnation point flow and heat transfer of nanofluid past a shrinking/stretching sheet: Metaheuristic approach." Chinese Journal of Physics 73 (2021): 442-461. https://doi.org/10.1016/j.cjph.2021.07.018
Bang, In Cheol, and Gyunyoung Heo. "An axiomatic design approach in development of nanofluid coolants." Applied Thermal Engineering 29, no. 1 (2009): 75-90. https://doi.org/10.1016/j.applthermaleng.2008.02.004
Che Sidik, Nor Azwadi, N. G. Yen Cheong, and Alireza Fazeli. "Computational Analysis of Nanofluids in Vehicle Radiator." In Applied Mechanics and Materials, vol. 695, pp. 539-543. Trans Tech Publications Ltd, 2015. https://doi.org/10.4028/www.scientific.net/AMM.695.539
Buongiorno, Jacopo. "Convective transport in nanofluids." Journal of Heat Transfer 128, no. 3 (2006): 240-250. https://doi.org/10.1115/1.2150834
Tiwari, Raj Kamal, and Manab Kumar Das. "Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids." International Journal of Heat and Mass Transfer 50, no. 9-10 (2007): 2002-2018. https://doi.org/10.1016/j.ijheatmasstransfer.2006.09.034
Devi, SP Anjali, and Julie Andrews. "Laminar boundary layer flow of nanofluid over a flat plate." International Journal of Applied Mathematics and Mechanics 7, no. 6 (2011): 52-71.
Srinivasacharya, D., Upendar Mendu, and K. Venumadhav. "MHD boundary layer flow of a nanofluid past a wedge." Procedia Engineering 127 (2015): 1064-1070. https://doi.org/10.1016/j.proeng.2015.11.463
Malvandi, A., F. Hedayati, and D. D. Ganji. "Nanofluid flow on the stagnation point of a permeable non-linearly stretching/shrinking sheet." Alexandria Engineering Journal 57, no. 4 (2018): 2199-2208. https://doi.org/10.1016/j.aej.2017.08.010
Jafar, Ahmad Banji, Sharidan Shafie, and Imran Ullah. "MHD radiative nanofluid flow induced by a nonlinear stretching sheet in a porous medium." Heliyon 6, no. 6 (2020): e04201. https://doi.org/10.1016/j.heliyon.2020.e04201
Ferdows, M., S. O. Adesanya, Faris Alzahrani, and T. A. Yusuf. "Numerical investigation of a boundary layer water-based nanofluid flow with induced magnetic field." Physica A: Statistical Mechanics and its Applications 570 (2021): 125492. https://doi.org/10.1016/j.physa.2020.125492
Johan, Nurul Aisyah, and Syahira Mansur. "Boundary Layer Flow of Dusty Nanofluid over Stretching Sheet with Partial Slip Effects." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 87, no. 2 (2021): 118-126. https://doi.org/10.37934/arfmts.87.2.118126
Nasir, Faiza Mohamed, Mohd Zulkifly Abdullah, and Mior Firdaus Mior Abdul Majid. "Empirical Viscosity Modeling for SiO2 and Al2O3 Nanofluids using the Response Surface Method." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 75, no. 3 (2020): 63-72. https://doi.org/10.37934/arfmts.75.3.6372
Sannad, Mohamed, Btissam Abourida, and El Houcine Belarche. "Numerical Study of the Effect of the Nanofluids Type and The Size of the Heating Sections on Heat Transfer for Cooling Electronic Components." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 75, no. 2 (2020): 168-184. https://doi.org/10.37934/arfmts.75.2.168184
Jahan, Sultana, M. Ferdows, M. D. Shamshuddin, and Khairy Zaimi. "Effects of Solar Radiation and Viscous Dissipation on Mixed Convective Non-Isothermal Hybrid Nanofluid over Moving Thin Needle." Journal of Advanced Research in Micro and Nano Engineering 3, no. 1 (2021): 1-11.
Uddin, Ziya, Rishi Asthana, Mukesh Kumar Awasthi, and Hamdy Hassan. "A metaheuristic approach for the comparative study of MHD flow of nano liquids in a semi-porous channel." International Journal for Computational Methods in Engineering Science and Mechanics (2021): 1-8. https://doi.org/10.1080/15502287.2021.1916700
Uddin, Ziya, Souad Harmand, and Shahid Ahmed. "Computational modeling of heat transfer in rotating heat pipes using nanofluids: A numerical study using PSO." International Journal of Thermal Sciences 100, no. 112 (2017): 44-54. https://doi.org/10.1016/j.ijthermalsci.2016.09.035
Uddin, Z., R. Asthana, M. Kumar Awasthi, and S. Gupta. "Steady MHD flow of nano-fluids over a rotating porous disk in the presence of heat generation/absorption: a numerical study using PSO." Journal of Applied Fluid Mechanics (JAFM) 10, no. 3 (2017): 871-879. https://doi.org/10.18869/acadpub.jafm.73.240.26650
Uddin, Ziya, Manoj Kumar, and Souad Harmand. "Influence of thermal radiation and heat generation/absorption on MHD heat transfer flow of a micropolar fluid past a wedge considering hall and ion slip currents." Thermal Science 18, no. suppl. 2 (2014): 489-502. https://doi.org/10.2298/TSCI110712085U
Uddin, Ziya, and Souad Harmand. "Natural convection heat transfer of nanofluids along a vertical plate embedded in porous medium." Nanoscale Research Letters 8, no. 1 (2013): 1-19. https://doi.org/10.1186/1556-276X-8-64
Wang, C. Y. "Flow due to a stretching boundary with partial slip-an exact solution of the Navier-Stokes equations." Chemical Engineering Science 57, no. 17 (2002): 3745-3747. https://doi.org/10.1016/S0009-2509(02)00267-1
Jamil, Muhammad, and Najeeb Alam Khan. "Slip effects on fractional viscoelastic fluids." International Journal of Differential Equations 2011 (2011). https://doi.org/10.1155/2011/193813
Mukhopadhyay, Swati. "Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation." Ain Shams Engineering Journal 4, no. 3 (2013): 485-491. https://doi.org/10.1016/j.asej.2012.10.007
Pandey, Alok Kumar, and Manoj Kumar. "Boundary layer flow and heat transfer analysis on Cu-water nanofluid flow over a stretching cylinder with slip." Alexandria Engineering Journal 56, no. 4 (2017): 671-677. https://doi.org/10.1016/j.aej.2017.01.017
Hussain, Azad, Sana Afzal, Rizwana Rizwana, and M. Y. Malik. "MHD stagnation point flow of a Casson fluid with variable viscosity flowing past an extending/shrinking sheet with slip effects." Physica A: Statistical Mechanics and Its Applications 553 (2020): 124080. https://doi.org/10.1016/j.physa.2019.124080
Aladdin, Nur Adilah Liyana, Norfifah Bachok, and I. Pop. "Boundary layer flow and heat transfer of Cu-Al2O3/water over a moving horizontal slender needle in presence of hydromagnetic and slip effects." International Communications in Heat and Mass Transfer 123 (2021): 105213. https://doi.org/10.1016/j.icheatmasstransfer.2021.105213
Merkin, J. H. "On dual solutions occurring in mixed convection in a porous medium." Journal of Engineering Mathematics 20, no. 2 (1986): 171-179. https://doi.org/10.1007/BF00042775
Weidman, P. D., D. G. Kubitschek, and A. M. J. Davis. "The effect of transpiration on self-similar boundary layer flow over moving surfaces." International Journal of Engineering Science 44, no. 11-12 (2006): 730-737. https://doi.org/10.1016/j.ijengsci.2006.04.005
Harris, S. D., D. B. Ingham, and I. Pop. "Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip." Transport in Porous Media 77, no. 2 (2009): 267-285. https://doi.org/10.1007/s11242-008-9309-6
Mahapatra, Tapas Ray, Samir Kumar Nandy, Kuppalapalle Vajravelu, and Robert A. Van Gorder. "Stability analysis of the dual solutions for stagnation-point flow over a non-linearly stretching surface." Meccanica 47, no. 7 (2012): 1623-1632. https://doi.org/10.1007/s11012-012-9541-6
Roşca, Natalia C., and Ioan Pop. "Mixed convection stagnation point flow past a vertical flat plate with a second order slip: heat flux case." International Journal of Heat and Mass Transfer 65 (2013): 102-109. https://doi.org/10.1016/j.ijheatmasstransfer.2013.05.061
Ishak, Anuar. "Dual Solutions in Mixed Convection Boundary Layer Flow: A Stability Analysis." International Journal of Mathematical and Computational Sciences 8, no. 9 (2014): 1216-1219.
Bhattacharyya, Krishnendu, Swati Mukhopadhyay, and G. C. Layek. "Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet." International Journal of Heat and Mass Transfer 54, no. 1-3 (2011): 308-313. https://doi.org/10.1016/j.ijheatmasstransfer.2010.09.041
Oztop, Hakan F., and Eiyad Abu-Nada. "Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids." International Journal of Heat and Fluid Flow 29, no. 5 (2008): 1326-1336. https://doi.org/10.1016/j.ijheatfluidflow.2008.04.009
Shampine, Lawrence F., Ian Gladwell, and S. Thompson. Solving ODEs with MATLAB. Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511615542
Aghamajidi, Mohammad, MohammadEftekhari Yazdi, Saeed Dinarvand, and Ioan Pop. "Tiwari-Das nanofluid model for magnetohydrodynamics (MHD) natural-convective flow of a nanofluid adjacent to a spinning down-pointing vertical cone." Propulsion and Power Research 7, no. 1 (2018): 78-90. https://doi.org/10.1016/j.jppr.2018.02.002
Mousavi, Seyed Mahdi, Mohammadreza Nademi Rostami, Mohammad Yousefi, Saeed Dinarvand, Ioan Pop, and Mikhail A. Sheremet. "Dual solutions for Casson hybrid nanofluid flow due to a stretching/shrinking sheet: A new combination of theoretical and experimental models." Chinese Journal of Physics 71 (2021): 574-588. https://doi.org/10.1016/j.cjph.2021.04.004
Mustafa, Irfan, Zaheer Abbas, Ayesha Arif, Tariq Javed, and Abuzar Ghaffari. "Stability analysis for multiple solutions of boundary layer flow towards a shrinking sheet: analytical solution by using least square method." Physica A: Statistical Mechanics and its Applications 540 (2020): 123028. https://doi.org/10.1016/j.physa.2019.123028
