Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet
Abstract
1. Introduction
2. Problem Development
3. Exact Solution
3.1. Initial Steady Flow
3.2. Final Unsteady State Flow
4. Implicit Finite Difference Method (IFDM)
5. Numerical Results and Discussions
6. Concluding Remarks
- A decline in skin friction and the heat transfer rate was noted with a rise in values of the relaxation time parameter.
- It was depicted that skin friction decreased with rising values of mixed convection parameter and , while the heat transmission rate showed the totally opposite behavior.
- The rise in radiation parameter reduced the local skin friction coefficient and led to enhance the heat transfer rate at the surface.
- It was noticed that higher values of the Prandtl number reduced the temperature and thermal boundary layer thickness rapidly.
- A rise in velocity and decline in heat transfer rate were witnessed with lower values.
- The magnetic field parameter turned down the local skin friction.
- The thermal diffusion rate could be controlled by varying the Prandtl number.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Prandtl number | Magnetic field | ||
Radiative heat flux | t | Time | |
Skin friction coefficient | Radiation parameter | ||
Local Nusselt number | T | Temperature | |
Rex | Local Reynolds number | Re | Reynold number |
Specific heat capacity | M | Magnetic field parameter | |
Local Grashof number | Kinematic viscosity |
Greek Symbols
Relaxation time parameter | Stefan–Boltzmann constant | ||
Fluid density | Stream function | ||
Thermal diffusivity | Wall shear stress | ||
Mean absorption coefficient | Mixed convection parameter |
References
- Ishak, A.; Nazar, R.; Pop, I. Hydro magnetic Flow and Heat Transfer Adjacent to a Stretching Vertical Sheet. Heat Mass Transf. 2008, 44, 921–927. [Google Scholar] [CrossRef]
- Sakiadis, B.C. Boundary layer behavior on continuous solid surfaces. II: The boundary layer on a continuous flat surface. AICHE J. 1961, 7, 221–225. [Google Scholar] [CrossRef]
- Crane, L.J. Flow Past a Stretching Plate. ZAMP 1970, 21, 645–647. [Google Scholar] [CrossRef]
- Erickson, L.E.; Fan, L.T.; Fox, V.G. Heat and mass transfer on a moving continuous flat plate with suction or injection. Ind. Eng. Chem. Res. Fund. 1966, 5, 19–25. [Google Scholar] [CrossRef]
- Gupta, P.S.; Gupta, A.S. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 1977, 55, 744–746. [Google Scholar] [CrossRef]
- Grubka, J.; Bobba, K.M. Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J. Heat Transf. 1985, 107, 248–250. [Google Scholar] [CrossRef]
- Chen, C.K.; Char, M.I. Heat transfer of a continuous stretching surface with suction and injection. J. Math. Anal. Appl. 1988, 135, 568–580. [Google Scholar] [CrossRef]
- Ali, M.E. Heat transfer characteristics of a continuous stretching surface. Heat Mass transf. 1994, 29, 227–234. [Google Scholar] [CrossRef]
- Chen, C.H. Laminar mixed convection adjacent to vertical continuous stretching sheets. Heat Mass transf. 1998, 33, 471–476. [Google Scholar] [CrossRef]
- Wang, C.Y. Analysis of viscous flow due to stretching sheet with surface slip and suction. Nonlinear Anal. RWA 2009, 10, 375–380. [Google Scholar] [CrossRef]
- Andersson, H.I.; Bech, K.H.; Dandapat, B.S. Magneto hydrodynamic Flow of a Power Law Fluid over a Stretching Sheet. Int. J. Non Linear Mech. 1992, 27, 929–936. [Google Scholar] [CrossRef]
- Hassanien, I.A. Flow and Heat Transfer on a Continuous Flat Surface Moving in a Parallel Free Stream of Power-Law Fluid. Appl. Model. 1996, 20, 779–784. [Google Scholar] [CrossRef]
- Sadeghy, K.; Sharifi, M. Local Similarity Solution for the Flow of a ‘Second-Grade’ Viscoelastic Fluid above a Moving Plate. Int. J. Non Linear Mech. 2004, 39, 1265–1273. [Google Scholar] [CrossRef]
- Serdar, B.; SalihDokuz, M. Three-Dimensional Stagnation Point Flow of a Second Grade Fluid Towards a Moving Plate. Int. J. Eng. Sci. 2006, 44, 49–58. [Google Scholar]
- Haroun, M.H. Effect of Deborah Number and Phase Difference on Peristaltic Transport of a Third-Order Fluid in an Asymmetric Channel. Commun. Nonlinear Sci. Numer. Simul. 2007, 12, 1464–1480. [Google Scholar] [CrossRef]
- Siddiqui, A.M.; Zeb, A.; Ghori, Q.K.; Benharbit, A.M. Homotopy Perturbation Method for Heat Transfer Flow of a Third Grade Fluid between Parallel Plates. Chaos Solitons Fractals 2008, 36, 182–192. [Google Scholar] [CrossRef]
- Sajid, M.; Ahmad, I.; Hayat, T.; Ayub, M. Unsteady Flow and Heat Transfer of a Second Grade Fluid over a Stretching Sheet. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 96–108. [Google Scholar] [CrossRef]
- Heyhat, M.M.; Khabazi, N. Non-Isothermal Flow of Maxwell Fluids above Fixed Flat Plates under the Influence of a Transverse Magnetic Field. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 2010, 225, 909–916. [Google Scholar] [CrossRef]
- Kayvan, S.; Hajibeygi, H.; Taghavi, S.-M. Stagnation-point flow of upper-convected Maxwell fluids. Int. J. Non Linear Mech. 2006, 41, 1242–1247. [Google Scholar]
- Aliakbar, V.; Alizadeh-Pahlavan, A.; Sadeghy, K. The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 779–794. [Google Scholar] [CrossRef]
- Alizadeh-Pahlavan, A.; Sadeghy, K. On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 1355–1365. [Google Scholar] [CrossRef]
- Hayat, T.; Abbas, Z.; Sajid, M. Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys. Lett. A 2006, 358, 396–403. [Google Scholar] [CrossRef]
- Abel, M.S.; Tawade, J.V.; Nandeppanavar, M.M. MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet. Meccanica 2012, 47, 385–393. [Google Scholar]
- Cess, R.D. The interaction of thermal radiation with free convection heat and mass transfer. Int. J. Heat Mass Transf. 1966, 9, 1269–1277. [Google Scholar] [CrossRef]
- Hossain, M.A.; Takhar, H.S. Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transf. 1996, 31, 243–248. [Google Scholar] [CrossRef]
- Cheng, E.H.; Özişik, M.N. Radiation with free convection in an absorbing, emitting and scattering medium. Int. J. Heat Mass Transf. 1972, 15, 1243–1252. [Google Scholar] [CrossRef]
- Raptis, A. Radiation and free convection flow through a porous medium. Int. J. Commun. Heat Mass Transf. 1998, 25, 289–295. [Google Scholar] [CrossRef]
- Hossain, M.A.; Alim, M.A.; Rees, D.A.S. The effect of radiation on free convection from a porous vertical plate. Int. J. Heat Mass Transf. 1999, 42, 181–191. [Google Scholar] [CrossRef]
- Hayat, T.; Abbas, Z.; Pop, I.; Asghar, S. Effects of radiation and magnetic field on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium. Int. J. Heat Mass Transf. 2010, 53, 466–474. [Google Scholar] [CrossRef]
- Duwairi, H.M.; Duwairi, R.M. Thermal radiation effects on MHD-Rayleigh flow with constant surface heat flux. Heat Mass Transf. 2004, 41, 51–57. [Google Scholar] [CrossRef]
- Gorla, R.S.R. Radiative effect on conjugate forced convection and conductive heat transfer in a circular pin. Int. J. Heat Fluid Flow 1988, 9, 49–51. [Google Scholar] [CrossRef]
- Pop, S.R.; Grosan, T.; Pop, I. Radiation effects on the flow near the stagnation point of a stretching sheet. Tech. Mech. 2004, 25, 100–106. [Google Scholar]
- Al-Odat, M.Q.; Damseh, R.A.; Al-Azab, T.A. Thermal boundary layer on an exponentially stretching continuous surface in the presence of magnetic field effect. Int. J. Appl. Mech. Eng. 2006, 11, 289–299. [Google Scholar]
- Sajid, M.; Hayat, T. Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet. Int. Commun. Heat Mass Transf. 2008, 35, 347–356. [Google Scholar] [CrossRef]
- Prasad, V.R.; Reddy, N.B.; Muthucumaraswamy, R. Radiation and mass transfer effects on two-dimensional flow past an impulsively started infinite vertical plate. Int. J. Therm. Sci. 2007, 46, 1251–1258. [Google Scholar] [CrossRef]
- Andersson, H.I.; Aarseth, J.B.; Dandapat, B.S. Heat Transfer in a Liquid Film on an Unsteady Stretching Surface. Int. J. Heat Mass Transf. 2000, 43, 69–74. [Google Scholar] [CrossRef]
- Elbashbeshy, E.M.A.; Bazid, M.A.A. Heat Transfer over an Unsteady Stretching Surface. Heat Mass Transf. 2004, 41, 1–4. [Google Scholar] [CrossRef]
- Sharidan, S.; Mahmood, T.; Pop, I. Similarity Solutions for the Unsteady Boundary Layer Flow and Heat Transfer due to a Stretching Sheet. Int. J. Appl. Mech. Eng. 2006, 11, 647–654. [Google Scholar]
- Ali, M.E.; Magyari, E. Unsteady Fluid and Heat Flow Induced by a Submerged Stretching Surface While Its Steady Motion is Slowed Down Gradually. Int. J. Heat Mass Transf. 2007, 50, 188–195. [Google Scholar] [CrossRef]
- Dandapat, B.S.; Santra, B.; Vajravelu, K. The Effects of Variable Fluid Properties and Thermo-capillarity on the Flow of a Thin Film on an Unsteady Stretching Sheet. Int. J. Heat Mass Transf. 2007, 50, 991–996. [Google Scholar] [CrossRef]
- Tsai, R.; Huang, K.H.; Huang, J.S. Flow and Heat Transfer over an Unsteady Stretching Surface with a Non-Uniform Heat Source. Int. Commun. Heat Mass Transf. 2008, 35, 1340–1343. [Google Scholar] [CrossRef]
- Liu, I.C.; Andersson, H.I. Heat Transfer in a Liquid Film on an Unsteady Stretching Sheet. Int. J. Therm. Sci. 2008, 47, 766–772. [Google Scholar] [CrossRef]
- Mukhopadhyay, S. Effect of Thermal Radiation on Unsteady Mixed Convection Flow and Heat Transfer over a Porous Stretching Surface in Porous Medium. Int. J. Heat Mass Transf. 2009, 52, 3261–3265. [Google Scholar] [CrossRef]
- Mukhopadhyay, S. Effects of Slip on Unsteady Mixed Convective Flow and Heat Transfer Past a Stretching Surface. Chin. Phys. Lett. 2010, 27, 124401. [Google Scholar] [CrossRef]
- Mukhopadhyay, S. Heat Transfer Analysis for Unsteady MHD Flow Past a Non-Isothermal Stretching Surface. Nucl. Eng. Des. 2011, 241, 4835–4839. [Google Scholar] [CrossRef]
- Mukhopadhyay, S. Heat Transfer Analysis for Unsteady Flow of a Maxwell Fluid over a Stretching Surface in the Presence of a Heat Source/Sink. Chin. Phys. Lett. 2012, 29, 054703. [Google Scholar] [CrossRef]
- Chamkha, A.J.; Aly, A.M.; Mansour, M.A. Similarity Solution for Unsteady Heat and Mass Transfer from a Stretching Surface Embedded in a Porous Medium with Suction/Injection and Chemical Reaction Effects. Chem. Eng. Commun. 2010, 197, 846–858. [Google Scholar] [CrossRef]
- Bhattacharyya, K.; Mukhopadhyay, S.; Layek, G.C. Slip Effects on an Unsteady Boundary Layer Stagnation-Point Flow and Heat Transfer Towards a Stretching Sheet. Chin. Phys. Lett. 2011, 28, 094702. [Google Scholar] [CrossRef]
- Ali, B.; Nie, Y.; Khan, S.A.; Sadiq, M.T.; Tariq, M. Finite element simulation of multiple slip effects on MHD unsteady Maxwell nanofluid flow over a permeable stretching sheet with radiation and thermo-diffusion in the presence of chemical reaction. Processes 2019, 7, 628. [Google Scholar] [CrossRef]
- Na, T.Y. (Ed.) Computational Methods in Engineering Boundary Value Problems; Academic Press: New York, NY, USA, 1980. [Google Scholar]
Sadeghy et al. [19] | Subhas et al. [23] | Present Results | Subhas et al. [21] | Present Results | |
---|---|---|---|---|---|
0.0 | −1.0000000 | −0.999962 | −0.999962 | −1.095445 | −1.095151 |
0.2 | −1.0549000 | −1.051948 | −1.052001 | −1.188270 | −1.188321 |
0.4 | −1.1008400 | −1.101850 | −1.101843 | −1.275878 | −1.275888 |
0.6 | −1.0015016 | −1.150163 | −1.150133 | −1.358733 | −1.358771 |
0.8 | −1.1987200 | −1.196692 | −1.196634 | −1.437369 | −1.437357 |
1.2 | − | −1.285257 | −1.285269 | −1.512280 | −1.512280 |
1.6 | − | −1.368641 | −1.368630 | −1.095445 | −1.095445 |
2.0 | − | −1.447617 | −1.447654 | −1.188270 | −1.188119 |
0.0 | −0.56784 | 0.86725 |
0.1 | −0.59342 | 0.97241 |
0.2 | −0.61735 | 1.06797 |
0.3 | −0.64177 | 1.16383 |
0.4 | −0.66621 | 1.25733 |
0.5 | −0.69096 | 1.35039 |
0.6 | −0.71573 | 1.44164 |
0.7 | −0.74067 | 1.53248 |
0.8 | −0.76554 | 1.62206 |
0.9 | −0.79034 | 1.71184 |
1.0 | −0.81459 | 1.80201 |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Haider, S.; Syed Muhammad, I.; Li, Y.-Z.; Faraz; Saeed Butt, A. Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet. Coatings 2020, 10, 208. https://doi.org/10.3390/coatings10030208
Haider S, Syed Muhammad I, Li Y-Z, Faraz, Saeed Butt A. Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet. Coatings. 2020; 10(3):208. https://doi.org/10.3390/coatings10030208
Chicago/Turabian StyleHaider, Sajjad, Imran Syed Muhammad, Yun-Zhang Li, Faraz, and Adnan Saeed Butt. 2020. "Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet" Coatings 10, no. 3: 208. https://doi.org/10.3390/coatings10030208
APA StyleHaider, S., Syed Muhammad, I., Li, Y.-Z., Faraz, & Saeed Butt, A. (2020). Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet. Coatings, 10(3), 208. https://doi.org/10.3390/coatings10030208