Computational Analysis of the Magnetized Second Grade Fluid Flow Using Modified Fourier and Fick’s Law towards an Exponentially Stretching Sheet
Abstract
:1. Introduction
2. Mathematical Modelling
Physical Quantities
3. Graphical Results and Discussion
4. Concluding Remarks
- The enhancement in the second order fluid parameter decreases the velocity profile.
- The variation in the magnetic effect parameter declines the velocity profile due to the resistive force.
- The growing values of porosity parameter boost the velocity sketch.
- The stronger variations of the radiation parameters and temperature difference parameter are prominent in the temperature profile.
- The growing in the estimations of and decline the profile.
- The result of Eckert numbers and thermal slip parameter enhance the temperature profile of the prescribed domain.
- Improving the value of the Schmidt number and thermophoretic parameter raise the concentration field.
- The higher values of and are prominent in the quantity .
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
List of Parameters
Nusselt number | |
Sharwood number | |
Second grade fluid parameter | |
Magnetic force parameter | |
Porous medium parameter | |
Thermal radiation parameter | |
Temperature ratio parameter | |
Pr | Prandtl number |
Thermal relaxation parameter | |
Eckert number | |
Schmidt number | |
Chemical reaction parameter |
References
- Passerini, A.; Thater, G. Boussinesq type approximation for second grade. Int. J. Non Linear Mech. 2005, 40, 821–831. [Google Scholar] [CrossRef] [Green Version]
- Ishak, A. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. SainsMalaysiana 2011, 40, 391–395. [Google Scholar]
- Mukhopadhyay, S. MHD boundary layer flow and heat transfer over an exponentially stretching sheet embedded in a thermally stratified medium. Alex. Eng. J. 2013, 52, 259–265. [Google Scholar] [CrossRef] [Green Version]
- Mukhopadhyay, S. Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation. Ain Shams Eng. J. 2013, 4, 485–491. [Google Scholar] [CrossRef] [Green Version]
- Hayat, T.; Imtiaz, M.; Alsaedi, A.; Mansoor, R. MHD flow of nanofluids over an exponentially stretching sheet in a porous medium with convective boundary conditions. Chin. Phys. B 2014, 23, 054701. [Google Scholar] [CrossRef]
- Hayat, T.; Shafiq, A.; Alsaedi, A.; Shahzad, S.A. Unsteady MHD flow over exponentially stretching sheet with slip conditions. Appl. Math. Mech. 2016, 37, 193–208. [Google Scholar] [CrossRef]
- Ahmad, R.; Mustafa, M.; Hayat, T.; Alsaedi, A. Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet. J. Magn. Magn. Mater. 2016, 407, 69–74. [Google Scholar] [CrossRef]
- Nayak, M.K.; Akbar, N.S.; Tripathi, D.; Khan, Z.H.; Pandey, V.S. MHD 3D free convective flow of nanofluid over an exponentially stretching sheet with chemical reaction. Adv. Powder Technol. 2017, 28, 2159–2166. [Google Scholar] [CrossRef]
- Senapati, M.; Swain, K.; Parida, S.K. Numerical analysis of three-dimensional MHD flow of Casson nano fluid past an exponentially stretching sheet. Karbala Int. J. Mod. Sci. 2020, 6, 13. [Google Scholar] [CrossRef] [Green Version]
- Reddy, N.N.; Rao, V.S.; Reddy, B.R. Chemical reaction impact on MHD natural convection flow through porous medium past an exponentially stretching sheet in presence of heat source/sink and viscous dissipation. Case Stud. Therm. Eng. 2021, 25, 100879. [Google Scholar] [CrossRef]
- Mandal, C.I.; Mukhopadhyay, S.; Vajravelu, K. Melting Heat Transfer of MHD Micropolar Fluid Flow Past An Exponentially Stretching Sheet with Slip and Thermal Radiation. Int. J. Appl. Comput. Math. 2021, 7, 31. [Google Scholar] [CrossRef]
- Shamshuddin; Khan, S.U.; Bég, O.A.; Bég, T.A. Hall current, viscous and Joule heating effects on steady radiative 2-D magneto-power-law polymer dynamics from an exponentially stretching sheet with power-law slip velocity: A numerical study. Therm. Sci. Eng. Prog. 2020, 20, 100732. [Google Scholar] [CrossRef]
- Hayat, T.; Qasim, M. Influence of thermal radiation and Joule heating on MHD flow of a Maxwell fluid in the presence of thermophoresis. Int. J. Heat Mass Transf. 2010, 53, 4780–4788. [Google Scholar] [CrossRef]
- Srinivasacharya, D.; Jagadeeshwar, P. Effect of Joule heating on the flow over an exponentially stretching sheet with convective thermal condition. Math. Sci. 2019, 13, 201–211. [Google Scholar] [CrossRef] [Green Version]
- Murugesan, T.; Kumar, M.D. Viscous dissipation and Joule heating effects on MHD flow of a Thermo-Solutal stratified nanofluid over an exponentially stretching sheet with radiation and heat generation/absorption. World Sci. News 2019, 129, 193–210. [Google Scholar]
- Sharada, K.; Shankar, B. Effect of partial slip and convective boundary condition on MHD mixed convection flow of Williamson fluid over an exponentially stretching sheet in the presence of joule heating. Glob. J. Pure Appl. Math. 2017, 13, 5965–5975. [Google Scholar]
- Yashkun, U.; Zaimi, K.; Ishak, A.; Pop, I.; Sidaoui, R. Hybrid nanofluid flow through an exponentially stretching/shrinking sheet with mixed convection and Joule heating. Int. J. Numer. Methods Heat Fluid Flow 2020, 31, 1930–1950. [Google Scholar] [CrossRef]
- Srinivasacharya, D.; Jagadeeshwar, P. MHD flow with Hall current and Joule heating effects over an exponentially stretching sheet. Nonlinear Eng. 2017, 6, 101–114. [Google Scholar] [CrossRef] [Green Version]
- Kumar, D.; Sinha, S.; Sharma, A.; Agrawal, P.; Dadheech, P.K. Numerical study of chemical reaction and heat transfer of MHD slip flow with Joule heating and Soret–Dufour effect over an exponentially stretching sheet. Heat Transf. 2021, 51, 1939–1963. [Google Scholar] [CrossRef]
- Agrawal, R.; Kaswan, P. Influence Of Thermal Radiation On Entropy Analysis Of Generalized Mhd Unsteady Viscous Fluid Flow On A Stretching Sheet With Joule Heating And Viscous Dissipation. Comput. Therm. Sci. Int. J. 2021, 13, 21–33. [Google Scholar] [CrossRef]
- Oyelakin, I.S.; Mondal, S.; Sibanda, P. Unsteady Casson nano fluid flow over a stretching sheet with thermal radiation, convective and slip boundary conditions. Alex. Eng. J. 2016, 55, 1025–1035. [Google Scholar] [CrossRef] [Green Version]
- Bakar, N.; Zaimi WM KA, W.; Hamid, R.; Bidin, B.; Ishak, A. Boundary layer flow over a stretching sheet with a convective boundary condition and slip effect. World Appl. Sci. J. 2012, 17, 49–53. [Google Scholar]
- Sajid, M.; Ali, N.; Abbas, Z.; Javed, T. Stretching Flows with General Slip Boundary Condition. Int. J. Mod. Phys. B 2010, 24, 5939–5947. [Google Scholar] [CrossRef]
- Andersson, H.I. Slip flow past a stretching surface. Acta Mech. 2002, 158, 121–125. [Google Scholar] [CrossRef]
- Hayat, T.; Javed, T.; Abbas, Z. Slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space. Int. J. Heat Mass Transf. 2008, 51, 4528–4534. [Google Scholar] [CrossRef]
- Afify, A.A. The influence of slip boundary condition on Casson nano fluid flow over a stretching sheet in the presence of viscous dissipation and chemical reaction. Math. Probl. Eng. 2017, 2017, 3804751. [Google Scholar] [CrossRef] [Green Version]
- Goud, B. Thermal Radiation Influences on MHD Stagnation Point Stream over a Stretching Sheet with Slip Boundary Conditions. Int. J. Thermofluid Sci. Technol. 2020, 7, 070201. [Google Scholar] [CrossRef]
- Khan, S.A.; Nie, Y.; Ali, B. Multiple slip effects on MHD unsteady viscoelastic nano-fluid flow over a permeable stretching sheet with radiation using the finite element method. SN Appl. Sci. 2020, 2, 66. [Google Scholar] [CrossRef] [Green Version]
- Wahab, H.A.; Hussain Shah, S.Z.; Ayub, A.; Sabir, Z.; Bilal, M.; Altamirano, G.C. Multiple characteristics of three-dimensional radiative Cross fluid with velocity slip and inclined magnetic field over a stretching sheet. Heat Transf. 2021, 50, 3325–3341. [Google Scholar] [CrossRef]
- Ahmad, U.; Ashraf, M.; Abbas, A.; Rashad, A.M.; Nabwey, H.A. Mixed convection flow along a curved surface in the presence of exothermic catalytic chemical reaction. Sci. Rep. 2021, 11, 12907. [Google Scholar] [CrossRef]
- Abbas, A.; Ashraf, M.; Chamkha, A.J. Combined effects of thermal radiation and thermophoretic motion on mixed convection boundary layer flow. Alex. Eng. J. 2021, 60, 3243–3252. [Google Scholar] [CrossRef]
- Ilyas, A.; Ashraf, M.; Rashad, A.M. Periodical Analysis of Convective Heat Transfer Along Electrical Conducting Cone Embedded in Porous Medium. Arab. J. Sci. Eng. 2021, 47, 8177–8188. [Google Scholar] [CrossRef]
- Ramzan, M.; Shaheen, N.; Chung, J.D.; Kadry, S.; Chu, Y.-M.; Howari, F. Impact of Newtonian heating and Fourier and Fick’s laws on a magnetohydrodynamic dusty Casson nanofluid flow with variable heat source/sink over a stretching cylinder. Sci. Rep. 2021, 11, 2357. [Google Scholar] [CrossRef] [PubMed]
- Abbas, S.Z.; Waqas, M.; Thaljaoui, A.; Zubair, M.; Riahi, A.; Chu, Y.M.; Khan, W.A. Modeling and analysis of unsteady second-grade nanofluid flow subject to mixed convection and thermal radiation. Soft Comput. 2022, 26, 1033–1042. [Google Scholar] [CrossRef]
- Baranovskii, E.S. Optimal Boundary Control of the Boussinesq Approximation for Polymeric Fluids. J. Optim. Theory Appl. 2021, 189, 623–645. [Google Scholar] [CrossRef]
K | M | Rd | βt | ||
---|---|---|---|---|---|
0.1 | 1.7620 | 1.4323 | |||
0.2 | 1.8703 | 1.5659 | |||
0.3 | 1.9532 | 1.5374 | |||
0.5 | 1.7572 | 1.5237 | |||
1.5 | 1.9332 | 1.7513 | |||
2.0 | 2.2716 | 2.0819 | |||
0.2 | 15.9371 | 15.7272 | |||
0.3 | 16.6546 | 16.3624 | |||
0.4 | 17.0134 | 17.0754 | |||
0.1 | 1.3450 | 1.7264 | |||
0.4 | 1.3571 | 1.8672 | |||
0.7 | 1.3634 | 1.9374 |
0.1 | 1.6532 | |||
0.2 | 2.8302 | |||
0.3 | 3.9293 | |||
2.1 | 1.7095 | |||
2.2 | 1.7683 | |||
2.3 | 1.8276 | |||
0.2 | 1.7695 | |||
0.4 | 1.6529 | |||
0.6 | 0.5724 | |||
0.1 | 0.8695 | |||
0.2 | 0.8802 | |||
0.3 | 0.8952 |
0.1 | ||||
0.2 | 0.3973 | |||
0.3 | 0.4896 | |||
0.1 | 0.3584 | |||
0.2 | 0.3642 | |||
0.3 | 0.3695 | |||
0.1 | 0.9700 | |||
0.2 | 1.0034 | |||
0.3 | 1.6007 | |||
0.1 | 6.5732 | |||
0.2 | 4.9534 | |||
0.3 | 3.0321 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 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 (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Nabwey, H.A.; Khan, A.A.; Ashraf, M.; Rashad, A.M.; Alshber, S.I.; Abu Hawsah, M. Computational Analysis of the Magnetized Second Grade Fluid Flow Using Modified Fourier and Fick’s Law towards an Exponentially Stretching Sheet. Mathematics 2022, 10, 4737. https://doi.org/10.3390/math10244737
Nabwey HA, Khan AA, Ashraf M, Rashad AM, Alshber SI, Abu Hawsah M. Computational Analysis of the Magnetized Second Grade Fluid Flow Using Modified Fourier and Fick’s Law towards an Exponentially Stretching Sheet. Mathematics. 2022; 10(24):4737. https://doi.org/10.3390/math10244737
Chicago/Turabian StyleNabwey, Hossam A., Aamir Abbas Khan, Muhammad Ashraf, Ahmad M. Rashad, Sumayyah I. Alshber, and Miad Abu Hawsah. 2022. "Computational Analysis of the Magnetized Second Grade Fluid Flow Using Modified Fourier and Fick’s Law towards an Exponentially Stretching Sheet" Mathematics 10, no. 24: 4737. https://doi.org/10.3390/math10244737
APA StyleNabwey, H. A., Khan, A. A., Ashraf, M., Rashad, A. M., Alshber, S. I., & Abu Hawsah, M. (2022). Computational Analysis of the Magnetized Second Grade Fluid Flow Using Modified Fourier and Fick’s Law towards an Exponentially Stretching Sheet. Mathematics, 10(24), 4737. https://doi.org/10.3390/math10244737