On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers
Abstract
:1. Introduction
2. Mathematical Formulation
Similarity Transformation
3. Method of Solution
3.1. Quasi-Linearization
3.2. Chebyshev Differentiation
4. Results and Discussions
5. Conclusions
- The SQLM is a very efficient and accurate method.
- The fluid velocity and the momentum boundary layer decrease with respective, increases in the Williamson parameter, unsteadiness parameter, magnetic parameter, Eckert number as well as the Prandtl and Schmidt numbers.
- The fluid velocity and the momentum boundary layer increase with increasing values of the electric parameter, buoyancy parameters, thermal radiation and chemical reaction parameter.
- The fluid temperature increases as the values of the magnetic parameter, thermal radiation parameter, electrical parameter and Eckert number increase.
- The fluid temperature is a decreasing function of the buoyancy parameter, Prandtl number, unsteadiness parameter as well as the Williamson number.
- The stretching parameter, chemical reaction parameter, suction, Schmidt number, buoyancy parameters and the Williamson number were found to reduce the concentration profiles.
- The concentration was observed to be increasing as the values of the magnetic parameter, injection and Eckert number increase.
- The skin friction increases with the increase of the unsteadiness parameter, magnetic parameter, Prandtl number, Schmidt number, chemical reaction parameter, and thermal radiation parameter.
- However, the skin friction decreases with increasing values of the Eckert number, buoyancy parameters, thermal radiation and the Williamson number.
- The wall temperature gradient decreases with the increasing values of the Williamson number, suction, magnetic parameter, chemical reaction parameter, Schmidt number and Eckert number.
- The study observed that the Nusselt number increases with the increase of the unsteadiness parameter, electric parameter, buoyancy parameters, Prandtl number, thermal radiation parameter, and the Williamson number.
- The unsteadiness parameter, magnetic parameter, the Prandtl number and the Williamson number cause the wall concentration gradient to decrease.
- Lastly, the Sherwood number increases as the values of the electric parameter, buoyancy parameters, chemical reaction, Schmidt number, thermal radiation and Eckert number increase.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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S | M | Pr | Sc | R | We | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.078951883204829 |
0.2 | 1.104506706276102 | ||||||||||
0.3 | 1.129271687273391 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.078951883204829 |
0.8 | 1.100265152109772 | ||||||||||
0.9 | 1.120409610249936 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.078951883204829 |
0.2 | 1.033110225329453 | ||||||||||
0.3 | 0.987166455664011 | ||||||||||
0.1 | 0.7 | 0.1 | 0.2 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.046544533993518 |
0.4 | 0.981494769194006 | ||||||||||
0.6 | 0.916019062597979 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.2 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.025760860799882 |
0.4 | 0.921685018418928 | ||||||||||
0.6 | 0.819005181818752 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 2.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.091693867285284 |
3.0 | 1.103983070497612 | ||||||||||
4.0 | 1.112467387854574 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 1.0 | 0.2 | 0.1 | 0.1 | 0.1 | 1.085984779187155 |
2.0 | 1.090279447513868 | ||||||||||
3.0 | 1.093114307914497 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.1 | 0.1 | 0.1 | 0.1 | 1.078951883204829 |
0.15 | 1.081939366249072 | ||||||||||
0.2 | 1.084395556441141 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.078951883204829 |
0.4 | 1.076840518443974 | ||||||||||
0.8 | 1.074763680141721 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 1.0 | 0.1 | 1.076389805314664 |
2.0 | 1.071224585593461 | ||||||||||
3.0 | 1.066003195448882 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.2 | 0.948884379906569 |
0.4 | 0.620322356942535 | ||||||||||
0.6 | 0.194330797445648 |
S | M | Pr | Sc | R | We | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.915860341811270 |
0.2 | 1.017901652122212 | ||||||||||
0.3 | 1.109121647214719 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.915860341811270 |
0.8 | 0.895915260713988 | ||||||||||
0.9 | 0.876756392212799 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.915860341811270 |
0.2 | 0.977894447068897 | ||||||||||
0.3 | 1.030403768637055 | ||||||||||
0.1 | 0.7 | 0.1 | 0.2 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.929041372189117 |
0.4 | 0.953439839063926 | ||||||||||
0.6 | 0.975582164474820 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.2 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.948908798318697 |
0.4 | 1.001281793736204 | ||||||||||
0.6 | 1.041796255284524 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 2.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 1.304303729032015 |
3.0 | 1.597280137353339 | ||||||||||
4.0 | 1.845769077311545 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 1.0 | 0.2 | 0.1 | 0.1 | 0.1 | 0.907214610540086 |
2.0 | 0.902204998128932 | ||||||||||
3.0 | 0.899080115679210 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.1 | 0.1 | 0.1 | 0.1 | 0.907214610540086 |
0.15 | 0.903342484386425 | ||||||||||
0.2 | 0.900478136478512 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.907214610540086 |
0.4 | 0.992636450654252 | ||||||||||
0.8 | 1.093677046062223 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 1.0 | 0.1 | 0.636725832004835 |
1.5 | 0.367594034183057 | ||||||||||
3.0 | 0.099808347796939 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.2 | 0.914843548830036 |
0.4 | 0.936472243301972 | ||||||||||
0.6 | 0.967724950819449 |
S | M | Pr | Sc | R | We | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.223598982783657 |
0.2 | 0.137162963897435 | ||||||||||
0.3 | 0.062147655092748 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.223598982783657 |
0.8 | 0.221909175374051 | ||||||||||
0.9 | 0.220302263370217 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.220302263370217 |
0.2 | 0.244356594673201 | ||||||||||
0.3 | 0.262718334286973 | ||||||||||
0.1 | 0.7 | 0.1 | 0.2 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.222825140750917 |
0.4 | 0.227598540998689 | ||||||||||
0.6 | 0.232071072563289 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.2 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.232796360927152 |
0.4 | 0.252429377039816 | ||||||||||
0.6 | 0.267886177341174 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 2.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.218996596175997 |
3.0 | 0.218382226977623 | ||||||||||
4.0 | 0.218017429115231 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 1.0 | 0.2 | 0.1 | 0.1 | 0.1 | 0.220049536513613 |
2.0 | 1.090279447513868 | ||||||||||
3.0 | 1.093114307914497 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.1 | 0.1 | 0.1 | 0.1 | 0.220049536513613 |
0.15 | 0.286846328785100 | ||||||||||
0.2 | 0.351830465481003 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.1 | 0.351830465481003 |
0.4 | 0.352601074008495 | ||||||||||
0.8 | 0.353544944638484 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 1.0 | 0.1 | 0.352768392930626 |
2.0 | 0.353807745690575 | ||||||||||
3.0 | 0.354844653039502 | ||||||||||
0.1 | 0.7 | 0.1 | 0.1 | 0.1 | 1.0 | 0.5 | 0.2 | 0.1 | 0.1 | 0.2 | 0.349345275097305 |
0.4 | 0.343567754574247 | ||||||||||
0.6 | 0.337874422565249 |
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Shateyi, S.; Muzara, H. On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers. Computation 2020, 8, 55. https://doi.org/10.3390/computation8020055
Shateyi S, Muzara H. On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers. Computation. 2020; 8(2):55. https://doi.org/10.3390/computation8020055
Chicago/Turabian StyleShateyi, Stanford, and Hillary Muzara. 2020. "On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers" Computation 8, no. 2: 55. https://doi.org/10.3390/computation8020055
APA StyleShateyi, S., & Muzara, H. (2020). On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers. Computation, 8(2), 55. https://doi.org/10.3390/computation8020055