A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis
Abstract
:1. Introduction
2. Mathematical Modeling
3. The Governing Equations
4. Physical Parameters of Interest
4.1. Skin Friction
4.2. Nusselt Number
4.3. Sherwood Number
5. Mathematical Expression for Entropy Generation
6. Numerical Procedure and Stability Analysis
7. Error Analysis and Confirmation of the RK4 Method
8. Table Discussion
9. Results and Discussion
10. Conclusions
- The velocity field of the nanoparticle’s fluid film upsurges when the unsteadiness factor is enhanced, but the velocity profile of the nanofluid film decreases due to increases in the magnetic factor.
- The coefficient of skin friction significantly develops when and are increased; however, the coefficient of skin friction reduces as the stretching and thickness parameters are increased.
- The Brownian factor has a direct impact on the temperature profile.
- With rising values of the and , the Nusseltnnumber, the thermal boundary-layer thickness reduces.
- With higher Prandtl numbers, the fluid’s surface temperature rises, but for higher values , the temperature field has a reverse effect.
- Observing the impact of the thermophoresis parameter, a similar consequence is observed for the temperature profile.
- The mass flow rate decreases when the Brownian factor is increased, but the thermophoretic factor displays the reverse tendency.
- For the validation of the numerical approach, the implemented technique convergence is quantitatively illustrated. Moreover, the present is compared with the published work reported by Nehad et al. [20], and a good agreement is established.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Order | |||
---|---|---|---|
1 | −0.050000 | −1.218600 | 0.064860 |
2 | −0.215767 | −0.218600 | 0.1967944 |
4 | −0.226793 | −0.217991 | 0.1946782 |
6 | −0.229531 | −0.217928 | 0.1966267 |
8 | −0.229191 | −0.217919 | 0.1196998 |
10 | −0.229372 | −0.217917 | 0.1972274 |
12 | −0.229414 | −0.217918 | 0.1972568 |
14 | −0.229424 | −0.217917 | 0.1972641 |
16 | −0.229426 | −0.217917 | 0.1972659 |
18 | −0.229427 | −0.217917 | 0.1972663 |
22 | −0.229427 | −0.217917 | 0.1972665 |
24 | −0.229427 | −0.217917 | 0.1972665 |
M | Gr | St | Nehad et al. [20] | ||
---|---|---|---|---|---|
0.2 | 0.4 | 0.1 | 1.4 | −0.2819 | −0.2815 |
0.4 | −0.3145 | −0.3142 | |||
1.2 | −0.3571 | −0.3570 | |||
1.8 | 0.2 | −0.2887 | −0.2881 | ||
0.4 | −0.2819 | −0.2812 | |||
1.2 | −0.2819 | −0.2813 | |||
1.7 | 0.2 | −0.4545 | −0.4548 | ||
0.4 | −0.8574 | −0.8570 | |||
1.2 | −7.9971 | −7.9973 | |||
1.7 | 0.2 | −2.9783 | −2.9787 | ||
0.4 | −0.3328 | −0.3329 | |||
1.2 | −0.2431 | −0.2435 |
M | St | Pr | Nu | Nehad et al. [20] | |
---|---|---|---|---|---|
0.2 | 0.2 | 1.4 | 1.5 | 0.4634 | 0.4636 |
0.4 | 0.4632 | 0.4637 | |||
1.2 | 0.4629 | 0.4623 | |||
1.8 | 0.2 | 0.4634 | 0.4630 | ||
0.4 | 2.6256 | 2.6252 | |||
1.2 | 2.3727 | 2.3725 | |||
1.7 | 0.2 | 2.7359 | 2.7353 | ||
0.4 | 0.3395 | 0.3390 | |||
1.2 | 0.3915 | 0.3913 | |||
1.7 | 1.2 | 0.4634 | 0.4638 | ||
1.8 | 0.4354 | 0.4350 | |||
4.0 | 0.4198 | 0.4197 | |||
6.0 | 0.4123 | 0.4122 |
Nb | Nt | Sc | St | Pr | Nehad et al. [20] | |
---|---|---|---|---|---|---|
0.2 | 0.4 | 0.2 | 1.4 | 1.4 | −2.4693 | −2.4693 |
0.4 | −0.3499 | −0.3492 | ||||
1.2 | −0.1988 | −0.1983 | ||||
1.7 | 0.2 | −0.1334 | −0.1338 | |||
0.4 | −2.4693 | −2.4690 | ||||
1.2 | −3.8648 | −3.8642 | ||||
1.7 | 0.2 | −5.2566 | −5.2566 | |||
0.4 | −4.9778 | −4.9772 | ||||
1.2 | −4.8564 | −4.8566 | ||||
1.7 | 0.2 | −2.9216 | −2.9214 | |||
0.4 | −3.4169 | −3.4160 | ||||
1.2 | −3.9294 | −3.9290 | ||||
1.7 | 1.4 | −4.6374 | −4.6372 | |||
2.5 | −6.7241 | −6.7244 | ||||
4.0 | −7.8817 | −7.8812 | ||||
6.0 | −8.4434 | −8.4438 |
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Zeeshan; Attaullah; Ahammad, N.A.; Shah, N.A.; Chung, J.D. A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis. Mathematics 2023, 11, 1078. https://doi.org/10.3390/math11051078
Zeeshan, Attaullah, Ahammad NA, Shah NA, Chung JD. A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis. Mathematics. 2023; 11(5):1078. https://doi.org/10.3390/math11051078
Chicago/Turabian StyleZeeshan, Attaullah, N. Ameer Ahammad, Nehad Ali Shah, and Jae Dong Chung. 2023. "A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis" Mathematics 11, no. 5: 1078. https://doi.org/10.3390/math11051078
APA StyleZeeshan, Attaullah, Ahammad, N. A., Shah, N. A., & Chung, J. D. (2023). A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis. Mathematics, 11(5), 1078. https://doi.org/10.3390/math11051078