Second Law Analysis of Dissipative Nanofluid Flow over a Curved Surface in the Presence of Lorentz Force: Utilization of the Chebyshev–Gauss–Lobatto Spectral Method
Abstract
:1. Introduction
2. Description of the Mathematical Formulation
3. Analysis of Entropy Production
4. Solution Methodology
5. Analysis of Results
6. Conclusions
- The enhancement in dimensionless radius of curvature κ (reducing bending of the curved sheet), solid volume fraction of nanoparticles φ, and magnetic parameter reduced the velocity of both types of nanofluids. Furthermore, velocity dominated for Cu–water nanofluid.
- A rise in temperature was observed with increasing values of magnetic parameter M, solid volume fraction of nanoparticles φ, variable thermal conductivity parameter ε, and Ecker number Ec. Moreover, the temperature inside the boundary layer containing silver nanoparticles was high, as compared to copper nanoparticles.
- Decrement in the temperature distribution was observed with decreasing bending in the curved surface (i.e., increasing κ).
- The thermal boundary layer thickness dominated for Ag–water nanofluid due to high effective thermal conductivity.
- One can reduce the entropy generation Ns by decreasing the operating temperature difference (Tw − Tb) and curvature of the curved boundary (i.e., by increasing the dimensionless radius of curvature κ).
- Entropy generation Ns was enhanced with rising values of Eckert number Ec and magnetic parameter M.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Properties | Base Fluid (Water) | Ag (Silver) | Cu (Copper) |
---|---|---|---|
4179 | 235 | 385 | |
0.613 | 429 | 401 | |
997.1 | 10,500 | 8933 | |
5.5 × 10−6 | 6.3 × 107 | 5.96 × 107 | |
6.8 | - | - |
Present Numerical Results | Present Exact Results | ||
---|---|---|---|
0 | 21 | 1.0000000000 | 1.0000000000 |
0.25 | 35 | 1.1180339887 | 1.1180339887 |
1 | 24 | 1.4142135623 | 1.4142135623 |
2.25 | 38 | 1.8027756377 | 1.8027756377 |
5 | 18 | 2.4494897427 | 2.4494897427 |
10 | 24 | 3.3166247903 | 3.3166247903 |
50 | 20 | 7.1414284285 | 7.1414284285 |
100 | 9 | 10.0498756210 | 10.0498756211 |
500 | 8 | 22.3830292855 | 22.3830292856 |
1000 | 6 | 31.6385840390 | 31.6385840391 |
Present Numerical Results | Present Exact Results | ||||
---|---|---|---|---|---|
0.0 | 7.0 | 0.5 | 14 | 3.9133020001 | 3.9133020001 |
0.3 | 15 | 3.1448005650 | 3.1448005650 | ||
0.5 | 15 | 2.6324662749 | 2.6324662749 | ||
0.7 | 16 | 2.1201319848 | 2.1201319848 | ||
0.1 | 0.7 | 0.2 | 35 | 1.0090352173 | 1.0090352173 |
1.0 | 24 | 1.2621850959 | 1.2621850959 | ||
3.0 | 18 | 2.3816005234 | 2.3816005234 | ||
7.0 | 18 | 3.7598314882 | 3.7598314882 | ||
0.4 | 3.0 | 0.0 | 26 | 2.2038900906 | 2.2038900906 |
1.0 | 21 | 1.6226583382 | 1.6226583382 | ||
1.5 | 21 | 1.3788073543 | 1.3788073543 | ||
2.0 | 18 | 1.1547005383 | 1.1547005383 |
Present Numerical Results | |||
---|---|---|---|
*CGLSM | *GDQM | *RKFM | |
5 | 1.1576312 | 1.1576312 | 1.1576312 |
10 | 1.0734886 | 1.0734886 | 1.0734886 |
20 | 1.0356098 | 1.0356098 | 1.0356098 |
30 | 1.0235310 | 1.0235310 | 1.0235310 |
40 | 1.0175866 | 1.0175866 | 1.0175866 |
50 | 1.0140492 | 1.0140492 | 1.0140492 |
100 | 1.0070384 | 1.0070384 | 1.0070384 |
200 | 1.0035641 | 1.0035641 | 1.0035641 |
1000 | 1.0007993 | 1.0007993 | 1.0007993 |
Rosca and Pop [11] | Present Results | |
---|---|---|
5 | 1.15076 | 1.1576312 |
10 | 1.07172 | 1.0734886 |
20 | 1.03501 | 1.0356098 |
30 | 1.02315 | 1.0235310 |
40 | 1.01729 | 1.0175866 |
50 | 1.01380 | 1.0140492 |
100 | 1.00687 | 1.0070384 |
200 | 1.00342 | 1.0035641 |
1000 | 1.00068 | 1.0007993 |
φ | ε | κ | M | Ec | Ag–Water Nanofluid | Cu–Water Nanofluid | ||
---|---|---|---|---|---|---|---|---|
0.00 | 0.2 | 10 | 0.2 | 0.3 | 1.1846573 | 1.2160859 | 1.1846573 | 1.2160859 |
0.05 | 1.4906065 | 1.4464253 | 1.4590642 | 1.4461851 | ||||
0.10 | 1.8083488 | 1.6262670 | 1.7485023 | 1.6414604 | ||||
Slope (Linear Regression) | 6.2369150 | 4.1018110 | 5.6384500 | 4.2537450 | ||||
0.10 | 0.0 | 10 | 0.2 | 0.3 | 1.8083488 | 3.1662365 | 1.7485023 | 3.2955617 |
0.5 | 1.8083488 | 0.1863395 | 1.7485023 | 0.1001680 | ||||
1.5 | 1.8083488 | −1.8979797 | 1.7485023 | −2.1188446 | ||||
Slope (Linear Regression) | 0.0000000 | −3.1915977 | 0.0000000 | −3.4109482 | ||||
0.10 | 0.2 | 5 | 0.2 | 0.3 | 1.9290412 | 1.6103249 | 1.8713011 | 1.6288812 |
15 | 1.7708298 | 1.6303497 | 1.7104317 | 1.6445620 | ||||
1000 | 1.7006379 | 1.6369517 | 1.6393829 | 1.6494311 | ||||
Slope (Linear Regression) | −0.0001520 | 0.0000169 | −0.0001542 | 0.0000129 | ||||
0.10 | 0.2 | 10 | 0.0 | 0.3 | 1.6874096 | 1.6522005 | 1.6222921 | 1.6601904 |
0.5 | 1.9707792 | 1.5674102 | 1.9166084 | 1.5918967 | ||||
1.5 | 2.4186108 | 1.2500458 | 2.3753224 | 1.2990534 | ||||
Slope (Linear Regression) | 0.4818052 | −0.2751404 | 0.4958336 | −0.2481987 | ||||
0.1 | 0.2 | 10 | 0.2 | 0.1 | 1.8083488 | 1.6617211 | 1.7485023 | 1.6571309 |
0.4 | 1.8083488 | 1.5613071 | 1.7485023 | 1.5909710 | ||||
0.7 | 1.8083488 | 1.1758866 | 1.7485023 | 1.2674579 | ||||
Slope (Linear Regression) | 0.0000000 | −0.8097241 | 0.0000000 | −0.6494550 |
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Afridi, M.I.; Qasim, M.; Wakif, A.; Hussanan, A. Second Law Analysis of Dissipative Nanofluid Flow over a Curved Surface in the Presence of Lorentz Force: Utilization of the Chebyshev–Gauss–Lobatto Spectral Method. Nanomaterials 2019, 9, 195. https://doi.org/10.3390/nano9020195
Afridi MI, Qasim M, Wakif A, Hussanan A. Second Law Analysis of Dissipative Nanofluid Flow over a Curved Surface in the Presence of Lorentz Force: Utilization of the Chebyshev–Gauss–Lobatto Spectral Method. Nanomaterials. 2019; 9(2):195. https://doi.org/10.3390/nano9020195
Chicago/Turabian StyleAfridi, Muhammad Idrees, Muhammad Qasim, Abderrahim Wakif, and Abid Hussanan. 2019. "Second Law Analysis of Dissipative Nanofluid Flow over a Curved Surface in the Presence of Lorentz Force: Utilization of the Chebyshev–Gauss–Lobatto Spectral Method" Nanomaterials 9, no. 2: 195. https://doi.org/10.3390/nano9020195