Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation
Abstract
:1. Introduction
2. Problem Formulation
3. Solution by Homotopy Analysis Method
4. Results and Discussion
5. Conclusions
- Smaller values of the Prandtl number enhance the thermal boundary layer.
- An increasing value of the magnetic field stops the fluid motion.
- Larger amounts of the thermal radiation parameter and thermophoretic parameter enhances the thermal boundary layer.
- The Casson fluid parameter produces a resistance force and its increasing value decreases the fluid motion.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Nb | Brownian motion parameter |
T | Cauchy stress tensor |
C | Concentration of the fluid |
h | Film thickness |
M | Magnetic parameter |
Nu | Nusselt number |
Pr | Prandtl number |
Nt | Thermoporetic parameter |
R | Radiation parameter |
x, y, z | Rectangular coordinates |
Re | Reynold number |
Sh | Sherwood number |
Cf | Skin friction coefficient |
W | Spraying velocity |
T∞ | Temperature of the fluid at large distance |
f | Transformed dependent variable |
u | Velocity component in x-direction |
v | Velocity component in y-direction |
μ | Coefficient of viscosity |
γ | Casson fluid parameter |
ρ | Density of the fluid |
p | Fluid pressure |
I | Identity tensor chord |
v | Kinematic viscosity |
Ω | Rotation parameter |
τ | Shearing stress |
ψ | Stream function |
β | Thickness of the fluid |
η | Transformed independent variable |
∞ | Condition at infinity |
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f(η) | HAM Solution | Numerical Solution | Absolute Error |
---|---|---|---|
0.0 | 0.000000 | −2.812710 × 10−10 | 2.812710 × 10−10 |
0.1 | 0.003258 | 0.003258 | 5.703240 × 10−8 |
0.2 | 0.012408 | 0.012408 | 2.280140 × 10−7 |
0.3 | 0.026573 | 0.026573 | 5.104210 × 10−7 |
0.4 | 0.044948 | 0.044948 | 8.995600 × 10−7 |
0.5 | 0.066795 | 0.066793 | 1.388520 × 10−6 |
0.6 | 0.091431 | 0.091429 | 1.965710 × 10−6 |
0.7 | 0.118224 | 0.118221 | 2.615740 × 10−6 |
0.8 | 0.146586 | 0.146583 | 3.327870 × 10−6 |
0.9 | 0.175966 | 0.175962 | 4.078740 × 10−6 |
1.0 | 0.205842 | 0.205837 | 4.847430 × 10−6 |
k(η) | HAM Solution | Numerical Solution | Absolute Error |
---|---|---|---|
0.0 | 0.000000 | 3.431260 × 10−8 | 3.431260 × 10−8 |
0.1 | 0.082367 | 0.082366 | 2.022930 × 10−7 |
0.2 | 0.155134 | 0.155134 | 4.388100 × 10−7 |
0.3 | 0.218705 | 0.218705 | 6.667190 × 10−7 |
0.4 | 0.273439 | 0.273438 | 8.749510 × 10−7 |
0.5 | 0.319620 | 0.319619 | 1.071030 × 10−6 |
0.6 | 0.357437 | 0.357436 | 1.231030 × 10−6 |
0.7 | 0.386981 | 0.386980 | 1.356500 × 10−6 |
0.8 | 0.408247 | 0.408246 | 1.432420 × 10−6 |
0.9 | 0.421138 | 0.421137 | 1.471570 × 10−6 |
1.0 | 0.425484 | 0.425483 | 1.467400 × 10−6 |
g(η) | HAM Solution | Numerical Solution | Absolute Error |
---|---|---|---|
0.0 | 1.000000 | 1.000000 | 1.286450 × 10−8 |
0.1 | 0.950141 | 0.950139 | 2.377450 × 10−6 |
0.2 | 0.903441 | 0.903436 | 4.705070 × 10−6 |
0.3 | 0.860810 | 0.860803 | 6.954740 × 10−6 |
0.4 | 0.822957 | 0.822948 | 9.080750 × 10−6 |
0.5 | 0.790417 | 0.790406 | 0.000011 |
0.6 | 0.763566 | 0.763553 | 0.000013 |
0.7 | 0.742641 | 0.742627 | 0.000014 |
0.8 | 0.727756 | 0.727740 | 0.000015 |
0.9 | 0.718908 | 0.718892 | 0.000016 |
1.0 | 0.715997 | 0.715981 | 0.000016 |
s(η) | HAM Solution | Numerical Solution | Absolute Error |
---|---|---|---|
0.0 | 0.000000 | −7.170820 × 10−9 | 7.170820 × 10−9 |
0.1 | −0.019540 | −0.019539 | 1.154300 × 10−6 |
0.2 | −0.038351 | −0.038349 | 2.304890 × 10−6 |
0.3 | −0.055850 | −0.055846 | 3.427710 × 10−6 |
0.4 | −0.071593 | −0.071588 | 4.501730 × 10−6 |
0.5 | −0.085251 | −0.085246 | 5.509380 × 10−6 |
0.6 | −0.096596 | −0.096590 | 6.416000 × 10−6 |
0.7 | −0.105479 | −0.105471 | 7.187270 × 10−6 |
0.8 | −0.111819 | −0.111811 | 7.792270 × 10−6 |
0.9 | −0.115595 | −0.115587 | 8.183170 × 10−6 |
1.0 | −0.116839 | −0.116831 | 8.326040 × 10−6 |
θ(η) | HAM Solution | Numerical Solution | Absolute Error |
---|---|---|---|
0.0 | 0.000000 | −1.777730 × 10−9 | 1.777730 × 10−9 |
0.1 | 0.116628 | 0.116628 | 1.151200 × 10−7 |
0.2 | 0.232964 | 0.232964 | 2.309850 × 10−7 |
0.3 | 0.348690 | 0.348690 | 3.439950 × 10−7 |
0.4 | 0.463468 | 0.463469 | 4.517690 × 10−7 |
0.5 | 0.576946 | 0.576946 | 5.526490 × 10−7 |
0.6 | 0.688763 | 0.688763 | 6.424670 × 10−7 |
0.7 | 0.798558 | 0.798559 | 7.163620 × 10−7 |
0.8 | 0.905975 | 0.905976 | 7.751320 × 10−7 |
0.9 | 1.010670 | 1.010670 | 8.130730 × 10−7 |
1.0 | 1.112300 | 1.112300 | 8.304690 × 10−7 |
ϕ(η) | HAM Solution | Numerical Solution | Absolute Error |
---|---|---|---|
0.0 | 0.000000 | −2.935480 × 10−9 | 2.935480 × 10−9 |
0.1 | 0.109309 | 0.109309 | 8.676730 × 10−8 |
0.2 | 0.218448 | 0.218448 | 1.752770 × 10−7 |
0.3 | 0.327225 | 0.327226 | 2.613220 × 10−7 |
0.4 | 0.435439 | 0.435440 | 3.434820 × 10−7 |
0.5 | 0.542882 | 0.542883 | 4.241530 × 10−7 |
0.6 | 0.649351 | 0.649352 | 4.969910 × 10−7 |
0.7 | 0.754655 | 0.754656 | 5.581250 × 10−7 |
0.8 | 0.858619 | 0.858620 | 6.055710 × 10−7 |
0.9 | 0.961090 | 0.961091 | 6.356540 × 10−7 |
1.0 | 1.061940 | 1.061940 | 6.465200 × 10−7 |
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Saeed, A.; Shah, Z.; Islam, S.; Jawad, M.; Ullah, A.; Gul, T.; Kumam, P. Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation. Coatings 2019, 9, 248. https://doi.org/10.3390/coatings9040248
Saeed A, Shah Z, Islam S, Jawad M, Ullah A, Gul T, Kumam P. Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation. Coatings. 2019; 9(4):248. https://doi.org/10.3390/coatings9040248
Chicago/Turabian StyleSaeed, Anwar, Zahir Shah, Saeed Islam, Muhammad Jawad, Asad Ullah, Taza Gul, and Poom Kumam. 2019. "Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation" Coatings 9, no. 4: 248. https://doi.org/10.3390/coatings9040248
APA StyleSaeed, A., Shah, Z., Islam, S., Jawad, M., Ullah, A., Gul, T., & Kumam, P. (2019). Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation. Coatings, 9(4), 248. https://doi.org/10.3390/coatings9040248