Numerical Computation of Hybrid Carbon Nanotubes Flow over a Stretching/Shrinking Vertical Cylinder in Presence of Thermal Radiation and Hydromagnetic
Abstract
:1. Introduction
2. Mathematical Framework
- A uniform magnetic field ${B}_{0}$ is employed in the radial direction. Assuming that we have the low Reynold number, R_{m}, then the magnetic field formed by induction is trivial compared to the applied magnetic field.
- The fluid velocities are expressed by u and $v$ that are in axial coordinate, x and radial coordinate, r, accordingly.
- The surface temperature, ${T}_{w}$ is higher than the ambient temperature, ${T}_{\infty}$.
- The cylinder with radius R is assumed to move in linear velocity, ${u}_{w}=\frac{{U}_{w}x}{L}$ while the free stream velocity is ${U}_{e}=\frac{{U}_{\infty}x}{L}$ in which L resembles the cylinder’s characteristic length.
- The flow is characterized by the Buoyancy parameter, $\mathsf{\lambda}=\frac{G{r}_{x}}{R{e}_{x}^{2}L}$ in which Gr denotes Grashoff number, and Re denotes Reynolds number. Here, the mixed convection regime is generally expressed as the range of ${\mathsf{\lambda}}_{min}\le \mathsf{\lambda}\le {\mathsf{\lambda}}_{max}$, in which ${\mathsf{\lambda}}_{max}$ and ${\mathsf{\lambda}}_{min}$ are the upper and lower bounds of the mixed convection flow regime.
3. Solutions on Stability Analysis
4. Numerical Computation
5. Analysis of Results
6. Conclusions
- The double solutions presence is apparent at the shrinking sheet, $\epsilon <0$ and opposing flow, $\lambda <0$, whereas a unique solution is found at the stretching sheet, $\epsilon >0$ and assisting flow, $\lambda >0$.
- Throughout the stability analysis, it was determined that not only is the first solution stable, but also some of the positive values in the second solution.
- An increment of ${\phi}_{1}$ (SWCNT) hybrid nanofluid and $\mathsf{\Upsilon}$ in the flow proneness to improve the ${C}_{f}{(R{e}_{x})}^{\frac{1}{2}}$ and $N{u}_{x}{(R{e}_{x})}^{-\frac{1}{2}}$. Meanwhile the opposite trend is noted when M is exists.
- The low Nr helps in enhance the heat transfer of the flow.
- The range of the solutions widen broadly with an augmentation in ${\phi}_{1}$(SWCNT) and $\mathsf{\Upsilon}$, therefore slowing down the boundary layer separation. Furthermore, the adding amount of M fastens the separation of boundary layer.
- In addition, the hybrid carbon nanotubes possess excellent result in skin friction and local Nusselt number compared to the SWCNT-water and MWCNT-water. This will give an insight to scientists or technologists in improving industrial and bio-medical production with regards to their expertise.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ODE | Ordinary Differential Equation |
Pr | Prandtl number |
T | Temperature |
U | Uniform free stream |
q_{w} | Plate heat flux |
Nu_{x} | Local Nusselt number |
Re_{x} | Reynolds number |
C_{f} | Skin friction coefficient |
w | Condition on plate |
∞ | Ambient condition |
hnf | Hybrid nanofluid |
nf | Nanofluid |
f | Fluid |
α | Thermal diffusivity |
μ | Dynamic viscosity |
ρ | Density |
ψ | Stream function |
η | Similarity variables |
θ | Dimensionless temperature |
(ρC_{p})_{f} | Specific heat for base fluid |
C_{p} | Specific heat at constant pressure |
τ | Dimensionless time |
k | Thermal conductivity |
$\phi $ | Concentration of nanoparticles |
$\gamma $ | Eigenvalues |
υ | Kinematic viscosity |
$\lambda $ | Mixed convection parameter |
$\epsilon $ | Velocity ratio parameter |
ϒ | Curvature |
Nr | Thermal radiation |
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Thermophysical Properties | K (W/mK) | C_{p} (J/kgK) | ρ (kg/m^{3}) | $\mathit{\beta}$ × 10^{−6} (1/K) |
---|---|---|---|---|
SWCNT | 6600 | 425 | 2600 | 19 |
MWCNT | 3000 | 796 | 1600 | 21 |
Water | 0.613 | 4179 | 997 | 210 |
Thermophysical | Hybrid Nanofluids |
---|---|
Density | ${\rho}_{hnf}=\left(1-{\phi}_{2}\right)\left[\left(1-{\phi}_{1}\right){\rho}_{f}+{\phi}_{1}{\rho}_{s1}\right]+{\phi}_{2}{\rho}_{s2}$ |
Heat capacity | ${\left(\rho {C}_{p}\right)}_{hnf}=\left(1-{\phi}_{2}\right)\left[\left(1-{\phi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}+{\phi}_{1}{\left(\rho {C}_{p}\right)}_{s1}\right]+{\phi}_{2}{\left(\rho {C}_{p}\right)}_{s2}$ |
Viscosity | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{(1-{\phi}_{1})}^{2.5}{(1-{\phi}_{2})}^{2.5}}$ |
Thermal conductivity | ${k}_{hnf}=\left[\frac{1-{\phi}_{2}+2{\phi}_{2}\left(\frac{{k}_{S}}{{k}_{S}-{k}_{nf}}\right)\mathrm{ln}\left(\frac{{k}_{S}+{k}_{nf}}{2{k}_{nf}}\right)}{1-{\phi}_{2}+2{\phi}_{2}\left(\frac{{k}_{nf}}{{k}_{S}-{k}_{nf}}\right)\mathrm{ln}\left(\frac{{k}_{S}+{k}_{nf}}{2{k}_{nf}}\right)}\right]{k}_{nf}$ ${k}_{nf}=\left[\frac{{k}_{s1}+2{k}_{f}-2{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}\right]{k}_{f}$ |
Thermal expansion coefficient | ${\left(\rho \beta \right)}_{hnf}=\left(1-{\phi}_{2}\right)\left[\left(1-{\phi}_{1}\right){\left(\rho \beta \right)}_{f}+{\phi}_{1}{\left(\rho \beta \right)}_{s1}\right]+{\phi}_{2}{\left(\rho \beta \right)}_{s2}$ |
ε | [75] | [76] | [77] | Present Result | ||||
---|---|---|---|---|---|---|---|---|
(Keller Box) | (Shooting Method) | (bvp4c Solution) | (bvp4c Solution) | |||||
−0.25 | 1.402241 | 1.402242 | 1.402241 | 1.402241 | ||||
−0.75 | 1.489298 | 1.489296 | 1.489298 | 1.489298 | ||||
−1 | 1.328817 | [0] | 1.328819 | [0] | 1.328817 | [0] | 1.328817 | [0] |
−1.2 | 0.932474 | [0.233650] | 0.932470 | [0.233648] | 0.932473 | [0.233650] | 0.932473 | [0.233650] |
−1.24 | 0.706605 | [0.435672] | ||||||
−1.246 | 0.584374 | [0.554215] | 0.609826 | [0.529035] | 0.609826 | [0.529035] | ||
−1.2465 | 0.584295 | [0.554283] | 0.584281 | [0.554296] | 0.584281 | [0.554296] |
${\mathit{\phi}}_{1}$ | M | $\mathit{\lambda}$ | First Solution | Second Solution |
---|---|---|---|---|
0 | −2.4771 | 0.0047 | 0.0897 | |
−2.477 | 0.0073 | 0.0869 | ||
−2.47 | 0.0474 | 0.0452 | ||
0.1 | −2.3069 | 0.0094 | 0.0853 | |
0.01 | −2.306 | 0.0196 | 0.0747 | |
−2.30 | 0.0488 | 0.0445 | ||
0.2 | −2.0558 | 0.1712 | −0.0776 | |
−2.055 | 0.1721 | −0.0784 | ||
−2.05 | 0.1774 | −0.0836 | ||
0 | −2.6421 | 0.0009 | 0.0949 | |
−2.642 | 0.0048 | 0.0893 | ||
−2.64 | 0.0247 | 0.0685 | ||
0.1 | −2.4598 | 0.0164 | 0.0779 | |
0.03 | −2.459 | 0.0228 | 0.0713 | |
−2.45 | 0.0579 | 0.0350 | ||
0.2 | −2.2781 | 0.0111 | 0.0842 | |
−2.278 | 0.0125 | 0.0828 | ||
−2.27 | 0.0532 | 0.0407 |
$\mathsf{\Upsilon}$ | M | $\mathit{\epsilon}$ | First Solution | Second Solution |
---|---|---|---|---|
0 | −1.1226 | 0.0134 | 0.1240 | |
−1.122 | 0.6931 | 0.0960 | ||
−1.12 | 0.7018 | 0.0518 | ||
0.1 | −1.0252 | 0.0254 | 0.0954 | |
0.2 | −1.025 | 0.0326 | 0.0883 | |
−1.02 | 0.8995 | 0.0076 | ||
0.2 | −0.8958 | 0.1932 | − 0.1988 | |
−0.895 | 0.1949 | − 0.2028 | ||
−0.89 | 0.2047 | − 0.2274 | ||
0 | −1.2325 | 0.0131 | 0.8797 | |
−1.232 | 0.6885 | 0.1083 | ||
−1.23 | 0.7062 | 0.0661 | ||
0.1 | −1.1345 | 0.0224 | 0.8997 | |
0.4 | −1.134 | 0.0388 | 0.0909 | |
−1.13 | 0.0913 | 0.7122 | ||
0.2 | −1.0395 | 0.6913 | 0.0824 | |
−1.039 | 0.7011 | 0.9168 | ||
−1.03 | 0.9290 | 0.5533 |
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Aladdin, N.A.L.; Bachok, N.; Rosali, H.; Wahi, N.; Abd Rahmin, N.A.; Arifin, N.M. Numerical Computation of Hybrid Carbon Nanotubes Flow over a Stretching/Shrinking Vertical Cylinder in Presence of Thermal Radiation and Hydromagnetic. Mathematics 2022, 10, 3551. https://doi.org/10.3390/math10193551
Aladdin NAL, Bachok N, Rosali H, Wahi N, Abd Rahmin NA, Arifin NM. Numerical Computation of Hybrid Carbon Nanotubes Flow over a Stretching/Shrinking Vertical Cylinder in Presence of Thermal Radiation and Hydromagnetic. Mathematics. 2022; 10(19):3551. https://doi.org/10.3390/math10193551
Chicago/Turabian StyleAladdin, Nur Adilah Liyana, Norfifah Bachok, Haliza Rosali, Nadihah Wahi, Nor Aliza Abd Rahmin, and Norihan Md Arifin. 2022. "Numerical Computation of Hybrid Carbon Nanotubes Flow over a Stretching/Shrinking Vertical Cylinder in Presence of Thermal Radiation and Hydromagnetic" Mathematics 10, no. 19: 3551. https://doi.org/10.3390/math10193551