Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel
Abstract
:1. Introduction
2. Mathematical Formulation
- Flow is uni-directional, incompressible, and unsteady
- Neglecting the pressure gradient
- A permeable medium is considered
- Thermal conductivity is considered a variable
- Applying magnetic field (neglecting induced magnetic field)
- Viscous dissipation is significant.
3. Skin Friction and Nusselt Number
4. Numerical Procedure
5. Numerical Analysis and Discussion
6. Conclusions
- a
- The addition of a nanoparticle (copper) to the base fluid (blood) enhances the thermal conductivity of fractional Maxwell bio-nanofluid; this causes the temperature profile to decrease by the increase in volumetric fractional nanoparticles (Copper).
- b
- The temperature profile gains high value by increasing the values of the variable thermal conductivity constant
- c
- The thermal radiation parameter increases the heat transfer rate of the aforementioned physical problem.
- d
- The permeable medium decreases the velocity profile of the viscoelastic fractional bio-nanofluid.
- e
- There is an increase in the temperature profile observed by increasing the values of the viscous dissipation parameter Ec.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Physical Quantities | Symbol (Unit) |
Velocity | u (m/s) |
Temperature | (k) |
Density of nanofluid | (kg/m3) |
Gravitational acceleration | g (m/s2) |
Heat capacity of nanofluid | (Cp)nf (JK−1) |
Time relaxation parameter | (s) |
Time retardation parameter | (s) |
Extra stress tensor component | Sxy (Nm−2) |
Prandtl number | Pr |
Eckert number | Ec |
Current density | J |
Dynamic viscosity of nanofluid | µnf (kg/ms) |
Thermal conductivity of nanofluid | knf (W/mK) |
Volumetric expansion | βθ (K−1) |
The electric conductivity of nanofluid | σnf (s/m) |
Kinematic viscosity of nanofluid | νnf (m2/s) |
Gharshof number | Gr |
Applied magnetic field | M |
Cauchy stress tensor | T |
Porosity parameter | |
Caputo fractional derivative | |
Darcy resistance | R |
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Material | Base Fluid (Blood) | Nanoparticle (Copper) |
---|---|---|
1053 | 8933 | |
3594 | 385 | |
0.492 | 401 | |
0.8 | 1.67 | |
0.18 | ||
Pr | 21 | - |
Ha | Gr | |||||||
---|---|---|---|---|---|---|---|---|
0.4 | 0.7 | 1 | ||||||
0.0 | 0.3 | 5 | 10 | 0.1 | 0.1 | 0.4980 | 0.4721 | 0.4162 |
0.1 | 0.5270 | 0.5073 | 0.4579 | |||||
0.2 | 0.5527 | 0.5387 | 0.4959 | |||||
0.2 | 0.0 | 0.6633 | 0.6771 | 0.6686 | ||||
0.1 | 0.6377 | 0.6433 | 0.6224 | |||||
0.2 | 0.6140 | 0.6131 | 0.5835 | |||||
0 | 1.2025 | 1.1485 | 1.0248 | |||||
2 | 1.0207 | 0.9813 | 0.8841 | |||||
5 | 0.5527 | 0.5387 | 0.4959 | |||||
5 | 0 | 0.0553 | 0.0539 | 0.0496 | ||||
2 | 0.1105 | 0.1077 | 0.0992 | |||||
5 | 0.2764 | 0.2694 | 0.2479 | |||||
10 | 0.7679 | 0.7581 | 0.7125 | |||||
1 | 0.7422 | 0.7316 | 0.6862 | |||||
0.1 | 0.5527 | 0.5387 | 0.4959 | |||||
0.01 | 0.3454 | 0.3488 | 0.3355 | |||||
0.1 | 0.5527 | 0.5387 | 0.4959 | |||||
0.2 | 0.4959 | 0.4661 | 0.4111 |
Nr | |||||
---|---|---|---|---|---|
0.4 | 0.7 | 1 | |||
0.0 | 5 | 1.5 | 2.6285 | 2.6285 | 2.6285 |
0.1 | 2.1938 | 2.2232 | 2.2236 | ||
0.2 | 1.9283 | 1.9324 | 1.8666 | ||
0.2 | 0.1 | 0.0164 | 0.0130 | 0.0090 | |
1 | 0.2633 | 0.2276 | 0.1770 | ||
10 | 1.9608 | 2.0404 | 2.0565 | ||
5 | 0 | 0.2614 | 0.2650 | 0.2564 | |
1 | 0.4245 | 0.4196 | 0.3932 | ||
2 | 0.7304 | 0.7011 | 0.6333 |
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Kaleem, M.M.; Usman, M.; Asjad, M.I.; Eldin, S.M. Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel. Fractal Fract. 2022, 6, 712. https://doi.org/10.3390/fractalfract6120712
Kaleem MM, Usman M, Asjad MI, Eldin SM. Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel. Fractal and Fractional. 2022; 6(12):712. https://doi.org/10.3390/fractalfract6120712
Chicago/Turabian StyleKaleem, Muhammad Madssar, Muhammad Usman, Muhammad Imran Asjad, and Sayed M. Eldin. 2022. "Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel" Fractal and Fractional 6, no. 12: 712. https://doi.org/10.3390/fractalfract6120712
APA StyleKaleem, M. M., Usman, M., Asjad, M. I., & Eldin, S. M. (2022). Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel. Fractal and Fractional, 6(12), 712. https://doi.org/10.3390/fractalfract6120712