Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer
Abstract
:1. Introduction
2. Formulation
3. The Steady Solution
4. The Linear Stability Problem
5. Method of Solution
6. Numerical Results and Discussion
7. Conclusions
- The effect of is to stabilize the system, while the parameter has a destabilizing effect on the system.
- The system is most stable for IMP & CON boundary condition, while it is least stable for IMP, CON & CHF boundary condition.
- The water-based ferrofluids are less stable than the ester-based ferrofluids.
- The value of is higher in the case when the heat supply function is increasing, while it is the least in the case when the heat supply function heats and cools the layer in a non-uniform way.
Acknowledgments
Author Contributions
Conflicts of Interest
Abbreviations
d | Thickness of the ferrofluid layer (m) |
g | Acceleration due to gravity (m/s) |
H | Magnetic field (T) |
Unit vector in the z-direction | |
Thermal conductivity (W/m K) | |
K | Permeability of porous medium (m) |
Magnetization (Amp/m) | |
Magnetic saturation | |
Boltzmann’s constant | |
p | Pressure (Pa) |
Q | Volumetric heat source of strength (W/m) |
t | Time (s) |
T | Temperature (K ) |
Temperature at the lower and upper surfaces (K) | |
Filtration velocity of the ferrofluid (m/s) | |
Prandtl number | |
Darcy number | |
Vadasz number | |
Internal Rayleigh number | |
, | Magnetic parameters |
Nonlinearity of magnetization | |
Greek symbols | |
Coefficient of thermal expansion (1/K) | |
Langevin parameter | |
Thermal diffusivity (m/s) | |
Viscosity of ferrofluid (kg/ms) | |
Magnetic permeability of vacuum (H/m) | |
Density (kg/m) | |
The perturbation in temperature (K) | |
Tangent magnetization susceptibility | |
Chord magnetization susceptibility | |
Porosity | |
Operators | |
∇ | |
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IMP & CON | |||||
---|---|---|---|---|---|
Nouri-Borujerdi et al. [49] | Nield and Kuznetsov [32] | Present Study | |||
4.67519 | 471.3787 | 4.67519 | 471.3847 | 4.67518 | 471.3846 |
IMP & CON | IMP, CON & FRE | IMP, CON & CHF | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Water | Ester | Water | Ester | Water | Ester | ||||||||
Case A | 1 | 5.37 | 107.90 | 5.21 | 134.62 | 3.11 | 67.83 | 3.01 | 81.23 | 2.98 | 63.95 | 2.92 | 77.41 |
2 | 5.48 | 87.39 | 5.34 | 111.96 | 3.20 | 56.48 | 3.09 | 69.89 | 3.04 | 53.06 | 2.97 | 66.21 | |
4 | 5.35 | 108.67 | 5.19 | 135.43 | 3.10 | 68.07 | 3.00 | 81.45 | 2.98 | 64.66 | 2.92 | 78.11 | |
6 | 5.17 | 138.03 | 5.03 | 164.86 | 3.00 | 82.63 | 2.91 | 94.34 | 2.92 | 79.39 | 2.87 | 91.59 | |
8 | 5.05 | 160.66 | 4.92 | 185.04 | 2.93 | 92.60 | 2.86 | 102.08 | 2.87 | 89.77 | 2.83 | 99.95 | |
10 | 4.96 | 177.21 | 4.85 | 198.37 | 2.88 | 99.17 | 2.83 | 106.69 | 2.85 | 96.80 | 2.81 | 105.05 | |
Case B | 1 | 5.94 | 111.45 | 5.75 | 140.36 | 3.37 | 69.67 | 3.26 | 84.28 | 3.21 | 65.47 | 3.14 | 80.03 |
2 | 6.08 | 89.70 | 5.90 | 115.79 | 3.47 | 57.58 | 3.35 | 71.89 | 3.27 | 53.94 | 3.20 | 67.89 | |
4 | 5.92 | 112.27 | 5.73 | 141.24 | 3.36 | 69.92 | 3.25 | 84.52 | 3.21 | 66.23 | 3.14 | 80.80 | |
6 | 5.71 | 144.10 | 5.53 | 174.07 | 3.24 | 85.84 | 3.14 | 99.06 | 3.14 | 82.22 | 3.07 | 95.88 | |
8 | 5.55 | 169.31 | 5.39 | 197.34 | 3.16 | 97.06 | 3.08 | 108.07 | 3.09 | 93.81 | 3.04 | 105.53 | |
10 | 5.44 | 188.22 | 5.30 | 213.12 | 3.11 | 104.65 | 3.05 | 113.56 | 3.06 | 101.86 | 3.02 | 111.57 | |
Case C | 1 | 4.62 | 105.71 | 4.50 | 129.92 | 2.81 | 67.65 | 2.73 | 79.79 | 2.71 | 64.15 | 2.67 | 76.51 |
2 | 4.71 | 86.51 | 4.60 | 109.45 | 2.87 | 56.97 | 2.79 | 69.55 | 2.76 | 53.79 | 2.71 | 66.26 | |
4 | 4.60 | 106.43 | 4.49 | 130.63 | 2.80 | 67.87 | 2.72 | 79.99 | 2.71 | 64.80 | 2.67 | 77.12 | |
6 | 4.48 | 132.94 | 4.37 | 156.03 | 2.71 | 81.04 | 2.65 | 91.12 | 2.66 | 78.26 | 2.62 | 88.88 | |
8 | 4.39 | 152.49 | 4.30 | 172.57 | 2.66 | 89.66 | 2.62 | 97.50 | 2.63 | 87.33 | 2.60 | 95.83 | |
10 | 4.33 | 166.24 | 4.26 | 183.06 | 2.63 | 95.13 | 2.59 | 101.18 | 2.61 | 93.24 | 2.58 | 99.94 | |
Case D | 1 | 4.25 | 81.73 | 4.15 | 99.47 | 2.66 | 55.72 | 2.59 | 65.02 | 2.58 | 53.04 | 2.54 | 62.58 |
2 | 4.33 | 67.36 | 4.23 | 84.50 | 2.72 | 47.32 | 2.65 | 57.21 | 2.61 | 44.82 | 2.57 | 54.68 | |
4 | 4.24 | 82.27 | 4.14 | 100.00 | 2.65 | 55.92 | 2.59 | 65.20 | 2.58 | 53.53 | 2.54 | 63.03 | |
6 | 4.12 | 101.67 | 4.03 | 118.06 | 2.58 | 65.99 | 2.53 | 73.43 | 2.53 | 63.89 | 2.50 | 71.79 | |
8 | 4.05 | 115.59 | 3.97 | 129.46 | 2.54 | 72.36 | 2.50 | 78.00 | 2.51 | 70.65 | 2.48 | 76.81 | |
10 | 3.99 | 125.14 | 3.93 | 136.51 | 2.51 | 76.32 | 2.49 | 80.58 | 2.49 | 74.95 | 2.47 | 79.70 |
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Mahajan, A.; Sunil; Sharma, M.K. Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer. Fluids 2017, 2, 22. https://doi.org/10.3390/fluids2020022
Mahajan A, Sunil, Sharma MK. Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer. Fluids. 2017; 2(2):22. https://doi.org/10.3390/fluids2020022
Chicago/Turabian StyleMahajan, Amit, Sunil, and Mahesh Kumar Sharma. 2017. "Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer" Fluids 2, no. 2: 22. https://doi.org/10.3390/fluids2020022