# Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer

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## 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 $d,\phantom{\rule{4pt}{0ex}}{\alpha}_{L},\phantom{\rule{4pt}{0ex}}K$ is to stabilize the system, while the parameter ${M}_{3}$ has a destabilizing effect on the system.
- The system is most stable for IMP${}_{LU}$ & CON${}_{LU}$ boundary condition, while it is least stable for IMP${}_{LU}$, CON${}_{L}$ & CHF${}_{U}$ boundary condition.
- The water-based ferrofluids are less stable than the ester-based ferrofluids.
- The value of $R{a}_{c}$ 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${}^{2}$) |

H | Magnetic field (T) |

$\mathit{k}$ | Unit vector in the z-direction |

${k}_{1}$ | Thermal conductivity (W/m K) |

K | Permeability of porous medium (m${}^{2}$) |

$\mathit{M}$ | Magnetization (Amp/m) |

${M}_{s}$ | Magnetic saturation |

${k}_{B}$ | Boltzmann’s constant |

p | Pressure (Pa) |

Q | Volumetric heat source of strength (W/m${}^{3}$) |

t | Time (s) |

T | Temperature (K ) |

${T}_{0}$ | Temperature at the lower and upper surfaces (K) |

$\mathit{u}$ | Filtration velocity of the ferrofluid (m/s) |

${P}_{r}$ | Prandtl number |

${D}_{a}$ | Darcy number |

${V}_{a}$ | Vadasz number |

$Ra$ | Internal Rayleigh number |

${M}_{1}$, ${M}_{2}$ | Magnetic parameters |

${M}_{3}$ | Nonlinearity of magnetization |

Greek symbols | |

$\alpha $ | Coefficient of thermal expansion (1/K) |

${\alpha}_{L}$ | Langevin parameter |

$\kappa $ | Thermal diffusivity (m${}^{2}$/s) |

$\mu $ | Viscosity of ferrofluid (kg/ms) |

${\mu}_{0}$ | Magnetic permeability of vacuum (H/m) |

$\rho $ | Density (kg/m${}^{3}$) |

$\theta $ | The perturbation in temperature (K) |

$\chi $ | Tangent magnetization susceptibility |

${\chi}_{2}$ | Chord magnetization susceptibility |

$\u03f5$ | Porosity |

Operators | |

${\nabla}^{2}$ | $\frac{{\partial}^{2}}{\partial {x}^{2}}}+{\displaystyle \frac{{\partial}^{2}}{\partial {y}^{2}}}+{\displaystyle \frac{{\partial}^{2}}{\partial {z}^{2}}$ |

∇ | $\frac{\partial}{\partial x}}+{\displaystyle \frac{\partial}{\partial y}}+{\displaystyle \frac{\partial}{\partial z}$ |

${\nabla}_{1}^{2}$ | $\frac{{\partial}^{2}}{\partial {x}^{2}}}+{\displaystyle \frac{{\partial}^{2}}{\partial {y}^{2}}$ |

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**Figure 2.**Neutral curves for different values of the Langevin parameter ${\alpha}_{L}$ for (

**a**) water-based ferrofluid and (

**b**) ester-based ferrofluid for Case A to Case D. The fixed parameter values are $d=\phantom{\rule{3.33333pt}{0ex}}0.001$ m, $\phantom{\rule{4pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.35$, and K = 2.0 × 10${}^{-7}$.

**Figure 3.**Neutral curves for different values of the width of ferrofluid layer d for (

**a**) water-based ferrofluid and (

**b**) ester-based ferrofluid for Case A to Case D. The fixed parameter values are ${\alpha}_{L}=\phantom{\rule{3.33333pt}{0ex}}2,\phantom{\rule{4pt}{0ex}}\u03f5=0.35$, and K = 2.0 × 10${}^{-7}$.

**Figure 4.**Neutral curves for different values of the permeability parameter K for (

**a**) water-based ferrofluid and (

**b**) ester-based ferrofluid for Case A to Case D. The fixed parameter values are ${\alpha}_{L}=\phantom{\rule{3.33333pt}{0ex}}2,\phantom{\rule{4pt}{0ex}}\u03f5=0.35$, and d = 0.001 m.

**Figure 5.**Neutral curves for different values of the nonlinearity of magnetization parameter ${M}_{3}$ for (

**a**) water-based ferrofluid and (

**b**) ester-based ferrofluid for Case A to Case D. The fixed parameter values are ${\alpha}_{L}=2,\phantom{\rule{4pt}{0ex}}\u03f5=0.35$, K = 2.0× 10${}^{-7}$, and d = 0.001 m.

**Table 1.**Comparison of critical Rayleigh number $R{a}_{c}$ and the critical wave number ${k}_{c}$ for IMP${}_{LU}$ & CON${}_{LU}$ boundary condition.

IMP${}_{\mathit{LU}}$ & CON${}_{\mathit{LU}}$ | |||||
---|---|---|---|---|---|

Nouri-Borujerdi et al. [49] | Nield and Kuznetsov [32] | Present Study | |||

${k}_{c}$ | $R{a}_{c}$ | ${k}_{c}$ | $2R{a}_{c}$ | ${k}_{c}$ | $2R{a}_{c}$ |

4.67519 | 471.3787 | 4.67519 | 471.3847 | 4.67518 | 471.3846 |

**Table 2.**The values of the critical internal Rayleigh number $R{a}_{c}$ and the critical wave number ${k}_{c}$ for water- and ester-based ferrofluids.

IMP${}_{\mathit{LU}}$ & CON${}_{\mathit{LU}}$ | IMP${}_{\mathit{L}}$, CON${}_{\mathit{LU}}$ & FRE${}_{\mathit{U}}$ | IMP${}_{\mathit{LU}}$, CON${}_{\mathit{L}}$ & CHF${}_{\mathit{U}}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Water | Ester | Water | Ester | Water | Ester | ||||||||

${\mathit{\alpha}}_{\mathit{L}}$ | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{Ra}}_{\mathit{c}}$ | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{Ra}}_{\mathit{c}}$ | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{Ra}}_{\mathit{c}}$ | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{Ra}}_{\mathit{c}}$ | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{Ra}}_{\mathit{c}}$ | ${\mathit{k}}_{\mathit{c}}$ | ${\mathit{Ra}}_{\mathit{c}}$ | |

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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Mahajan, 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