# Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Primary Stationary and Oscillatory Instabilities

#### 3.1. Infinite Aspect Ratios

#### 3.2. Effect of Lateral Confinement on Pattern Selection

## 4. Secondary Instabilities

#### 4.1. Nonlinear Solution and Formulation of Its Linear Stability

#### 4.2. Results for Newtonian Fluids

#### 4.3. Results for Viscoelastic Fluids

#### 4.3.1. Hopf Bifurcation to Transverse Rolls

#### 4.3.2. Bifurcation to Steady or Oscillatory Longitudinal Rolls

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Neutral stability curves: (

**a**) stationary instability, which exists independently of viscoelastic parameters; (

**b**) oscillatory instability, which may develop first depending on viscoelastic parameters.

**Figure 4.**(

**a**) Critical Rayleigh number and (

**b**) critical frequency at the onset of oscillatory convection as a function of ${\lambda}_{1}$ for different values of $\mathsf{\Gamma}$. The line $Ra=12$ in (a) corresponds to the critical Rayleigh number at the onset of stationary convection.

**Figure 5.**Critical Rayleigh number against the lateral aspect ratio with different numbers of rolls ($L=1$: red dashed curve; $L=2$: green dotted curve; $L=3$: black dash dotted curve; and $L=4$: blue densely-dotted curve): (

**a**) steady longitudinal rolls that exist independently of viscoelastic parameters; (

**b**) oscillatory longitudinal rolls for $\mathsf{\Gamma}=0.1$ and ${\lambda}_{1}=0.5$. In both figures, the horizontal lines indicate the corresponding critical Rayleigh number for transverse rolls.

**Figure 6.**Newtonian fluids: (

**a**) neutral stability curve at the onset of oscillatory transverse rolls; (

**b**) critical Rayleigh number at the onset of steady longitudinal rolls against the lateral aspect ratio with different numbers of rolls ($L=1$: red dashed curve; $L=2$: green dotted curve; $L=3$: black dash dotted curve; and $L=4$: blue densely-dotted curve). The horizontal line corresponds to the threshold of oscillatory transverse rolls.

**Figure 7.**Critical Rayleigh number for the destabilization of fully-developed flow against the wave number k with $l=0$ for Newtonian fluids (Newt) and for viscoelastic solutions with: (

**a**) ${\lambda}_{1}=0.1$ and $\mathsf{\Gamma}=0.75;0.5;0.3$; (

**b**) $\mathsf{\Gamma}=0.75$ and ${\lambda}_{1}=0.1;0.3;0.5$.

**Figure 8.**Critical Rayleigh number for the onset of steady (ultra thick curves) and oscillatory (thick curves) longitudinal rolls as a function of aspect ratio a for different numbers L of rolls ($L=1$: red dashed curve; $L=2$: green dotted curve; $L=3$: black dash-dotted curve; $L=4$: blue densely-dotted curve). (

**a**) $\mathsf{\Gamma}=0.75$ and ${\lambda}_{1}=0.3$; (

**b**) $\mathsf{\Gamma}=0.5$ and ${\lambda}_{1}=0.1$. The horizontal line corresponds to the threshold of oscillatory transverse rolls.

**Table 1.**Critical Rayleigh number $R{a}_{c2}^{T}$, frequency ${\omega}_{c2}^{T}$ and wave number ${k}_{c2}^{T}$ at the onset of moving transverse rolls as a secondary instability for ${\lambda}_{1}=0.1$ and different values of $\mathsf{\Gamma}$.

$\mathsf{\Gamma}$ | ${\mathit{Ra}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ | ${\mathit{\omega}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ | ${\mathit{k}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ |
---|---|---|---|

Newtonian | 506.27 | 138.24 | 4.8 |

0.75 | 358.62 | 115.209 | 4.660 |

0.70 | 329.48 | 110.448 | 4.630 |

0.65 | 300.89 | 105.918 | 4.610 |

0.60 | 272.90 | 101.395 | 4.590 |

0.55 | 245.58 | 96.897 | 4.570 |

0.50 | 219.04 | 92.825 | 4.570 |

0.45 | 193.39 | 88.902 | 4.575 |

0.40 | 168.81 | 85.603 | 4.610 |

0.35 | 145.47 | 83.112 | 4.685 |

0.30 | 123.55 | 81.578 | 4.805 |

**Table 2.**Critical Rayleigh number $R{a}_{c2}^{T}$, frequency ${\omega}_{c2}^{T}$ and wave number ${k}_{c2}^{T}$ at the onset of moving transverse rolls as a secondary instability for $\mathsf{\Gamma}=0.75$ and different values of ${\lambda}_{1}$.

${\mathit{\lambda}}_{\mathbf{1}}$ | ${\mathit{Ra}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ | ${\mathit{\omega}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ | ${\mathit{k}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ |
---|---|---|---|

0.7 | 354.21 | 110.819 | 4.545 |

0.6 | 354.31 | 110.979 | 4.550 |

0.5 | 354.45 | 111.042 | 4.550 |

0.4 | 354.66 | 111.251 | 4.555 |

0.3 | 355.05 | 111.642 | 4.565 |

0.2 | 355.83 | 112.584 | 4.590 |

0.1 | 358.62 | 115.209 | 4.660 |

**Table 3.**Critical Rayleigh number $R{a}_{c2}^{L}$, frequency ${\omega}_{c2}^{L}$ and wave number ${k}_{c2}^{L}$ at the onset of oscillatory longitudinal rolls as the secondary instability for different values of $\mathsf{\Gamma}$ and ${\lambda}_{1}$.

${\mathit{\lambda}}_{\mathbf{1}}$ | $\mathsf{\Gamma}$ | ${\mathit{Ra}}_{\mathit{c}\mathbf{2},\mathbf{osc}}^{\mathit{L}}$ | ${\mathit{\omega}}_{\mathit{c}\mathbf{2}}^{\mathit{L}}$ | ${\mathit{k}}_{\mathit{c}\mathbf{2}}^{\mathit{L}}$ | ${\mathit{R}}_{\mathit{c}\mathbf{2}}^{\mathit{T}}$ | ${\mathit{R}}_{\mathit{c}\mathbf{2},\mathit{s}}^{\mathit{L}}$ |
---|---|---|---|---|---|---|

Newtonian | - | - | - | 506.27 | 313.107 | |

0.1 | 0.75 | 426.27 | 1.53 | 5.8 | 358.62 | 313.107 |

0.3 | 0.75 | 317.55 | 3.58 | 4.5 | 355.03 | 313.107 |

0.5 | 0.75 | 291.34 | 2.65 | 3.9 | 354.45 | 313.107 |

0.1 | 0.6 | 333.47 | 12.35 | 6.3 | 272.90 | 313.107 |

0.1 | 0.5 | 288.08 | 17.53 | 6.5 | 219.04 | 313.107 |

0.1 | 0.3 | 194.20 | 33.62 | 7.0 | 123.55 | 313.107 |

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**MDPI and ACS Style**

Gueye, A.; Ouarzazi, M.N.; Hirata, S.C.; Hamed, H.B.
Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux. *Fluids* **2017**, *2*, 42.
https://doi.org/10.3390/fluids2030042

**AMA Style**

Gueye A, Ouarzazi MN, Hirata SC, Hamed HB.
Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux. *Fluids*. 2017; 2(3):42.
https://doi.org/10.3390/fluids2030042

**Chicago/Turabian Style**

Gueye, Abdoulaye, Mohamed Najib Ouarzazi, Silvia C. Hirata, and Haikel Ben Hamed.
2017. "Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux" *Fluids* 2, no. 3: 42.
https://doi.org/10.3390/fluids2030042