# Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Stability Analysis

## 3. Results and Discussion

#### 3.1. Sample Neutral Curves

#### 3.2. The Modal Map

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Latin letters | |

$\mathcal{A}$ | imaginary constant |

$A,C,E$ | particular integral coefficients for ${g}_{1}$ |

$\mathcal{B}$ | complementary function coefficient |

$B,D,F$ | particular integral coefficients for ${f}_{1}$ |

d | height of the channel |

$f,g$ | reduced pressure and temperature |

g | gravity |

k | wave number |

K | permeability |

L | channel aspect ratio |

${\mathcal{L}}_{1},{\mathcal{L}}_{2}$ | differential operators in Equation (A41) |

m | wave number |

n | number of rolls |

N | number of intervals |

p | pressure |

$\mathrm{Ra}$ | Darcy-Rayleigh number |

${\mathcal{R}}_{1},{\mathcal{R}}_{2}$ | right hand sides in Equation (A41) |

t | time |

u | Darcy velocity along the layer |

v | Darcy velocity across the layer |

w | spanwise Darcy velocity |

x | coordinate along the layer |

y | coordinate across the layer |

z | spanwise coordinate |

Greek letters | |

$\alpha $ | inclination angle |

$\beta $ | coefficient of cubical expansion |

$\mathsf{\Delta}T$ | temperature difference |

$\theta $ | temperature |

$\kappa $ | thermal diffusivity |

$\lambda $ | exponential growth rate |

$\mu $ | dynamic viscosity |

$\rho $ | reference density |

$\varphi $ | orientation of oblique roll |

Other symbols | |

$0,1,2$ | terms in series expansion |

i | imaginary component |

r | real component |

${\phantom{\rule{3.33333pt}{0ex}}}^{\prime}$ | differentiation with respect to y |

## Appendix A

## References

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**Figure 1.**Critical values of $Ra$ against $\varphi $ for $L=0.6$, $0.8$ ($0.2$) ⋯ 5 when the inclination from the horizontal is $\alpha ={5}^{\circ}$ and for the $n=1$, $n=2$, $n=3$ and $n=4$ modes. The red curves denote those values of L which are less than n. The dotted lines correspond to integer values of L while the rightmost curve is for $L=5$. The blue horizontal line corresponds to transverse rolls.

**Figure 2.**Critical values of $Ra$ against $\varphi $ for the $n=2$ mode, the following values of L: $L=0.6$, $0.8$ ($0.2$) ⋯ 5, and for the indicated inclinations from the horizontal. The red curves denote those values of L which are less than n. The dotted lines correspond to integer values of L while the rightmost curve is for $L=5$. The blue horizontal line corresponds to transverse rolls.

**Figure 3.**Critical values of $Ra$ against $\varphi $ for $\alpha ={20}^{\circ}$ and for the given values of L. The following modes are shown: $n=1$, $n=2$, $n=3$, $n=4$, $n=5$, $n=6$ and $n=7$. The horizontal line corresponds to transverse rolls.

**Figure 4.**Showing the variation of $\mathrm{Ra}$ and $\varphi $ against L when $\alpha ={10}^{\circ}$. For the curves for $\mathrm{Ra}$, blue indicates longitudinal rolls, while red indicates oblique rolls and black the transverse roll. The disks denote the beginning and ending of the ranges within which oblique rolls are favoured.

**Figure 7.**Showing the orientations of the preferred modes (black dotted lines) when the layer is horizontal and for which ${\mathrm{Ra}}_{c}=4{\pi}^{2}$. The given values of n correspond to the number of cells in the spanwise direction. The red line depicts the preferred mode when $\left|\alpha \right|\ll 1$.

**Figure 8.**Modal map showing the form of the most dangerous disturbance. Blue signifies transverse rolls, orange signifies oblique rolls and white corresponds to longitudinal rolls. The vertical lines shows the boundary between longitudinal rolls with n cells and with $n+1$ cells. The lines within each orange region denote the following values of $\varphi $: ${10}^{\circ}$ (dashed), ${20}^{\circ}$, ${30}^{\circ}$, ${40}^{\circ}$ and ${50}^{\circ}$.

$\mathit{L}=1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{\alpha}=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{\varphi}=0$ | $\mathit{L}=1.2,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{\alpha}{=25}^{\circ},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{\varphi}{=10}^{\circ}$ | |||
---|---|---|---|---|

N | ${\mathrm{Ra}}_{\mathbf{c}}\left(\mathbf{N}\right)$ | Error | ${\mathrm{Ra}}_{c}\left(\mathbf{N}\right)$ | Error |

25 | $39.478\phantom{\rule{0.166667em}{0ex}}580\phantom{\rule{0.166667em}{0ex}}76$ | 0.000 163 16 | 45.024 341 41 | 0.000 201 46 |

50 | $39.478\phantom{\rule{0.166667em}{0ex}}427\phantom{\rule{0.166667em}{0ex}}84$ | 0.000 010 24 | 45.024 152 58 | 0.000 012 63 |

100 | $39.478\phantom{\rule{0.166667em}{0ex}}418\phantom{\rule{0.166667em}{0ex}}25$ | 0.000 000 65 | 45.024 140 74 | 0.000 000 79 |

200 | $39.478\phantom{\rule{0.166667em}{0ex}}417\phantom{\rule{0.166667em}{0ex}}64$ | 0.000 000 04 | 45.024 140 00 | 0.000 000 05 |

400 | $39.478\phantom{\rule{0.166667em}{0ex}}417\phantom{\rule{0.166667em}{0ex}}61$ | 0.000 000 00 | 45.024 139 95 | 0.000 000 00 |

800 | $39.478\phantom{\rule{0.166667em}{0ex}}417\phantom{\rule{0.166667em}{0ex}}61$ | 0.000 000 00 | 45.024 139 95 | 0.000 000 00 |

n | $\mathit{\alpha}$(Degrees) | |
---|---|---|

1 | $34.167212$ | |

2 | $28.006732$ | |

3 | $24.143052$ | |

4 | $21.499921$ | |

5 | $19.557799$ | |

10 | $14.306414$ | |

20 | $10.288103$ | |

50 | $6.572284$ | |

100 | $4.662749$ |

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**MDPI and ACS Style**

Rees, D.A.S.; Barletta, A.
Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection. *Fluids* **2020**, *5*, 83.
https://doi.org/10.3390/fluids5020083

**AMA Style**

Rees DAS, Barletta A.
Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection. *Fluids*. 2020; 5(2):83.
https://doi.org/10.3390/fluids5020083

**Chicago/Turabian Style**

Rees, D. Andrew S., and Antonio Barletta.
2020. "Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection" *Fluids* 5, no. 2: 83.
https://doi.org/10.3390/fluids5020083