# Horizontal Cellular Oscillations Caused by Time-Periodic Resonant Thermal Forcing in Weakly Nonlinear Darcy-Bénard Convection

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Weakly Nonlinear Analysis

## 4. Real Solutions of the Amplitude Equations

#### 4.1. Initial Considerations and Context

#### 4.2. Numerical solutions

- (1)
- a single-signed oscillation, which we call Type I,
- (2)
- solutions taking both positive and negative amplitudes, which we call Type II.

#### 4.3. Analysis of the Real Solution Profiles for Large Frequencies ($\omega \gg 1$)

#### 4.4. Analysis of the Basic Flow for Low Frequencies ($\omega \ll 1$)

- (1)
- ${R}_{2}<0$,
- (2)
- $0<{R}_{2}<3/{2}^{2/3}$,
- (3)
- ${R}_{2}\approx 3/{2}^{2/3}$,
- (4)
- ${R}_{2}>3/{2}^{2/3}$.

#### 4.4.1. The Quasi-Static Regimes, ${R}_{2}<0$ and ${R}_{2}>3/{2}^{3/2}$

#### 4.4.2. Sudden Transitions Between Branches

#### 4.4.3. Transitional Regime (${R}_{2}\approx 3/{2}^{2/3}$)

## 5. Stability Analysis for Real Solutions

#### 5.1. General Linear Stability Analysis

#### 5.2. Numerical Solutions

#### 5.3. Stability for Large Frequencies ($\omega \gg 1$)

## 6. Complex Solutions

#### 6.1. Numerical Simulations

#### 6.2. Solutions for Large Values of ${R}_{2}$

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

A | amplitude of convection |

B | real part of A |

c | heat capacity |

C | imaginary part of A |

d | height of the porous layer |

D | disturbance |

g | gravity |

k | wavenumber |

K | permeability |

p | pressure |

Ra | Darcy-Rayleigh number |

${\mathcal{R}}_{1},{\mathcal{R}}_{2}$ | Right hand sides of Equations (28) and (29) |

t | time |

$\mathcal{T}$ | nondimensional forcning period |

T | dimensional temperature |

T_{c} | upper (cold) boundary temperature |

T_{h} | lower (hot) boundary temperature |

u,v | horizontal velocities |

w | vertical velocity |

x,y | horizontal coordinates |

z | vertical coordinate |

Greek symbols | |

β | thermal expansion coefficient |

δ | amplitude of thermal imperfection |

ϵ | amplitude of convection |

θ | temperature |

μ | dynamic viscosity |

ρ | density |

σ | heat capacity ratio |

τ | scaled time |

ψ | streamfunction |

ϕ | porosity |

ω | frequency |

Subscripts, superscripts, and other symbols | |

c | critical value |

f | fluid |

s | solid |

- | dimensional quantity |

0, 1, 2, ⋯ | terms in a series expansion |

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**Figure 1.**Neutral curve for the classical Darcy-Bénard problem with $n=1$; see Equation (1). the critical values are given by $R{a}_{c}=4{\pi}^{2}$ and ${k}_{c}=\pi $.

**Figure 2.**The steady solution curves for the amplitude equation Equation (37). The upper branch is stable. The other two branches are unstable with respect to perturbations in phase, while the middle branch is also unstable with respect to perturbations in amplitude.

**Figure 3.**Numerical solutions of Equation (37) for initial conditions placed on a unit circle in the complex plane, for $Ra=-5$, $-1$, 0, 1, 2 and 5. Showing how solutions evolve to the unique solution on the positive real axis.

**Figure 4.**The effect of different Rayleigh numbers on real solutions of Equation (36) for the following selection of forcing periods: $\mathcal{T}=1$, 5, 20 and 100. The respective frequencies are $\omega =2\pi $, $0.4\pi $, $0.1\pi $ and $0.02\pi $. The uppermost curve corresponds to ${R}_{2}=3$; the dash-dotted curve to $\mathrm{Ra}=0$ with intermediate curves corresponding to intervals of $0.2$ in $Ra$; the dotted curves correspond to $Ra=-1$, $-2$, $-5$ and $-10$.

**Figure 5.**The bifurcation diagram for the given values frequencies/periods. The curves correspond to ${B}_{\mathrm{max}}$ and ${B}_{\mathrm{min}}$ over period. The dashed lines correspond to solutions for which $B(t=0)$ is negative, while continuous lines have $B(t=0)>0$.

**Figure 6.**Depiction of the bifurcation diagram corresponding to quasi-static real solutions of Equation (40) over half a period. The five frames correspond to $\omega t=0$, $\pi /4$, $\pi /2$, $3\pi /4$ and $\pi $. In each frame, and from left to right, the symbols indicate potential solutions for (i) ${R}_{2}<0$, (ii) $0<{R}_{2}<3/{2}^{2/3}$ and (iii) ${R}_{2}>3/{2}^{2/3}$. The solid circles depict how one might predict how an unsteady solution might evolve from the initial conditions given when $\omega t=0$, while the circles depict possible solutions but these do not arise with the chosen initial conditions.

**Figure 7.**An example of a catastrophic change in the qualitative nature of the solution for the small frequency, $\omega =\pi /100$. The solid line corresponds to ${R}_{2}=1.871666$ and the dashed line to ${R}_{2}=1.871665$.

**Figure 8.**Solution curves (continuous lines) for $\omega =\pi /100$ ($\mathcal{T}=200$) and for ${R}_{2}=1.4$ (lowest), $1.8$ and $2.0$ (uppermost). Also shown are the corresponding quasi-static solutions.

**Figure 9.**Solution of Equation (66) for $\mathcal{S}=-0.5$ (continuous line). The corresponding quasi-static solution of $3{\widehat{\widehat{B}}}^{2}=-\mathcal{S}\widehat{\widehat{t}}$ is represented by the dashed line.

**Figure 10.**Solution of Equation (73) for $\widehat{R}=-0.3$, $-1$, 0 and 1 (continuous lines). The corresponding quasi-static solutions are represented by the dashed lines.

**Figure 11.**Solutions of Equation (73) for values of $\widehat{R}$ which are close to the critical value, i.e., $\widehat{R}=-0.3054955\pm n\times {10}^{-7}$ with $n=-5,-4,\cdots ,4,5$ (continuous lines). The corresponding quasi-statics solution is represented by the dashed line.

**Figure 12.**Showing the value of ${R}_{2}$ as a function of $\omega $ at which solutions of Type II undergo a supercritical bifurcation to solutions of Type I (continuous line), and the neutral stability curve with respect to imaginary disturbances (dashed line).

**Figure 13.**Critical values of ${R}_{2}$ above which real solutions are unstable to iaginary disturbances (dashed line). Also shown are one-term (thin line) and two-term (thick line) approximations using a large-$\omega $ analysis.

**Figure 14.**The effect of different values of ${R}_{2}$ and $\omega $ on solution trajectories. Bold values of ${R}_{2}$ indicate values for which solutions are real. All solution trajectories follow that of the arrow shown in the top right subfigure.

**Figure 15.**Continuous curves show the solution trajectories for the given values of ${R}_{2}$. The left hand column corresponds to $\omega =\pi /2$ and the right hand column corresponds to $\omega =\pi /20$. Dashed lines indicate the large-${R}_{2}$ asymptotic solution.

$\mathit{\omega}/\mathit{\pi}$ | ${\mathit{R}}_{2}^{\mathbf{zero}}$ | ${\mathit{R}}_{2}^{\mathbf{bif}}$ |
---|---|---|

0 | $1.889881$ | $1.889881$ |

1/100 | $1.871665$ | $1.871665$ |

1/50 | $1.853286$ | $1.853286$ |

1/20 | $1.797093$ | $1.797080$ |

1/10 | $1.699355$ | $1.699006$ |

1/5 | $1.488797$ | $1.466053$ |

1/4 | $1.378595$ | $1.321533$ |

1/3 | $1.192159$ | $1.050448$ |

2/5 | $1.043338$ | $0.837670$ |

1/2 | $0.829189$ | $0.583587$ |

2/3 | $0.540125$ | $0.339267$ |

4/5 | $0.388046$ | $0.236835$ |

1 | $0.251850$ | $0.151874$ |

4/3 | $0.142320$ | $0.085479$ |

2 | $0.063326$ | $0.037995$ |

4 | $0.015856$ | $0.009499$ |

$\mathit{\omega}/\mathit{\pi}$ | ${\mathit{R}}_{2\mathit{c}}$ |
---|---|

0 | 1.88988158 |

0.001 | 1.84449388 |

0.002 | 1.83042922 |

0.005 | 1.79452727 |

0.01 | 1.74209700 |

0.02 | 1.64886998 |

0.05 | 1.41741816 |

0.1 | 1.12923353 |

0.2 | 0.74433223 |

0.3 | 0.56289546 |

$\mathit{\omega}/\mathit{\pi}$ | ${\mathit{R}}_{2\mathit{c}}$ | ${\mathit{R}}_{2\mathit{c},\mathbf{asymp}}$ | ${\mathit{R}}_{2\mathit{c}}-{\mathit{R}}_{2\mathit{c},\mathbf{asymp}}$ |
---|---|---|---|

0.4 | 0.30410142 | 0.30211086 | $1.99\times {10}^{-3}$ |

0.5 | 0.20030754 | 0.20020688 | $1.01\times {10}^{-4}$ |

0.6 | 0.14016556 | 0.14015740 | $8.16\times {10}^{-6}$ |

0.8 | 0.07910061 | 0.07910046 | $1.48\times {10}^{-7}$ |

1 | 0.05065108 | 0.05065108 | $7\times {10}^{-9}$ |

1.5 | 0.02251545 | 0.02251545 | |

2 | 0.01266511 | 0.01266511 | |

3 | 0.00562895 | 0.00562895 | |

5 | 0.00202642 | 0.00202642 | |

10 | 0.00050661 | 0.00050661 |

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**MDPI and ACS Style**

Jais, I.M.; Rees, D.A.S.
Horizontal Cellular Oscillations Caused by Time-Periodic Resonant Thermal Forcing in Weakly Nonlinear Darcy-Bénard Convection. *Fluids* **2017**, *2*, 60.
https://doi.org/10.3390/fluids2040060

**AMA Style**

Jais IM, Rees DAS.
Horizontal Cellular Oscillations Caused by Time-Periodic Resonant Thermal Forcing in Weakly Nonlinear Darcy-Bénard Convection. *Fluids*. 2017; 2(4):60.
https://doi.org/10.3390/fluids2040060

**Chicago/Turabian Style**

Jais, Ibrahim M., and D. Andrew S. Rees.
2017. "Horizontal Cellular Oscillations Caused by Time-Periodic Resonant Thermal Forcing in Weakly Nonlinear Darcy-Bénard Convection" *Fluids* 2, no. 4: 60.
https://doi.org/10.3390/fluids2040060