# Onset of Convection in an Inclined Anisotropic Porous Layer with Internal Heat Generation

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and the Basic State

## 3. Linear Stability Analysis for Two-Dimensional Disturbances

#### 3.1. Reduction to ODE Eigenvalue Form

#### 3.2. Numerical Results

## 4. Linear Stability Analysis for Three-Dimensional Disturbances

#### 4.1. Reduction to ODE Eigenvalue Form

#### 4.2. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

c | phase velocity |

g | gravitational acceleration |

h | thickness of layer |

k | wavenumber |

${K}_{L}$ | longitudinal permeability |

${K}_{T}$ | transverse permeability |

p | pressure |

q | rate of heat generation |

Ra | Darcy-Rayleigh number |

T | dimensional temperature |

u | velocity in the x-direction |

v | velocity in the y-direction |

w | velocity in the z-direction |

x | coordinate up the layer |

y | coordinate across the layer |

z | spanwise coordinate |

Greek symbols | |

$\alpha $ | inclination angle |

$\beta $ | coefficient of thermal expansion |

$\theta $ | nondimensional temperature |

$\kappa $ | thermal diffusivity |

$\lambda $ | growth rate |

$\mu $ | dynamic viscosity |

$\xi $ | anisotropy ratio |

$\rho $ | density |

$\sigma $ | heat capacity ratio |

$\varphi $ | orientation of roll |

$\psi $ | streamfunction |

Subscripts, superscripts, and other symbols | |

* | dimensional |

′ | differentiation with respect to y |

_ | amplitude |

0 | reference quantity |

1 | perturbation |

b | steady basic flow |

c | critical value |

## References

- Horton, C.W.; Rogers, F.T. Convection currents in a porous medium. J. Appl. Phys.
**1945**, 16, 367–370. [Google Scholar] [CrossRef] - Lapwood, E.R. Convection of a fluid in a porous medium. Proc. Camb. Philos. Soc.
**1948**, 44, 508–521. [Google Scholar] [CrossRef] - Kulacki, F.A.; Ramchandani, R. Hydrodynamic instability in porous layer saturated with heat-generating fluid. Wärme-Stoffübertrag
**1975**, 8, 179–185. [Google Scholar] [CrossRef] - Gasser, R.D.; Kazimi, M.S. Onset of convection in a porous medium with internal heat generation. ASME J. Heat Transf.
**1976**, 98, 49–54. [Google Scholar] [CrossRef] - Buretta, R.J.; Berman, A.S. Convective heat transfer in a liquid saturated porous layer. ASME J. Appl. Mech.
**1976**, 43, 249–253. [Google Scholar] [CrossRef] - Hardee, H.C.; Nilson, R.H. Natural convection in porous media with heat generation. Nucl. Sci. Eng.
**1977**, 63, 119–132. [Google Scholar] [CrossRef] - Hwang, I.T.; Marr, W.W. Onset of thermal-convection in a fluid-saturated porous layer with heat source. Trans. Am. Nuclear Soc.
**1977**, 27, 655–656. [Google Scholar] - Tveitereid, M. Thermal convection in a horizontal porous layer with internal heat sources. Int. J. Heat Mass Transf.
**1977**, 20, 1045–1050. [Google Scholar] [CrossRef] - Rhee, S.J.; Dhir, V.K.; Catton, I. Natural convection heat transfer in beds of inductively heated particles. ASME J. Heat Transf.
**1978**, 100, 78–85. [Google Scholar] [CrossRef] - Kulacki, F.A.; Freeman, R.G. A note on thermal convection in a saturated, heat generating porous layer. ASME J. Heat Transf.
**1979**, 101, 169–171. [Google Scholar] [CrossRef] - Barletta, A.; Celli, M.; Nield, D.A. Unstable buoyant flow in an inclined porous layer with an internal heat source. Int. J. Therm. Sci.
**2014**, 79, 176–182. [Google Scholar] [CrossRef] - Nandal, R.; Mahajan, A. Linear and nonlinear stability analysis of a Horton-Rogers-Lapwood problem with an internal heat sources and Brinkman effects. Transp. Porous Media
**2017**, 117, 261–280. [Google Scholar] [CrossRef] - Nouri-Borujerdi, A.; Noghrehabadi, A.R.; Rees, D.A.S. Influence of Darcy number on the onset of convection in a porous layer with a uniform heat source. Int. J. Therm. Sci.
**2008**, 47, 1020–1025. [Google Scholar] [CrossRef] - Nouri-Borujerdi, A.; Noghrehabadi, A.R.; Rees, D.A.S. The onset of convection in a horizontal porous layer with uniform heat generation using a thermal non-equilibrium model. Transp. Porous Media
**2007**, 69, 343–357. [Google Scholar] [CrossRef] - Nield, D.A.; Kuznetsov, A.V. The onset of convection in a horizontal porous layer with spatially non-uniform internal heating. Transp. Porous Media
**2016**, 111, 541–553. [Google Scholar] [CrossRef] - Nield, D.A.; Kuznetsov, A.V. Onset of convection in with internal heating in a weakly heterogeneous porous medium. Transp. Porous Media
**2013**, 98, 543–552. [Google Scholar] [CrossRef] - Kuznetsov, A.V.; Nield, D.A. The effect of strong heterogeneity on the onset of convection induced by internal heating in a porous medium: A layered model. Transp. Porous Media
**2013**, 99, 85–100. [Google Scholar] [CrossRef] - Matta, A.; Hill, A.A. Double-diffusive convection in an inclined porous layer with a concentration-based internal heat source. Contin. Mech. Thermodyn.
**2018**, 30, 165–173. [Google Scholar] [CrossRef] - Mahajan, A.; Nandal, R. Stability of an anisotropic porous layer with internal heat source and Brinkman effects. Spec. Top. Rev. Porous Media
**2019**, 10, 65–87. [Google Scholar] [CrossRef] - Yadav, D.; Wang, J.; Lee, J. Onset of Darcy-Brinkman convection in a rotating porous layer induced by purely internal heating. J. Porous Media
**2017**, 20, 691–706. [Google Scholar] [CrossRef] - Malashetty, M.S.; Swamy, M.S. The onset of convection in a viscoelastic liquid saturated anisotropic porous layer. Transp. Porous Media
**2007**, 67, 203–218. [Google Scholar] [CrossRef] - Yovogan, J.; Miwadinou, C.H.; Claude, E.V.; Degan, G. Effect of anisotropy in permeability on thermal convection of viscoelastic fluids in rotating porous layer heated from below. Aust. J. Mech. Eng.
**2018**. [Google Scholar] [CrossRef] - Raghunatha, K.R.; Shivakumara, I.S.; Sowbhagya, A.A. Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer. Appl. Math. Mech.
**2018**, 39, 653–666. [Google Scholar] [CrossRef] - Abdelhafez, M.A.; Tsybulin, V.G. Modeling of anisotropic convection for the binary fluid in porous medium. Comput. Res. Model.
**2018**, 10, 801–816. [Google Scholar] [CrossRef] - Yadav, D.; Kim, M.C. Theoretical and numerical analyses on the onset and growth of convective instabilities in a horizontal anisotropic porous medium. J. Porous Media
**2014**, 17, 1061–1074. [Google Scholar] [CrossRef] - Rees, D.A.S.; Storesletten, L. The linear instability of a thermal boundary layer with suction in an anisotropic porous medium. Fluid Dyn. Res.
**2002**, 30, 155–168. [Google Scholar] [CrossRef] - Rees, D.A.S.; Postelnicu, A. The onset of convection in an inclined anisotropic porous layer. Int. J. Heat Mass Transf.
**2001**, 44, 4127–4138. [Google Scholar] [CrossRef] - Postelnicu, A.; Rees, D.A.S. The onset of convection in an anisotropic porous layer inclined at a small angle from the horizontal. Int. Commun. Heat Mass Transf.
**2001**, 28, 641–650. [Google Scholar] [CrossRef] - Rees, D.A.S.; Storesletten, L.; Postelnicu, A. The onset of convection in an inclined anisotropic porous layer with oblique principle axes. Transp. Porous Media
**2006**, 62, 139–156. [Google Scholar] [CrossRef] - Capone, F.; Gentile, M.; Hill, A.A. Penetrative convection via internal heating in anisotropic porous media. Mech. Res. Commun.
**2010**, 37, 441–444. [Google Scholar] [CrossRef] - Rees, D.A.S.; Bassom, A.P. Onset of Darcy-Bénard convection in an inclined porous layer heated from below. Acta Mech.
**2000**, 144, 103–118. [Google Scholar] [CrossRef] - Storesletten, L.; Tveitereid, M. Onset of convection in an inclined porous layer with anisotropic permeability. Appl. Mech. Eng.
**1999**, 4, 575–587. [Google Scholar]

**Figure 1.**Definition sketch of the inclined layer showing the coordinate directions, the inclination angle, $\alpha $, and the direction of the rolls, $\varphi $, relative to the x-direction. (

**a**) side view; (

**b**) plan view.

**Figure 2.**Showing neutral curves for transverse rolls for $\xi =5$, $\xi =2$, $\xi =0.5$ and $\xi =0.2$, and for different inclinations, $\alpha $, from ${0}^{\circ}$ onwards in increments of ${1}^{\circ}$. Black curves correspond to mode 1 and blue curves to mode 2. Red dotted curves correspond to $\alpha $ being a multiple of ${10}^{\circ}$. The black dotted line is the locus of minima while the black disk corresponds to the isola point, i.e., the largest inclination for which there is marginal stability. The red disk corresponds to a saddle point.

**Figure 3.**Showing the variation of ${\mathrm{Ra}}_{c}$ for transverse roll disturbances at critical points on the neutral curve for $\xi =0.2$ (far right), $0.3$, $0.4$, $0.5$, $0.7$, 1, 2, 3, 5 and 10 (far left). The black disks correspond either to isola points or to saddle points. The dotted lines correspond to isolae (lower curve) and saddle points (upper curve) in the surface, $\alpha =\alpha (\mathrm{Ra},k)$.

**Figure 4.**Showing the variation of ${k}_{c}$ for transverse roll disturbances at critical points on the neutral curve for $\xi =0.2$ (uppermost), $0.3$, $0.4$, $0.5$, $0.7$, 1, 2, 3, 5 and 10 (lowest). The black disks correspond either to isola points or to saddle points.

**Figure 5.**Showing the variation of c, the phase velocity of transverse roll disturbances, at critical points on the neutral curve for $\xi =0.2$, $0.3$, $0.4$, $0.5$, $0.7$, 1, 2, 3, 5 and 10. The black disks correspond either to isola points or to saddle points.

**Figure 6.**Showing the disturbance isotherms for one spatial period for transverse rolls when $\xi =5$. The isotherms are at 20 equally-spaced intervals in each case.

**Figure 7.**For $\xi =0.5$. The variation with $\alpha $ of the critical value of $\mathrm{Ra}cos\alpha $ for the roll orientations, $\varphi ={0}^{\circ}$, ${5}^{\circ}$, ${10}^{\circ}$, …, ${90}^{\circ}$.

**Figure 8.**For $\xi =2$. (

**a**) the variation with $\alpha $ of the critical value of $\mathrm{Ra}cos\alpha $ for the roll orientations, $\varphi ={0}^{\circ}$, ${5}^{\circ}$, ${10}^{\circ}$, …, ${90}^{\circ}$. (

**b**) Showing a close-up view of the smooth transition between transverse and longitudinal rolls.

**Figure 9.**For $\xi =5$. (

**a**) the variation with $\alpha $ of the critical value of $\mathrm{Ra}cos\alpha $ for the roll orientations, $\varphi ={0}^{\circ}$, ${5}^{\circ}$, ${10}^{\circ}$, …, ${90}^{\circ}$. (

**b**) Showing a close-up view of the smooth transition between transverse and longitudinal rolls.

**Figure 10.**As functions of the inclination, $\alpha $: (

**a**) the critical value of $\mathrm{Ra}cos\alpha $ which has been minimised over both wavenuber and roll orientation; (

**b**) the corresponding wavenumber; (

**c**) the corresponding roll orientation in degrees. For $\xi =1$, $1.1$, $1.5$ and 2 (continuous lines) and for $\xi =3$, 5 and 10 (dotted lines).

**Figure 11.**Showing the different regions in $(\xi ,\alpha )$–space wherein longitudinal rolls, oblique rolls (between the two curves) and transverse rolls form the most unstable planform of convection.

$\mathit{\xi}$ | for $\mathit{\varphi}={90}^{\circ}$ | for $\mathit{\varphi}={0}^{\circ}$ |
---|---|---|

125 | $3.542304$ | $3.660913$ |

250 | $2.512439$ | $2.595616$ |

500 | $1.779291$ | $1.837828$ |

1000 | $1.259118$ | $1.300388$ |

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**MDPI and ACS Style**

Storesletten, L.; Rees, D.A.S.
Onset of Convection in an Inclined Anisotropic Porous Layer with Internal Heat Generation. *Fluids* **2019**, *4*, 75.
https://doi.org/10.3390/fluids4020075

**AMA Style**

Storesletten L, Rees DAS.
Onset of Convection in an Inclined Anisotropic Porous Layer with Internal Heat Generation. *Fluids*. 2019; 4(2):75.
https://doi.org/10.3390/fluids4020075

**Chicago/Turabian Style**

Storesletten, Leiv, and D. Andrew S. Rees.
2019. "Onset of Convection in an Inclined Anisotropic Porous Layer with Internal Heat Generation" *Fluids* 4, no. 2: 75.
https://doi.org/10.3390/fluids4020075