### Abstract

The present paper is the first to consider Darcy–Bénard–Bingham convection. A Bingham fluid saturates a horizontal porous layer that is subjected to heating from below. It is shown that this simple extension to the classical Darcy–Bénard problem is linearly stable to small-amplitude disturbances but nevertheless admits strongly nonlinear convection. The Pascal model for a Bingham fluid occupying a porous medium is adopted, and this law is regularized in a frame-invariant manner to yield a set of two-dimensional governing equations that are then solved numerically using finite difference approximations. A weakly nonlinear theory of the regularized Pascal model is used to show that the onset of convection is via a fold bifurcation. Some parametric studies are performed to show that this non-linear onset of convection arises at increasing values of the Darcy–Rayleigh number as the Rees–Bingham number increases and that, for a fixed Rees–Bingham number, the wavenumber at which the rate of heat transfer is maximized increases with the Darcy–Rayleigh number.

Original language | English |
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Article number | 084107 |

Number of pages | 11 |

Journal | Physics of Fluids |

Volume | 32 |

Issue number | 8 |

Early online date | 21 Aug 2020 |

DOIs | |

Publication status | Published - 31 Aug 2020 |

## Cite this

Rees, D. A. S. (2020). Darcy-Bénard-Bingham convection.

*Physics of Fluids*,*32*(8), [084107]. https://doi.org/10.1063/5.0018775