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**A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations.**
*(English)*
Zbl 1340.35264

In this paper, the authors face an important question concerning the validation of the Brinkman-Forchheimer equations. More precisely, the governing laws in porous media have been adapted based on the assumption of the validity of the continuum hypothesis. These conservation laws have been postulated based on upscaling the conservation laws at the fluid continuum level using the theory of volume averaging, the method of homogenization, the theory of mixtures, etc. Therefore, to obtain field equations that are workable, researchers suggested terms to the governing equations in addition to some properties of the porous material. Then, it remains to experimentalists and theoreticians to validate these terms. Darcy’s law suggests that the mass flux and the pressure gradient are proportional and the relationship between them is linear. This linear relationship has been later found to be valid for values of Reynolds number less than one. As the Reynolds number becomes greater than one, this linear relationship is no longer valid. To account for such nonlinearities, Forchheimer suggested a quadratic term of the velocity to be included when considering momentum balance. Furthermore, Brinkman considered another term to account for the possible no slip condition once a confining wall exists. The Brinkman model is believed to be accurate when the flow velocity is too large for Darcy’s law to be valid, and additionally the porosity is not too small. In this article, the authors prove the continuous dependence of the solution on Brinkman’s and Forchheimer’s coefficients as well as on the initial data and external forces. Moreover, they study a perturbed compressible system that approximates the Brinkman-Forchheimer equations. Finally, they propose a time discretization of the perturbed system by using a semi-implicit Euler scheme and a lowest-order Raviart-Thomas element for the spatial discretization.

Reviewer: Vincenzo Vespri (Firenze)

### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35K55 | Nonlinear parabolic equations |

76S05 | Flows in porous media; filtration; seepage |