We investigate a convective Brinkman--Forchheimer problem coupled with a heat equation. The investigated model considers thermal diffusion and viscosity depending on the temperature. We prove the existence of a solution without restriction on the data and uniqueness when the solution is slightly smoother and the data is suitably restricted. We also propose a finite element discretization scheme for the considered model and derive convergence results and a priori error estimates. Finally, we illustrate the theory with numerical examples.

翻译：我们研究了一个与热方程耦合的对流Brinkman-Forchheimer问题。所研究的模型考虑了依赖于温度的热扩散和粘度。我们证明了在不限制数据条件下解的存在性，以及在解略为光滑且数据适当限制条件下的唯一性。我们还为所考虑的模型提出了一个有限元离散化格式，并推导了收敛结果和先验误差估计。最后，我们通过数值算例对理论进行了验证。