Structural Stability of the Brinkman–Forchheimer Equations for Flows in Porous Media With Variable Porosity

ABSTRACT In this paper, we are concerned with the structural stability (continuous dependence of solutions) of the Brinkman–Forchheimer model for flows in porous media. Structural stability for the case of constant porosity has been established by Payne and Straughan. Herein, we extend these results to porous media of variable porosity. The main differences when the porosity is no longer constant but a time‐independent field variable are the following: (i) the velocity field is not divergence‐free, (ii) the term related to the shear viscous stresses does not reduce to the Laplacian operator but involves the full deviatoric strain‐rate tensor, and (iii) there is an additional term due to normal viscous stresses. Herein, we establish continuous dependence of solutions with respect to the physical parameters entering the equations, namely, the shear and bulk viscosities and the coefficients of the linear (Darcy) and quadratic (Forchheimer) terms for the interfacial drag. More specifically, we show continuous dependence in the weighted norm, with the porosity being the weight. Finally, we provide upper and lower bounds for the rate of decay of the kinetic energy. According to them, the kinetic energy decays exponentially with time, but the decay rate depends on both the minimum and maximum porosities of the medium.