Multiscale homogenization for dual porosity time-dependent Darcy-Brinkman/Darcy coupling and its application to the lymph node.

We study the coupling between time-dependent Darcy-Brinkman and the Darcy equations at the microscale subjected to inhomogeneous body forces and initial conditions to describe a double porosity problem. We derive the homogenized governing equations for this problem using the asymptotic homogenization technique, and as macroscopic results, we obtain a coupling between two Darcy equations, one of which with memory effects, with mass exchange between phases. The memory effects are a consequence of considering the time dependence in the Darcy-Brinkman equation, and they allow us to study in more detail the role of time in the problem under consideration. After the formulation of the model, we solve it in a simplified setting and we use it to describe the movement of fluid within a vascularized lymph node.