Multiscale computational analysis of the steady fluid flow through a lymph node.

Lymph Nodes (LNs) are crucial to the immune and lymphatic systems, filtering harmful substances and regulating lymph transport. LNs consist of a lymphoid compartment (LC) that forms a porous bulk region, and a subcapsular sinus (SCS), which is a free-fluid region. Mathematical and mechanical challenges arise in understanding lymph flow dynamics.

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 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.

A Mathematical Description of the Flow in a Spherical Lymph Node.

The motion of the lymph has a very important role in the immune system, and it is influenced by the porosity of the lymph nodes: more than 90% takes the peripheral path without entering the lymphoid compartment. In this paper, we construct a mathematical model of a lymph node assumed to have a spherical geometry, where the subcapsular sinus is a thin spherical shell near the external wall of the lymph node and the core is a porous material describing the lymphoid compartment. For the mathematical formulation, we assume incompressibility and we use Stokes together with Darcy-Brinkman equation for the flow of the lymph.