Effective governing equations for dual porosity Darcy–Brinkman systems subjected to inhomogeneous body forces and their application to the lymph node

Abstract

We derive the homogenized governing equations for a double porosity system where the fluid flow within the individual compartments is governed by the coupling between the Darcy and the Darcy–Brinkman equations at the microscale , and are subjected to inhomogeneous body forces. The homogenized macroscale results are obtained by means of the asymptotic homogenization technique and read as a double Darcy differential model with mass exchange between phases. The role of the microstructure is encoded in the effective hydraulic conductivities which are obtained by solving periodic cell problems whose properties are illustrated and compared. We conclude by solving the new model by means of a semi-analytical approach under the assumption of azimuthal axisymmetry to model the movement of fluid within a lymph node.

Multiscale computational analysis of the Steady Multiscale Fluid Flow through a Lymph Node

“…In Figure 4 we can see the interstitial pressure p m values in the LC varying the parameter L p . We have similar behavior to the one found in (Girelli et al, 2023): increasing L p decreases the minimum of the interstitial pressure p m (and increasing the maximum of the blood vessels pressure p v ) and moves the minimum towards the center of the node. This behavior is due to a combination of the pressure variation given by the pressure of the Stokes flow in the SCS and the fluid exchange between phases.…”

“…This behavior is due to a combination of the pressure variation given by the pressure of the Stokes flow in the SCS and the fluid exchange between phases. The values we found with these simulations are similar to the ones found in (Girelli et al, 2023) but slightly different: this is why it is important to take into account the coupling between the SCS and the LC.…”

“…Here we summarize the model: see (Girelli et al, 2023) for more details about the derivation of the model.…”

“…1.1 × 10 −6 mm 3 s mg Hydraulic conductivity of the blood vessels calculated using the Kozeny-Carman formula (Kozeny, 1927;Carman, 1997;Girelli et al, 2023). Km 3.65 × 10 −9 mm 3 s mg Macroscopic interstitial hydraulic conductivity (solving system (5)).…”

“…For a complete review, we refer to (Girelli et al, 2023, Appendix B) Fig. 2 The 3D simplified spherical geometry of our problem, inspired by a mouse popliteal lymph node as in (Giantesio et al, 2022;Girelli et al, 2023).…”