Analytic solutions of the interstitial fluid flow models

Abstract

In this paper, we present the analytic solutions of several continuum porous media models that describe the interstitial fluid flow in the interosseous membrane. We first compare the results of the Brinkman, Stokes and Darcy systems in describing the isotropic interstitial fluid flows. Our calculations show that the Stokes equations can well approximate the Brinkman equations when the Darcy number Da ≥ 0.2, while the Darcy model is an appropriate approximation to the Brinkman model in the interosseous membrane when Da ≤ 2×10−4. Yet, in most cases, the anisotropy dominates the interstitial fluid. Therefore, we build an anisotropic Darcy model and show that an isotropic model can be used as a suitable approximation when the ratio between the transverse and longitudinal permeabilities is no larger than 20. Lastly, we take the blood flow in capillaries into consideration as well and introduce the coupled Stokes-Darcy system to describe the cases comprising both the capillary and the interstitial domain. Our results reveal that the profile of the interface exchange flow is not exactly in the linear form as was widely adopted in the numerical simulation, instead, the flux near the artery and the vein is more significant, which in turn results in the increase of the maximum horizontal velocity in the interstitial space while the outflow rate remains the same.