An asymptotic study to explain the role of active transport in models with countercurrent exchangers

We study a solute concentrating mechanism that can be represented by coupled transport equations with specific boundary conditions. Our motivation for considering this system is urine concentrating mechanism in nephrons. The model consists in 3 tubes arranged in a countercurrent manner. Our equations describe a countercurrent exchanger, with a parameter V which quantifies the active transport. In order to understand the role of active transport in the mechanism, we consider the limit V −→ ∞. We prove that when V goes to infinity, the system converges to a profile which stays uniformly bounded in V and which presents a boundary layer at the border of the domain. The effect is that the solute is concentrated at a specific point in the tubes. When considering urine concentration, this is physilogically optimal because the composition of final urine is determined at this point. Mathematics Subject Classification : 34B18, 34E99, 92C30 Key-words : Countercurrent, active transport, asymptotic analysis, boundary layer, urine concentration, kidney physiology.