Optimal reservoir conditions for fluid extraction through permeable walls in the viscous limit

TL;DR

This paper analyzes how peristaltic pumping in leaky channels affects fluid extraction efficiency, deriving a formal solution for the flow and identifying optimal reservoir conditions for maximum material transfer.

Contribution

It introduces a formal power series solution for viscous flow with permeable, contracting walls driven by hydrostatic pressure, extending beyond weakly permeable assumptions.

Findings

Pumping can both enhance and impede fluid extraction.

Optimal reservoir conditions exist for maximum material flow.

Flow solution applicable to biological transport systems.

Abstract

In biological transport mechanisms such as insect respiration and renal filtration, fluid travels along a leaky channel allowing exchange with systems exterior the the channel. The channels in these systems may undergo peristaltic pumping which is thought to enhance the material exchange. To date, little analytic work has been done to study the effect of pumping on material extraction across the channel walls. In this paper, we examine a fluid extraction model in which fluid flowing through a leaky channel is exchanged with fluid in a reservoir. The channel walls are allowed to contract and expand uniformly, simulating a pumping mechanism. In order to efficiently determine solutions of the model, we derive a formal power series solution for the Stokes equations in a finite channel with uniformly contracting/expanding permeable walls. This flow has been well studied in the case of weakly permeable channel walls in which the normal velocity at the channel walls is proportional to the wall velocity. In contrast we do not assume weakly driven flow, but flow driven by hydrostatic pressure, and we use Dacry's law to close our system for normal wall velocity. We use our flow solution to examine flux across the channel-reservoir barrier and demonstrate that pumping can either enhance or impede fluid extraction across channel walls. We find that associated with each set of physical flow and pumping parameters, there are optimal reservoir conditions that maximizes the amount of material flowing from the channel into the reservoir.