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Osmotically driven flows and maximal transport rates in systems of long, linear porous pipes

Hanna Rademaker,K. Jensen,T. Bohr

Published 2016 in arXiv: Fluid Dynamics

ABSTRACT

We study the flow of water and solutes in linear cylindrical pipes with semipermeable walls (membranes), driven by concentration differences across the membranes, inspired by the sieve tubes in conifer needles. The aim is to determine the efficiency of such systems. For single pipes, we assume that the velocity at the entrance (the tip of the needle) is zero, and we determine the velocity profile throughout the pipe and the outflow at the end of the pipe, where the pressure is specified. This is done for the particular case where the concentration of the solute is constant inside the pipe, and it is shown that the system has a characteristic length scale $L_{\text{eff}}$ depending on the pipe radius, the permeability of the wall and the viscosity of the fluid such that pipes with lengths $L \gg L_{\text{eff}}$ will contain a stagnant zone from the entrance, where the velocity is very small. The outflow comes from a region of length $L_{\text{eff}}$ near the end, and the increase of velocity, if the pipe is made longer, is marginal. We show that relaxing the assumption of constant solute concentration $c_0$ cannot lead to larger outflows, as long as the local concentration never exceeds $c_0$ and we determine viscous dissipation, including bulk dissipation in the pipe and in the flow through the pores of the membrane. We generalise these results to systems of interacting parallel, cylindrical pipes with a power law distribution of lengths (as in the sieve tubes of conifer needles). For constant concentration we give an analytical solution for the velocity profile in terms of modified Bessel functions and show that the results are surprisingly similar to the single pipe results regarding the stagnant zone and value of $L_{\text{eff}}$.The biological context and some of the mathematical results have been described in [1].

PUBLICATION RECORD

  • Publication year

    2016

  • Venue

    arXiv: Fluid Dynamics

  • Publication date

    2016-10-28

  • Fields of study

    Physics, Environmental Science

  • Identifiers
    arXiv 1610.09175
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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