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Research Papers: Fundamental Issues and Canonical Flows

Numerical Modeling of Flow Through Phloem Considering Active Loading

[+] Author and Article Information
Prashanta Dutta

e-mail: dutta@mail.wsu.edu
School of Mechanical and Materials Engineering,
Washington State University,
Pullman, WA 99164

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 1, 2013; final manuscript received October 4, 2013; published online December 12, 2013. Assoc. Editor: Ali Beskok.

J. Fluids Eng 136(2), 021206 (Dec 12, 2013) (10 pages) Paper No: FE-13-1465; doi: 10.1115/1.4025869 History: Received August 01, 2013; Revised October 04, 2013

Transport through phloem is of significant interest in engineering applications, including self-powered microfluidic pumps. In this paper we present a phloem model, combining protein level mechanics with cellular level fluid transport. Fluid flow and sucrose transport through a petiole sieve tube are simulated using the Nernst–Planck, Navier–Stokes, and continuity equations. The governing equations are solved, using the finite volume method with collocated storage, for dynamically calculated boundary conditions. A sieve tube cell structure consisting of sieve plates is included in a two dimensional model by computational cell blocking. Sucrose transport is incorporated as a boundary condition through a six-state model, bringing in active loading mechanisms, taking into consideration their physical plant properties. The effects of reaction rates and leaf sucrose concentration are investigated to understand the transport mechanism in petiole sieve tubes. The numerical results show that increasing forward reactions of the proton sucrose transporter significantly promotes the pumping ability. A lower leaf sieve sucrose concentration results in a lower wall inflow velocity, but yields a higher inflow of water due to the active loading mechanism. The overall effect is a higher outflow velocity for the lower leaf sieve sucrose concentration because the increase in inflow velocity outweighs the wall velocity. This new phloem model provides new insights on mechanisms which are potentially useful for fluidic pumping in self-powered microfluidic pumps.

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Figures

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Fig. 1

(a) Schematic of a petiole section connecting the leaf to the stem. (b) Computational domain and associated boundary conditions for flow through a sieve tube. Schematic of flow penetration from the (c) bottom, and (d) top surfaces. For the bottom wall, water enters directly into the sieve. While for the top wall, in addition to entering directly, water will also first enter into the neighboring companion cell and then into the sieve tube.

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Fig. 2

Six-state model for potato STP adapted from Boorer et al. [17] for sucrose transport from outside (apoplast) to inside (leaf sieve). Each protein state is denoted as Pm (m = 1, 2,…,6). The forward reaction allows entry of sucrose into the sieve element from the apoplast. In the first state (P1) unbounded protein is open to the outside. Transition to the second state (P2) takes place when a proton (H+) from outside binds to P1. Next a sucrose molecule binds to P2 to form P3. This is followed by a conformational change from the protein being open to the outside to in (P3P4), bringing the proton and sucrose molecule inside the leaf sieve tube. The sucrose molecule leaves P4 to form P5 and, finally, the proton leaves from P5 resulting in P6. The cycle is completed by the change in confirmation from being open inwards P6 to outwards P1.

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Fig. 3

(a) Cross-stream (y) distributions far away from sieve plates for the u-velocity. (b) Vector plot for fluid flow past simulated sieve plate. (c) Cross-stream (y) distributions far away from sieve plates for petiole sieve sucrose concentration. In figures (a) and (c) the open circles are data points from simulations taken at x = 400 μm and the solid lines are parabolic curve fits.

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Fig. 4

Outflow velocity distribution for different proton dissociation rate constants k5 in units of s−1. All other simulation conditions are the same as in Fig. 3.

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Fig. 5

Effects of the proton dissociation rate constant k5 on (a) sucrose influx, and (b) sieve tube inflow, wall flow, and total outflow. All other simulation conditions are the same as in Fig. 3. Here Q is the volumetric flow per unit depth.

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Fig. 6

Petiole sieve tube wall inflow for different leaf sieve tube sucrose concentrations cLS. All other simulation conditions are the same as in Fig. 3. Here Qwall is the wall volumetric flow rate per unit depth.

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Fig. 7

Velocity distribution at the outlet for different leaf sieve tube sucrose concentrations cLS

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Fig. 8

Effect of sieve plates on flow through the petiole sieve tube. The outlet pressure difference between without and with sieve plates for different proton dissociation rate constants k5, is shown. All other simulation conditions are the same as in Fig. 3.

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