Journal of Experimental Botany, Vol. 53, No. 373, pp. 1411-1419,
June 2002
© 2002 Oxford University Press
Original Papers |
Revisiting the Münch pressure–flow hypothesis for long-distance transport of carbohydrates: modelling the dynamics of solute transport inside a semipermeable tube
Horticultural Research, Batchelor Research Centre, Private Bag 11030, Palmerston North, New Zealand
Received 5 September 2001; Accepted 25 January 2002
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Abstract |
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A mathematical model of the Münch pressure–flow hypothesis for long-distance transport of carbohydrates via sieve tubes is constructed using the Navier–Stokes equation for the motion of a viscous fluid and the van't Hoff equation for osmotic pressure. Assuming spatial dimensions that are appropriate for a sieve tube and ensuring suitable initial profiles of the solute concentration and solution velocity lets the model become mathematically tractable and concise. In the steady-state case, it is shown via an analytical expression that the solute flux is diffusion-like with the apparent diffusivity coefficient being proportional to the local solute concentration and around seven orders of magnitude greater than a diffusivity coefficient for sucrose in water. It is also shown that, in the steady-state case, the hydraulic conductivity over one metre can be calculated explicitly from the tube radius and physical constants and so can be compared with experimentally determined values. In the time-dependent case, it is shown via numerical simulations that the solute (or water) can simultaneously travel in opposite directions at different locations along the tube and, similarly, change direction of travel over time at a particular location along the tube.
Key words: Münch pressure–flow hypothesis, Navier–Stokes equation, phloem transport, van't Hoff equation.
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Introduction |
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Long-distance transport of carbohydrate, the major substrate for plant growth, occurs within the phloem vascular tissue. Phloem vasculature consists of sieve elements that are approximately 20–30 µm in diameter and 100–500 µm in length and are aligned end-to-end (20–100 sieve elements per cm) to form a continuous, membrane lined conduit (MacRobbie, 1971
; Canny, 1973). The total length of a file of connected sieve elements does not appear to have been measured, but is thought to extend from the leaves to all carbohydrate sinks of annual plants. Large perennial species, such as trees, require lengths of at least an order of magnitude greater than annuals which is possibly achieved by phloem unloading from one file and reloading into another (Lang, 1979), although this does not appear to have been demonstrated. Movement of material from one vascular trace to another appears to occur commonly (Eschrich, 1975; Thorpe and Minchin, 1996) so extremely long open files of conducting sieve tubes are probably unnecessary.
A number of qualitative experimental observations have been thought to be in conflict with existing mathematical models of the Münch pressure–flow hypothesis. Firstly, the long-distance transport of carbohydrates via the phloem appears diffusion-like in some respects (Mason and Maskell, 1928; MacRobbie, 1971), although molecular diffusion itself has been discarded as a transport mechanism. Both the estimated mass transfer rates (Milburn, 1975) and the observed dynamics of radioactive tracer profiles (Minchin and Troughton, 1980) have shown that molecular diffusion of carbohydrates is far too slow. Secondly, a radial flux of water across the sieve tube walls has been shown to be an important feature of the long-distance transport of carbohydrates (Minchin and Thorpe, 1982; Van Bel, 1990; Baker and Milburn, 1994). This flux of water is found to be, amongst other things, correlated with the concentration and movement of carbohydrates within the sieve tubes. Thirdly, it has been shown that carbohydrates can simultaneously travel in opposite directions at different locations inside a sieve tube (Trip and Gorham, 1968; Eschrich, 1975) and can change their direction of travel at one particular location inside a sieve tube over a period of time (Geiger, 1987; Turgeon, 1989). Such directionality observations could be attributed to the spatial and temporal dynamics of the competing sources and sinks of carbohydrates. It is the aim of this paper to construct a mathematical model based on the Münch pressure–flow hypothesis that can reproduce these qualitative observations of phloem transport.
This paper presents a model based on the Münch pressure–flow hypothesis constructed from the Navier–Stokes equation for the motion of a viscous fluid and the van't Hoff equation for osmotic pressure. The derivation is for a solution inside a long, narrow, rigid tube with no radial fluxes of solute through the tube walls. Spatial dimensions that are appropriate for a sieve tube and suitable initial profiles of the solute concentration and solution velocity are adopted to make the model more mathematically tractable and concise. Firstly, the solute flux, the longitudinal water flux, and the radial water flux for the steady-state case are calculated. The expression for the solute flux is then shown to be similar to Fick's law of diffusion and an expression for the hydraulic conductivity is calculated from the tube radius and physical constants and then compared to experimentally determined values. Secondly, two numerical simulations are used to illustrate some key temporal and spatial dynamics of the solute and water fluxes that arise from the relaxation of a non-uniform solute concentration profile.
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The mathematical model |
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Governing equations
The aim of this section is to derive a set of expressions to describe, analyse, and simulate the dynamics of solute transport inside a long, narrow, rigid tube immersed in a water reservoir that stays at a constant hydrodynamic pressure (c.f. the models of Christy and Ferrier, 1973
; Tyree and Dainty, 1975; Sheehy et al., 1995). A schematic diagram of the modelled tube is shown in Fig. 1. It is firstly assumed that the radial component of the velocity of the solution inside the tube can be regarded as negligible in comparison with the longitudinal component of the velocity of the solution. It is secondly assumed that, due to the narrowness of the tube, the radial component of molecular diffusion smoothes out radial variations in the solute concentration (Tyree and Dainty, 1975). The model presented here is focused on interpreting the mechanism and resulting dynamics of the solute per se so it will be further assumed that there is no radial flux of solute through the tube walls. Although the loading and unloading of carbohydrates from sieve tubes are of fundamental importance in plants, it is, for the intents and purposes of this paper, considered more appropriate to construct and then interpret a model of long-distance transport that does not take these fluxes into account.
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The longitudinal component of the velocity of the solution (i.e. the solute and water) inside the tube is denoted by v(r, z, t) and the solute concentration is denoted by C(z, t) where r and z are the radial coordinate and longitudinal coordinate of a cylindrical coordinate system and t represents time. The Navier–Stokes equation describes the motion of a viscous fluid and is typically regarded as the starting point for formulating a time-dependent fluid dynamics problem (Landau and Lifschitz, 1987
; Phillips and Dungan, 1993). The one-dimensional form of this partial differential equation in a cylindrical coordinate system is
 | (1) |
is the density of the solution, and µ is the viscosity of the solution. The terms on the left-hand-side account for the acceleration of the solution, whereas the terms on the right-hand-side account for the hydrodynamic pressure gradient and viscous nature of the solution. For simplicity, the solution viscosity will be treated as a constant and the force due to gravity has not been taken into account.
A convenient assumption for formulating the Münch hypothesis is that the hydrodynamic pressure P inside the tube is always maintained at an osmotic equilibrium with respect to the surrounding water reservoir (Thornley and Johnson, 1990; Minchin et al., 1993). The dynamic equilibrium thereby results in a local pressure difference across the tube walls and so the hydrodynamic pressure inside the tube can be expressed algebraically as
 | (2) |
is the osmotic pressure and P0 is the constant hydrodynamic pressure of the water surrounding the tube. It is noted that the effect of radial water fluxes on the local hydrodynamic pressure surrounding the tube is neglected. Assuming a dynamic equilibrium implies that the resistance to the radial flux of water due to the permeability of the tube wall is neglected (i.e. the hydraulic conductivity is infinite) and so, as shown later in the derivation, the accompanying radial flux of water across the tube wall can then be calculated independently. Including the effect of the permeability of the tube wall increases the complexity of the model and thereby makes it much more difficult to interpret both analytically and via numerical simulations (Goeschl et al., 1976; Smith et al., 1980; Phillips and Dungan, 1993). The van't Hoff equation for osmotic pressure of a dilute solution can be written as
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 | (4) |
. Secondly, it will be assumed that there are no large, localized variations in the solution velocity along the tube, which means that the longitudinal solute velocity profile is regarded as smooth. Hence the two terms involving derivatives of v along the tube (i.e. the derivatives of v with respect to z) in equation (4) can be treated as insignificant when compared to the other more dominant terms (e.g. the pressure term or the viscosity term) and can be dropped. Thus equation (4) can be approximated as
 | (5) |
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The dynamics of the solute concentration inside the tube can be expressed by way of a conservation-of-mass equation for the solute:
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Individual flux expressions
The volume flux passing over the cross-sectional area at a particular location along the tube Qvolume(z, t) is given by
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 | (8) |
 | (9) |
). This relationship can be expressed as
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water is the density of water and V is the partial molar volume of the solute. Thus the solute flux and longitudinal water flux are in the same direction at any given location along the tube. The radial flux of water passing through a unit length of the tube wall Jradial(z, t) can be calculated by considering the volume flux balance for a stationary volume element and then taking the limit (Fig. 1). This relationship can be expressed as
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The sign of the first-order derivative on the right-hand-side of this equation determines the direction of the radial flux of water at a particular location along the tube. Namely, if the sign is negative then Jradial(z, t) is positive and so water will be incoming at that location and, conversely, if the sign is positive then Jradial(z, t) is negative and so water will be outgoing at that location. Similarly, if the derivative itself is zero then Jradial(z, t) is also zero.
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Results |
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The steady-state case
The steady-state solution for the situation where the solute concentration is held constant at the two ends of the tubes can be expressed analytically. If the concentration at the left-hand end (i.e. z=0) is C0 and the concentration at the right-hand end (i.e. z=l) is Cl then the steady-state form of equation (5) can be integrated twice to give
 | (12a) |
 | (12b) |
 | (13) |
; Goeschl et al., 1976), implies that the apparent diffusivity coefficient is around seven orders of magnitude greater than the diffusivity coefficient for sucrose in stationary water. Meanwhile, the respective water fluxes are
 | (14a) |
 | (14b) |
water is the density of water.
Using solute concentration values of C0=300 kg m-3 (around 876 mol m-3/0.876 mol l-1) and Cl=150 kg m-3 (around 438 mol m-3/0.438 mol l-1) produces a maximum value of the longitudinal component of the solution velocity at around 5x10-2 m s-1 which is much greater than the typical solution speed of 1x10-3 m s-1 (MacRobbie, 1971). On a similar note, it has been estimated from experimental data that the apparent diffusivity coefficient is approximately four orders of magnitude greater than the diffusivity coefficient for sucrose in water (MacRobbie, 1971). Hence the apparent diffusivity coefficient for the model developed here is around three orders of magnitude greater than values that have been estimated from experimental data. These differences serve to highlight the simplifications made during the model development.
It is known that there are sieve plates inside the sieve tubes (MacRobbie, 1971; Spanner, 1978) which are likely to provide an additional resistance to the translocation of carbohydrates along a sieve tube. The conductivity for the flux of solution between adjacent sieve elements within the steady-state models developed previously (Christy et al., 1973; Goeschl et al., 1976) is assigned a value of 1.02x10-6 m4 J-1 s-1. Using the values given in Table 1 of Goeschl et al. (Goeschl et al., 1976), the conductivity of the tube for 1 m is
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 | (018) |
; Goeschl et al., 1976) produce slower solute and water fluxes because of the comparatively small value given to the hydraulic conductivity of the tube (Milburn, 1975).
A possible modification could be made to the model developed above by judiciously introducing a damping multiplier, 
say, in front of the viscosity term in the Navier–Stokes equation. The mathematical analysis thereafter would essentially remain unchanged except that the lumped parameters h and L would now have appearing in the denominators. Specifying a value of
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; Goeschl et al., 1976) and would also reduce both the maximum longitudinal component of the solution velocity and the apparent diffusivity coefficient.
The time-dependent case
Analytical solutions cannot be obtained in the time-dependent situation so numerical simulations will be used to help illustrate some key features of the spatial and temporal dynamics of the model. The solute is sucrose, the tube radius a is 1x10-5 m, the tube length l is 3x10-1 m, and the other constants are given in Table 1. Non-uniform profiles of the initial solute concentration inside the tube are then specified and key features of the relaxation toward a steady-state are observed. These profiles must be smooth and contain no large, localized variations in the gradient of the solute concentration profile (i.e. smooth profiles) so that approximations used in the model development are satisfied. It is noted that the damping coefficient defined earlier has not been included in these simulations. The inclusion of the damping factor defined in the steady-state case given earlier would slow down the relaxation process by between one to two orders of magnitude.
A numerical code based on the exact time-dependent equations using a Crank–Nicholson scheme (Smith, 1985; Press et al., 1988) is constructed in order to simulate the dynamics of the solution velocity and solute concentration. The tube is resolved with a large number of elements and the solute concentration for each element is then evaluated using equations (5) and (6), respectively. The total longitudinal water flux and the radial water flux per unit length for each element can be evaluated using discrete forms of equations (10) and (11), respectively. Here the concentration gradient at both ends is fixed at zero so that neither the solute nor the water enters or exits the tube across these boundaries. Since solute cannot enter or exit the tube, the steady-state situation will have a uniform solute concentration along the tube and both the accompanying water fluxes will be zero.
Simulation 1
The first simulation illustrates the diffusion-like characteristics of the solute flux and key features of the accompanying water fluxes. To achieve this result, an initial concentration profile is chosen such that the left-hand side of the tube starts off as a source-like region (i.e. a region of high concentration) and the right-hand side of the tube starts off as a sink-like region (i.e. a region of low concentration) as seen in Fig. 2a. In this case the initial profile has a minimum concentration of 150 kg m-3 (around 438 mol m-3/0.438 mol l-1) at the right-hand end the tube and a maximum concentration of 300 kg m-3 (around 876 mol m-3/0.876 mol l-1) at the left-hand end of the tube although there is no large localization in the solute concentration along the tube.
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Overlaid profiles of the solute concentration inside the tube at various times are shown in Fig. 2a
. The dynamics of the solute inside the tube behaves qualitatively similarly to diffusion, the initial solute concentration profile gets smoothed out until the concentration becomes uniform along the tube. Hence a net result is that solute travels from regions of high concentration to regions of low concentration.
Overlaid profiles of the total longitudinal water flux over a cross-sectional area inside the tube at various times are shown in Fig. 2b. Positive values for this flux signify that the water is travelling from left to right. The longitudinal flux of water inside the tube is also down gradients in the local solute concentration and decreases in magnitude as the solute concentration profile becomes more uniform. The flux of water is zero across the two ends of the tube (because the solute gradient has been set to zero at these boundaries) and has a local maximum near the centre of the tube.
Overlaid profiles of the radial water flux per unit length of tube wall at various times are shown in Fig. 2c. There is an influx of water in the regions of high solute concentration, i.e. the source-like region, and an outflux of water in the regions of low solute concentration, i.e. the sink-like region. The maximum influx is at the left-hand end of the tube, the maximum outflux is at the right-hand end of the tube, and the flux is zero near the centre of the tube. Hence the profile in Fig. 2c is qualitatively similar to the profile in Fig. 2a. Again the magnitude of the flux of water decreases as the solute concentration becomes uniform inside the tube.
Simulation 2
The second simulation illustrates the different aspects of the directionality of the solute and water fluxes. To achieve this result, a concentration profile that is initially twin-peaked is chosen to reproduce the desired directionality features (Fig. 3a). The starting profile is symmetrical and has a minimum concentration of 150 kg m-3 (around 438 mol m-3/0.438 mol l-1) at the two respective ends of the tube and a maximum concentration of approximately 300 kg m-3 (around 876 mol m-3/0.876 mol l-1) about one-third and two-thirds of the way along the tube, respectively.
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Overlaid profiles of the solute concentration inside the tube at various times are shown in Fig. 3a
. The two superimposed peaks first coalesce into a single peak and then this aggregated peak gets smoothed out until the solute concentration is uniform. As the solute gradient determines the direction of movement, the flux of the solute inside the tube displays several directionality features with this particular initial profile. Firstly, the solute flux near the left-hand end of the tube is always directed to the left and the flux near the right-hand end of the tube is always directed to the right. Hence the solute is able to travel in different directions at different locations along the tube at the same time. Secondly, the solute flux at a given location along the tube that is slightly to the left of the tube centre is initially directed to the right, eventually reaches zero, and then is directed to the left thereafter. Hence the flux of the solute can change direction at a specific location along the tube over time. Due to the symmetry of the starting profile, the difference in the solute concentration between the two ends of the tube is always zero. The solute flux occurs because it is the local gradients in the solute concentration along the tube which drive the dynamics inside the tube.
Overlaid profiles of the total longitudinal water flux over a cross-sectional area inside the tube at various times are shown in Fig. 3b. The longitudinal flux of water inside the tube has similar directionality features with the solute flux in Fig. 3a. Firstly, the water flux can be in different directions at different locations along the tube at the same time. Secondly, the water flux can change direction at a given location along the tube over time.
Overlaid profiles of the radial water flux per unit length of the tube wall at various times are shown in Fig. 3c. The radial flux of the water across the tube walls also has several directionality features of interest with this simulation. Firstly, the first profile shows that water flux around one-third the way along the tube is initially incoming and the water flux around two-thirds the way along the tube is initially outgoing. Hence there are locations where the flux is incoming, the flux is zero, and the flux is outgoing at the same time. Secondly, the overlaid profiles show that the water flux at the centre of the tube is initially incoming, reaches zero, and then is outgoing. Hence the radial water flux change can change direction at a specific location along the tube over time.
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Discussion |
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TopAbstractIntroductionThe mathematical modelResultsDiscussionReferences |
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A mathematical model of the Münch pressure–flow hypothesis for long-distance transport of carbohydrates via sieve tubes has been constructed from the Navier–Stokes equation for the motion of a viscous fluid and the van't Hoff equation for osmotic pressure. Assuming spatial dimensions that are appropriate for a sieve tube and ensuring suitable initial profiles of the solute concentration and solution velocity lets the model become more mathematically tractable and concise. Thus several key features of the solute flux and accompanying water fluxes can be interpreted directly from the resulting equations themselves without requiring simulations, although simulations are necessary to determine the model dynamics explicitly. These key features can be compared to known qualitative observations of the long-distance transport of carbohydrates inside the phloem.
In the steady-state case, the flux of solute inside the tube is given by an expression that is similar to Fick's law of molecular diffusion. This relationship with Fick's law is implicit in several other phloem transport models (Thornley and Johnson, 1990; Minchin et al., 1993; Sheehy et al., 1995), but has not been derived explicitly. Observations of diffusion-like behaviour for carbohydrate transport in the phloem tissue date back to early work (Mason and Maskell, 1928) and several papers since (Canny and Phillips, 1963; Passioura and Ashford, 1974). Here the apparent diffusivity coefficient is proportional to the local solute concentration and also depends on the radius of the tube and a set of physical constants. Using a realistic value for the sucrose concentration of 300 kg m-3 (around 876 mol m-3/0.876 mol l-1), a tube radius of 1x10-5 m, and a tube length of 3x10-1 m, the apparent diffusivity coefficient is around seven orders of magnitude greater than the diffusivity coefficient for sucrose in stationary water. It has been estimated from experimental data that the apparent diffusivity coefficient is around four orders of magnitude greater than the diffusivity coefficient for sucrose (MacRobbie, 1971).
The time-dependent model is able to demonstrate that it is possible to have a difference in solute concentration between two locations, but have no translocation of solution between these two locations. The flux of solute at a given location is driven by the local gradient in the solute concentration at that location. Therefore, a difference in the local solute concentration between two separate regions is not necessarily indicative of a net solute translocation between these two regions (Minchin et al., 1993). This feature can be readily seen in the second simulation given in this paper where the initial solute concentration profile along the tube is specified to be a twin-peaked, symmetrical function. Although the concentration at any location in a given half of the tube is always greater than the concentration at the opposing end, the solution in this half never travels toward this end. Hence there is no translocation of solution between the two halves of the modelled tube.
In the time-dependent case, a radial flux of water across the tube walls is intrinsically coupled with the longitudinal fluxes of solute and water all the way along the tube. Firstly, it was demonstrated with the aid of a numerical simulation that the water flux at a specific location along the tube can change from being incoming, zero, or outgoing over a period of time. Secondly, it was demonstrated with the aid of a numerical simulation that the water flux can be incoming, be zero, or be outgoing at different locations along the tube at a particular stage in time. The phloem transport literature suggests that a radial flux of water is an important feature of the long-distance transport of carbohydrates inside the sieve tubes (Wardlaw, 1969; Minchin and Thorpe, 1982; Van Bel, 1990). Hence it has been proposed that the long distance transport would ‘stall’ (Baker and Milburn, 1994) if significant radial flux of water did not take place with the result being that carbohydrate demands of distant sinks would not be met.
There has been historical interest in the concept of ‘bidirectional’ transport of both carbohydrates and water as a means of experimentally testing the various proposed mechanisms of phloem transport, some mechanisms would allow a bidirectional movement within a single sieve element and other mechanisms would not (e.g. flow en masse) (reviewed by Eschrich, 1975). It needs to be emphasized at this stage that there are two feasible interpretations of ‘bidirectional’ transport within a single sieve element or file of sieve tubes. The first is the transport of carbohydrate and/or water in opposite directions at different locations along a sieve tube (Trip and Gorham, 1968; Eschrich et al., 1972) whereas the second is the transport of carbohydrates and/or water in opposite directions at a particular location along the sieve tube (Chen, 1951; Biddulph and Cory, 1957; Eschrich, 1975). It has yet to be demonstrated experimentally that bidirectional transport of carbohydrate occurs at a particular location within a single file of sieve tubes, although it has been demonstrated within a stem and also within the same vascular bundle (Eschrich, 1975; Thorpe and Minchin, 1996).
In the time-dependent model presented here only bidirectional transport of the first interpretation is possible. Firstly, it was demonstrated that the solute (and water) can simultaneously travel in opposite directions at different locations along the model tube at a given time, i.e. arising from competing sources and sinks (Eschrich et al., 1972; Minchin et al., 1993). Secondly, it was demonstrated that the solute (and water) can change the direction of travel at a given location along the model tube over time, i.e. comparable to a sink/source transition (Geiger, 1987; Turgeon, 1989). Thus it can be concluded from the simulations given in this paper that these bidirectionality results for the transport of solute within a single tube are still consistent with the original Münch pressure–flow hypothesis.
The model presented in this paper does not take into account any flux of solute through the tube walls. The focus of this work was on the translocation of solute and the accompanying fluxes of water inside the tube which can be successfully modelled independently of any radial solute fluxes. Here the relaxation of a suitable non-uniform profile within the tube helped provide both a rapid and sufficient means of illustrating and interpreting the translocation process. Loading (Ho and Baker, 1982; Komor et al., 1996) or unloading (Oparka, 1990; Patrick, 1990) of solute from the tube could be incorporated by including radial flux terms into the derivation of the presented model. Such flux terms have been included in previous mathematical models of solute transport within a tube (Christy and Ferrier, 1973; Tyree and Dainty, 1975; Goeschl et al., 1976). However, a disadvantage of these models is that they are less amenable to being studied analytically and therefore more reliant on numerical simulations for making interpretations. The advantage of the analytical model developed in this paper is that key features of the model dynamics can be easily distinguished from the governing equations themselves and are less reliant on numerical simulations.
The model presented in this paper does not take into account the presence of sieve plates (MacRobbie, 1971; Spanner, 1978) or transcellular strands (MacRobbie, 1971; Johnson et al., 1976). The analysis presented here assumes that the tube is strictly one-dimensional with a constant radius and contains only the solute and water. As briefly discussed in the text, introducing a damping factor into the viscosity term in the Navier–Stokes equation is one possible way of incorporating the effect of the sieve plates into the proposed model. Similarly the elasticity of the sieve tube walls is not taken into account (Milburn, 1970; Lee, 1981). The analysis presented here assumes that the tube walls are always rigid. Thus a time-dependent model of solute transport that incorporates the effect of the sieve plates and the elastic nature of the tube walls could be viewed as future work.
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Acknowledgments |
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We would like to acknowledge the comments and advice kindly given by Alistair Hall, Alfred Sneyd, Ian Craig, Michael Spink, Michael Thorpe, Martin Hunt, and Adam Matich. Anonymous referees are thanked for pointing out several weaknesses in earlier manuscripts. We also gratefully acknowledge the service provided by our librarians, Sarah Nation, Ann Ainscough and Steven Northover. This work was funded by the PGSF CO6806.
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Notes |
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1 To whom correspondence should be addressed. Fax: +64 6 354 6731. E-mail: shenton{at}hortresearch.co.nz
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