Science in China Series G: Physics, Mechanics & Astronomy
© 2008
SCIENCE IN CHINA PRESS
Springer-Verlag
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Mathematical modelling study for water
uptake of steadily growing plant root
CHU JiaQiang1, JIAO WeiPing1 & XU JianJun1,2†
1
2
School of Mathematical Sciences, Nankai University, Tianjin 300071, China;
Department of Mathematics and Statistics, McGill University, Montreal H3A 2K6, Canada
The root system of plant is a vitally important organ for living plant. One of the
major functions of the root system is uptaking water and nutrients from the soil.
The present paper analyzes the whole process of water uptake from soil by a
steadily growing plant with a single slender root. We start from the basic principles
of physics and fluid-dynamics, consider the structure characteristics of the water
transport channel formed by the tiny xylems tubes inside plant, and establish a
simplified coherent mathematical model to describe the water transport in the
complete system consisting of soil, individual plant, including root, stem and
leaves-atmosphere, on the basis of the plant physiology. Moreover, we resolve the
proposed mathematical model for a simple artificial plant model under a variety of
conditions, in terms of the numerical approach as well as analytical approach. It is
shown that the results obtained by both approaches are in very good agreement;
the theoretical predictions are qualitatively consistent with the practical experiences very well. The simplified mathematical model established in the present paper may provide a basis for the further investigations on the more sophisticated
mathematical model.
plant root, water uptake, Poiseuille flow, asymptotic expansion solution
1 Introduction
Root is a vital organ in plant. The primary physiological function of root is uptaking water as well
as nutriments, and transporting them to the other parts of plant above the ground. The function of
root is vital for plant’s survival, maintaining its growth, development and evolution. The investigations and observations of the uptake of water and nutriments in plant can be traced back to many
years ago. The subject not only has an important theoretical significance for the scientific advance
of biology, but also possesses an important practical significance for agricultural production and
economic development. In traditional food planting and agriculture, the deep theoretical understanding of the dynamical mechanism of uptake of water and nutriments is invaluable for utilizing
Received September 26, 2006; accepted March 6, 2007
doi: 10.1007/s11433-008-0010-0
†
Corresponding author (email: [email protected])
Supported by the National Natural Science Foundation of China (Grant No. 10572062)
Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 184-205
water and fertilizer more effectively and increasing production. In recent years, some new industries of planting, growth of inedible plant are emerging. For example, to prevent the salinization
and desertification of soil, one may plant some special plants, whose roots possess a high drought
resistance, alkali and salt resistance; to clean pollution of heavy metals or radioelements, one may
plant some plants, whose roots have the capability of uptaking heavy metals, which may be called
“plant control of soil pollution”. Moreover, one also develops the so-called “plant’s mining”, to
collect the valuable metals, like gold, in soil by planting some plants whose roots possess a special
capability for absorbing the valuable metals. For all these newly developed industries, the deep
understanding of the process of uptake of water and microelements by the roots can become important theoretical guidance[1].
The observations indicate that the water uptake can be considered as water entering plant
through the surface of roots from soil; then being transported to the leaves through the internal
micro-tubes of plant, and being vaporized into air from the pores on leaves. Hence, the whole
process of water transport can be divided into three parts: (1) the water flows in the porous soil; (2)
the water flow through the micro-tubes inside the plant; (3) the two dynamical processes occurring
on the interfaces: One occurs on the interface between soil and surface of roots, which determines
the amount of water crossing the surface of root from soil; the other occurs at the interface between
leaves and air, which determines the amount of water vaporizing into air through the pores on
leaves. The flows of water in micro-tubes of plants and soils are coupled. The transport of water in
plant can affect the water flow in soil, and vice versa.
The above dynamical process involves the circulation of water in a soil-plant-air system. Some
botanists have put forward a concept named SPAC (soil-plant-air continuum) to describe the
－
connection between the soil, air condition and water transport in plant[2 4]. There is no doubt that
the concept of SPAC is correct, but the key is how to develop an appropriate mathematical model
with such a concept to carry out quantitative, high-level investigations. In recent years, a number of
researchers from various areas, such as physics, applied mathematics and plant physiology, have
paid more attention to this interdisciplinary subject. A piece of the outstanding work in this area is
－
done by Roose et al. (2000－2004)[5 8]. Roose proposed a simple mathematical model for the
uptake of water and nutriment on the basis of the early work by Nye, Tinker and Barber et al.[9,10].
Roose, assuming that the root is an infinitely long cylinder, derived the analytical solutions for the
problems. Obviously, such a model is not quite appropriate for the realistic roots. Moreover, Roose
et al. assumed that the roots were static without growing. Up to now, the mathematical modeling
study of the subject is still far from completion.
In this article, we consider the dynamical process of water uptake of an axi-symmetric slender
single plant. We apply the hydrodynamics to describe water flow in plant and soil, apply the
thermodynamics to describe the kinetics occurring between the interfaces of soil-root, as well as
leaves-air, and take into account of the realistic micro geometrical structure of root, in order to
establish the mathematical model for the process of water uptake of root. We then solve the model
and discuss the physiological implications of the mathematical results.
2 Mathematical model
2.1 Water motion in the exterior of root
We consider a single, axi-symmetric, slender plant root, growing in soil with a constant speed U.
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185
Convective water motion will be induced by the growing motion and the water uptake of the root.
We assume that the root shape r = R(z) in the moving coordinate frame fixed at the root tip is given
as shown in Figure 1.
For simplicity, we consider the soil as an ideal, uniform, homogeneous porous medium. So that,
the fluid motion outside the root is subject to the Darcy’s law. It implies that one may set
k
ua = − ∇p + ρ gkˆ ,
(1)
μ
(
)
where ua is absolute velocity of water, p is the dynamic pressure, k̂ is the unit vector pointing
vertically upwards, k is the permeability coefficient and μ is the viscosity of water. Note that ∇p
is the gradient of dynamic pressure caused by the flow in the soil. ρ gkˆ is the gradient of the static
pressure of flow due to the depth of water in soil. For simplification, one may employ the total
pressure p̂ to replace dynamic pressure p, where
pˆ = p + ρ gz.
Thus, eq. (1) can be written in the form:
ua = −
k
μ
∇pˆ.
(2)
In the moving coordinate system (Figure 1), the relative velocity of water u is expressed as
k
u = ua + Ukˆ = Ukˆ − ∇pˆ.
μ
(3)
Figure 1 The sketch of a plant root tip. The dark area in the root is the water pathway. (I) Inner region; (II) tip region; (III) outer
region of the flow field in soil.
One may introduce the potential function φ to describe the relative motion of water in the
moving frame. Namely,
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u = −∇φ , φ = −Uz +
k
μ
pˆ + φ0 ,
(4)
where φ0 is a constant. Due to the incompressibility of fluid, we have ∇ ⋅ u = 0. From eq. (4), it
follows that
∇ 2φ = 0.
(5)
The boundary conditions are as follows: (1) the boundary condition in the far field:
z → −∞, u → Ukˆ ;
(6)
(2) the boundary condition on the surface of roots, we can get: at r = R(z),
J w = −(u ⋅ n),
(7)
or
∂φ
∂φ
− R′( z ) ,
(8)
∂r
∂z
here Jw is the flux of water per unit area of the root’s surface, and n is the unit outer normal vector
on the root’s surface. The function Jw depends on the kinetics on the interface between the soil and
root and is also affected by the motion of water inside the root.
J w 1 + R ′2 ( z ) =
2.2 Water motion in the interior of root
Inside the root, water is transported upwards along a bunch of the micro tubes with the average
radius rm located in the cortex. Assume that the number of the micro-tubes is N. The total area of
2
2
cross-section of the micro-tubes can be written as Nπr m = πr 0. We may model the water path effectively by a single tiny circular tube with the radius r0(z). So that, the water motion in the interior
of root is described by the one-dimensional Poiseuille flow along the pipe with a variable
cross-section area. Hence, we have the velocity profile:
1 2
⎛ ∂p
⎞
(9)
uI ( r , z ) = −
[r0 − r02 ( z )] ⎜ I + ρ g ⎟ ,
4μ
⎝ ∂z
⎠
where uI is the velocity of the water in the pipe and pI is the water dynamic pressure. Notice that the
Poiseuille flow discussed here is different from the classical one, as there exists a flux crossing the
wall of the pipe here. Again, for simplification we may employ the total pressure p̂I to replace pI,
where
pˆ I = pI + ρ gz.
From eq. (9), the total flux of water passing the cross-section at the level z is calculated as:
r0 ( z )
πr 4 ( z ) ∂pˆ I
(10)
2πruI (r ) d r = − 0
.
qI = ∫
0
8μ ∂z
2.3
Description of the interfacial dynamics between leaves-air
We simplify the interface between leaves-air as a horizontal plane at z = L (see Figure 2). Here the
2
effective area of water pathway in plant is (πr 0), which is actually the total areas of all opened
micro pores on leaves, namely
∑ i =1 πri2 , here ri is the effective radius of the opened pores on
Na
leaves, and Na is the number of the total opened pores on the leaves.
In general, the liquid pressure of the opened pore on the leaves pi′0 is negative. It is caused by
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187
Figure 2 The model of water transport in plants.
the transpiration on the leaves. The essence and origin of the generation of the negative pi′0 are the
equilibrium conditions at the interface. Here we can distinct the two physical mechanisms.
(1) Assume that the liquid and the air are separated by a curved surface, then the negative pi′0 is
generated by the capillarity occurring on the curved surface. Hence pa − pi′0 =
σ
ri
, where pa is the
pressure of air, pi′0 is the liquid pressure inside the leaves, ri is the radius of the vessel, and σ is the
surface tension coefficient of the liquid at the interface.
(2) Assume that the liquid and the air are separated by a semi-permeable membrane, and water
on the atmosphere side of the membrane exists in the form of vapor, whose fractional pressure pa′
depends on the humidity of the air. Water on the side of the membrane that faces the interior of the
leaf is in the form of solution. Suppose that the concentration of the water there is C, according to
the solution theory, the thermodynamic equilibrium condition at the membrane is that the chemical
potentials on both sides of the membrane must be equal. One may write pi′0 = pa + pπ ( pπ < 0),
where pi′0 is the pressure of the solution in the plant; pπ is the osmotic pressure, which depends on
the difference of the concentrations (C- p0′ / pa ) of the water on both sides of the membrane. If the
liquid on one side of the membrane is pure water (the concentration of the water is C = 1), while it
is solution on the other side and the concentration of the solute is S, or say, the concentration of the
water is C′ = 1− S, then the osmotic pressure pπ can be deduced by the Van’t-Hoff formula[11].
Namely
pπ = C*RT,
here C* is the concentration of solute in the solution, T is the absolute temperature and R is the
universal gas constant.
The above condition is applicable for the system that the solutions on both sides of the membrane are in the thermodynamics equilibrium. If the solutions on both sides of the membrane are
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not in the thermodynamics equilibrium, then there exists a difference of pressures Δp ≠ 0 (in case
(1), Δp = pi′0 − pa −
σ
ri
, in case (2), Δp = pi′0 − pa + pπ). Such pressure difference is the driving force
for the water flux passing through the leaves-air interface. When Δp > 0, the water will vaporize
into air, otherwise water in air will enter leaves. According to irreversible thermodynamics, the flux
vaporizing into air can be expressed as qi = kiΔp. Here ki is the osmosis coefficient. As to leaves-air
the interface, we can consider ki 1. Hence, we can deduce pi′0 ≈ pa + pπ or pi′0 ≈ pa − ω.
Assuming that the average value of the pressures pi′0 at the top of micro-tubes over all the
leaves is p0, we can write the boundary condition at z = L above the ground as follows:
pˆ I = p0 + ρ gL,
(11)
here pressure p0 is a function of the osmotic pressue pπ, surface tension coefficient σ, effective
radius of pores in leaves ri, humidity of air pa′ , temperature T and percents of opening pores and so
on, namely
p0 = p0 (σ, pπ, ri, ω, T, pa′ ).
Such function may have a complicated nonlinear form; it varies for different plants and growth
conditions, and should be determined by plant’s anatomy and experimental measurements. In this
article, we suppose p0 is given.
2.4 Mathematical description of dynamics on soil-root interface
We simplify the soil-root interface as a semi-permeable membrane. With the analysis similar to the
membrane that separates leaves-air under the thermodynamic equilibrium, we can write the
boundary condition:
pˆ − ( pˆ I + pπ ) = 0
here pπ is the osmotic pressure of the solution inside the root. Under the non-equilibrium condition,
the difference of pressures of both sides of the membrane, Δp = pˆ − ( pˆ I + pπ ) will be the driving
force for water flux passing through the soil-root membrane. As a result, we have
(12)
J w = kw ⎡⎣ pˆ − ( pˆ I + pπ ) ⎤⎦ ,
here Jw is the water flux per unit area through the surface; kw is the osmosis coefficient of the
surface of root. For the special condition (the system has pure water outside the root and diluted
solution inside the root), as we mentioned before, the osmotic pressure pπ can be determined by
Van’t-Hoff formula. The osmosis coefficient kw depends on the physical characteristics of the
surface of root and may have a complicated function form. In this article, we suppose that kw is a
known constant and the osmotic pressure pπ is given.
2.5
Coupling of water flux outside and inside root
The mass conservation law yields that
2πRJ w =
∂qI
.
∂z
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(13)
189
3 Scaling and dimensionless system
3.1
The dimensionless system for water motion in the exterior of root
We use the characteristic length of the root
d
as the length scale, use the characteristic velocity V,
to be determined later as the scale of velocity, use [q] = V
d
as the scale of flux, and finally use the
viscous pressure [ p] = P0 = (pa − p0) as the scale of pressure. With the above scales, we may define
the following dimensionless variables:
r
z
r= , z= .
(14)
d
d
On the other hand, we suppose that the root is slender, needle-like; the tip radius of the root is
Then the parameter δ =
t
t
.
1 measures the slenderness of the needle. The shape function of the
d
needle can be written in the form:
r = δ R* ( z ),
( 0 < R* ( z ) ≤ 1) .
(15)
Thus, let δ → 0, the slender body theory can be applied. In the exterior of the root, the following
dimensionless quantities are utilized:
pˆ − pa
pˆ − pa
J
p
u
φ
, pˆ =
(16)
u = , Jw = w , φ =
, pˆ I = I
, pπ = π ,
P0
P0
V
V
P0
dV
where pa is the pressure of atmosphere. With these dimensionless quantities, one may derive the
dimensionless system for water motion in the exterior of root (refer to the appendix for the details).
3.2
The dimensionless system for water motion in the interior of root
Let α be the average value of r0 (z), the effective radius of water pathway in the interior of root and
a
define the parameter α = . We introduce dimensionless quantities:
t
r=
r ( z)
r
z
, z = , r0 ( z ) = 0
a
a
d
(17)
uI
q
, qI = I .
V
dV
(18)
and
uI =
Thus, one may derive the dimensionless system for water motion in the interior of root (refer to the
appendix for the details). The system contains five dimensionless parameters representing different
physical characteristics:
(1) δ, it measures the slenderness of the root;
a
(2) α = , it measures the vascular micro-structure of the root;
t
(3) U ∞ =
190
U 4 μ dU
, it measures the effect of root growth velocity;
= 2
V
a P0
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(4) Ca =
ρ gL
P0
, it measures the effect of pulling force generated on the top of plant;
⎛ k P ⎞ ⎛ 4k μ
(5) M = ⎜ w 0 ⎟ = ⎜ w 2
⎝ V ⎠ ⎝ a
⎞
⎟ , it measures the capability of osmosis of membrane.
⎠
For simplicity, we omit the bar “−” over the dimensionless quantities hereafter.
d
4 Slender body approximations of solution in the exterior of root
4.1 Uniformly valid asymptotic expansion solution of flow in the exterior of root
To find the solution for the flow in the exterior of root, we divide flow field into three sub- regions:
(I) the inner region near the side surface of root; (II) the tip region near the tip of root; (III) the outer
region away from the root (see Figure 1); then find the asymptotic local solution in each sub-region,
respectively; finally match these local asymptotic solutions to form a uniformly valid asymptotic
global solution in the whole flow field (refer to the appendix for the details). The potential function
of flow on the surface of root r*=R*(z) is obtained as
φ* [ R* ( z ), z ] = −U ∞ z −
δ 2 | ln δ |
2π
Q0 ( z )
⎫
⎡
⎤
R*2
⎥ + [Q0 ( z ) − Q0 (0) ] ln 2 z + G10 ( z ) ⎪⎬ +
⎨Q0 ( z ) ln ⎢
2
2 2
2 2 ⎥
4π ⎪
⎢
⎪⎭
⎣ ( z + δ aˆ ) + ( z + δ aˆ ) + δ R* ⎦
⎩
The pressure is
pˆ ( z ) = φ* ( R* , z ) + U ∞ z ,
−
δ 2 ⎧⎪
. (19)
(20)
here,
Q0 ( z ) = 2πδ −1MR* ( z )( pˆ − pˆ I − pπ ) − πU ∞ S*′ ( z ),
S ′ (0)
S * ( z ) = R*2 ( z ), aˆ = − * ,
4
z
∞
0
z
(21)
G10 ( z ) = ∫ Q′(ξ ) ln 2( z − ξ ) d ξ − ∫ Q′(ξ ) ln(ξ − z ) d ξ .
4.2
The water transport equation in the interior of root
One may derive the governing equation for the pressure function of flow in the interior of root as
(refer to the appendix for the details):
1 d ⎧ 4 d pˆ I ⎫
MR* ( z ) [ pˆ ( z ) − pˆ I ( z ) − pπ ] = −
(22)
⎨r0 ( z )
⎬.
4δ d z ⎩
dz ⎭
Eqs. (19)－(22) connect the four unknown functions:
{Q0 ( z ), φ* ( z ), pˆ ( z ), pˆ I ( z )}.
To find their solution, the following iterative procedure can be adopted:
(19)
(20)
(21)
(22)
Q0(0) ⎯⎯⎯
→ φ* ( z ) ⎯⎯⎯
→ pˆ ( z ) ⎯⎯⎯
→ pˆ I ( z ) ⎯⎯⎯
→ Q0(1) .
The value of the input for the next iteration can be set as
Q0(1)′ = θ Q0(0) + (1 − θ )Q0(1) ,
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191
here θ is an adjustable parameter.
4.3 The approximate solution for the water transport in the interior of root
If only keep the leading order terms of O(δ 2 | ln δ |) and omit all the higher order small terms of
O(δ 2 ) , eq. (20) can be simplified into the form:
pˆ ( z ) = φ* ( R* , z ) + U ∞ z ≈ −
δ 2 | ln δ |
2π
Q0 ( z ).
By combining (21) and (23), we derive
Q0 ( z )
1
1
⎡ −1
⎤
=−
δ MR* pˆ I + U ∞ S*′ ( z ) ⎥
⎢
2π
1 + MR*δ | ln δ | ⎣
2
⎦
and
pˆ ( z ) =
δ 2 ln δ
1
⎡ −1
⎤
δ MR* pˆ I + U ∞ S*′ ( z ) ⎥ .
⎢
1 + MR*δ | ln δ | ⎣
2
⎦
(23)
(24)
(25)
Moreover, by substituting (25) into (22), we derive the following 2nd order ODE for pˆ I ( z ) :
d ⎧ 4 d pˆ I ⎫
2
⎨r0 ( z )
⎬ − 4 Bpˆ I + 2 Bδ | ln δ | U ∞ S*′ ( z ) − 4 Bpπ = 0,
dz⎩
dz ⎭
(26)
where we define a parameter:
B=
M δ R* ( z )
.
[1 + M δ | ln δ | R* ( z )]
Eq. (26) is the governing equation for the water uptake pressure inside the root. Solution pˆ I ( z )
must satisfy the following boundary conditions:
(1) The boundary condition at the tip of root: at z = 0, there will be no water crossing the surface
of root from soil. So, we have
d pˆ I
= 0.
(27)
dz
(2) The boundary condition on surface of leaf: at z = L,
pˆ I = p0 .
(28)
It is seen that there are three driving forces for the transport of the water in plant: (1) due to the
growth of root, which is proportional to δ 2lnδ U∞S ′*(z); (2) due to the positive pressure of water in
the lower part of root, produced by the osmosis at the side surface of root, which is proportional to
pπ; (3) due to the negative pressure of water at the top of root induced by the transpirations on
leaves, which is proportional to p0. For any given plant, one may determine its interior structure and
geometric sizes of micro-tubes by anatomy and measurements. The parameters introduced in
mathematical model, such as osmosis coefficient should be determined by experimental measurements. Once these data are known, one can determine the water flux in the root and the pressure
distribution along the height of the plant by the solution of the above mathematical model. These
theoretical results then can be compared with the corresponding experimental data. To further
exam the validity of the above mathematical model and clarify the process of solution finding, in
what follows, we shall specifically consider a simple artificial plant model.
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5 Numerical study of some special cases
Suppose that the geometrical structure of the virtual plant is as follows: it has a very small rounded
tip region with height ε, where no water flux exists; the exterior shape of root and interior water
transport channel are cylinders. The radius of water transport channel from tip of root to ground is
unit one, and the radius from ground to leaves is ω ≥1 (see Figures 2 and 3). Namely,
⎧
⎪ω ,
⎪
⎨
r0 ( z ) = ⎪1,
⎪⎩0,
here ε
L⎞
⎛
⎜1 < z ≤ ⎟ ,
d ⎠
⎝
(ε < z ≤ 1),
(0 ≤ z ≤ ε ),
⎧
⎪1,
⎪
R* ( z ) = ⎨
⎪ 1
⎪⎩ δ
L⎞
⎛
⎜ε < z < ⎟,
d ⎠
⎝
S*′ (0) z ,
(0 ≤ z < ε ),
1. Moreover, we suppose that physical data about plant root are shown in Table 1.
Figure 3 The sketch of geometrical structure of an artificial plant.
Table 1 The physical parameters of the artificial plant
Parameter
Physical meaning
Viscosity
of
water
μ
pa
Pressure of the air
Surface tension of water
σ
Slenderness of the root
δ
Density of water
ρ
g
Gravity acceleration
U
Growth speed of the root
kw
Osmosis coefficient of soil
d
t
α
pπ
R
Length of the root
Value
1.002×10−3 N·s·m−2
1.0×10−5 Pa
7.27×10−2 N·m−1
0.01
1.0×103 kg·m3
9.8 m·s−2
7.0×10−8 m·s−1
3.47×10−12 m2
0.1 m
Radius of the tip of the root
0.001 m
Effective radius of water transport in the root
Osmosis pressure of plant root cells
Gas constant
1×10−4 m
−0.5×106 － −1.0×106 Pa
8.31 J ⋅ mol−1 K−1
It is presumed that under normal situation of water uptake, all coefficients in eq. (26) should
have the same order of magnitude. Namely, the coefficient B should have the same order of magnitude as the coefficient r04 ( z ) . Consequently, for the case under study, eq. (26) can be written as
follows:
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193
1
pˆ I = Λ0 = δ 2 | ln δ | U ∞ S*′ ( z ) − pπ , (0 ≤ z ≤ ε ),
2
(29)
and
d 2 pˆ I
1
L
⎡
⎤ ⎛
= 4 B ⎢ pˆ I − δ 2 | ln δ | U ∞ S*′ ( z ) + pπ ⎥ , ⎜ ε ≤ z ≤
2
dz
⎣
⎦ ⎝
d
The solution of eq. (30) is subject to the following two boundary conditions:
(1) at z = L, pˆ I = p0 < 0;
ω4
2
⎞
⎟.
⎠
(30)
(2) at z = ε, pˆ I = Λ0 > 0.
The parameter p0 represents the pulling force driven by the leaves, and Λ0 is the positive pressure
generated by the root. From Table 1, we can conclude that, in general, S*′ (0) ≈ 1, while both the
growth speed of root U ∞ = U / V
1 are very small; hence the term (δ 2 | ln δ | U ∞ S*′ (0))
1 and δ
is a very small parameter, whose effect is negligible.
The general solution of eq. (30) is
pˆ I = A e z
λ
+ B e− z
λ
+ Λ0 ,
(ε ≤ z < 1),
(31)
where
λ=
4B
ω4
(32)
.
The arbitrary constants A and B can be determined by the above two boundary conditions as
⎧
Λ −Λ
2ε
⎪ A = λ 1 (2ε −01) λ , B = − A e
e −e
⎪
⎪
1
p0 + 4 ( β − 1)ς λΛ 0
⎪
⎪
ω
,
⎨Λ1 =
1
⎪
1 + 4 ( β − 1)ς λ
⎪
ω
⎪
(2ε −1) λ
λ
⎪ς = e + e
,
⎪⎩
e λ − e(2ε −1) λ
here Λ1 = pˆ I (1) is the interior pressure in plant at z = 1, and β =
λ
,
(33)
L
is the ratio of the whole length
d
of plant vs. the length of root. From the above solution, we obtain (refer to eq. (a10) given in the
Appendix):
πω 4 d pˆ I
πω 4
λ A e z λ − λ B e− z λ .
qI = −
=−
(34)
2 dz
2
With eq. (33), the solutions (31) and (34) are completely determined. In this work, besides the
above approximate solutions, we also compute the numerical solutions for the system of (19)－(22)
under various conditions. The results show that the solutions obtained by these two approaches are
numerically so close that one hardly sees the differences between their graphs (see Figure 4).
We summarize the numerical results as follows.
Figure 4 shows the comparison of the results obtained by numerical solutions and approximate
analytical solutions. It is seen that these results are well consistent to each other. The results explore
how the pulling force p0 generated by the transpiration on the leaves affects the distributions of
(
194
)
CHU JiaQiang et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 184-205
water pressure and flux along the part of plant under ground. It is seen that with the increase of | p0|,
the transpiration flux qI will increase on the leaves of plant, as it is expected.
Figure 4 Given parameters β = 1, ε = 0.01, pπ = −1, M = 100, δ = 0.01, the pressure p̂I vs. z, from the top to the bottom,
corresponding to p0 = −0.15, −0.3, −0.45, −0.6, −0.75: (a) the results of numerical solutions; (c) the results of approximate analytical solutions. Given parameters β = 1, ε = 0.01, pπ = −1, M = 100, δ = 0.01, the flux qI vs. z, from the bottom to the top,
corresponding to p0 = −0.15, −0.3, −0.45, −0.6, −0.75: (b) the results of numerical solutions; (d) the results of analytical solutions.
Figure 5 shows how the ratio of height of plant and the length of the root, β =
L
affects the
d
distributions of the pressure and water flux along the plant, given parameters ω = 1, ε = 0.01, p0 =
−0.5, pπ = −1, M = 100, δ = 0.01. From the figure, one sees that, with the increase of the parameter
L
β = , the flux qI will decrease. It implies that at a fixed pulling force produced by top of plant,
d
increasing the height of over ground plant will decrease water uptake; contrarily at fixed height,
increasing the length of root will increase water uptake.
Figure 6 shows how the value of parameter ω affects the distributions of water pressure and flux
along the plant, given parameters β = 3, ε = 0.01, p0 = −0.5, pπ = −1, M = 100, δ = 0.01. From the
figure, one sees that since there is no water entering plant above the ground, as a consequence, the
water flux in plant is constant in the part of plant above the ground. Furthermore, with the increase
of ω, the flux qI will increase. It implies that the more the branches, leaves or percentages of
opening of pores, the larger the water flux above the ground. According to (10) or (a10), with a
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195
Figure 5 For the cases, ω = 1, ε = 0.01, p0 = −0.5, pπ = −1, M = 100, δ = 0.01, (a) the pressure p̂I vs. z, from left to right,
corresponding to β =
L
= 1, 2, 3, 4, 5; (b) the flux qI vs. z, from the top to the bottom, corresponding to β =
d
L
= 1, 2, 3, 4, 5.
d
The dashed lines show the parts of curves above the ground.
Figure 6 When the parameters β =3, ε = 0.01, p0 = −0.5, pπ = −1, M = 100, δ = 0.01, (a) the pressure p̂I vs. z, from the top to the
bottom, corresponding to ω = 1, 2, 3; (b) the flux qI vs. z, from the bottom to the top, corresponding to ω = 1, 2, 3. The dashed lines
show the parts of curves above the ground.
fixed flux, increasing r0 will decrease the gradient of the pressure rapidly. Therefore, if ω increases,
the pressure difference (Λ1 − p0) along the part of plant above the ground will decrease, accordingly
the pressure difference (Λ1− Λ0) along the part of plant under the ground will increase. In Figure
6(a), one sees that at z = 1, namely on the ground, the slope of the curve is becoming bigger, when
ω becomes larger. It then in term increases the flux in plant. Figures 5 and 6 illustrate the deepness
of root and denseness of leaves may remarkably increase water uptake of plant, which is well
consistent with our experience.
Figure 7 shows how the slenderness of root affects the distributions of pressure and water uptake
along the underground part of plant, given the parameters β = 1, ε = 0.01, p0 = −0.5, pπ = −1, M = 10.
It is seen that with the increase of δ, the flux qI in the part of plant above the ground will increase, in
other words, with the increase of the radius of root, plant can absorb more water.
Figure 8 shows how osmosis capability of root surface affects the distributions of water pressure
and flux along the underground part of plant, given parameters β = 1, ε = 0.01, p0 = −0.5, pπ = −1,
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δ = 0.01. It is seen that with the increase of M, the flux qI above the ground will increase. In other
words, with the increase of osmosis capability of root surface, plant carries up more water from soil
to leaves.
Figure 7 Given the parameters β = 1, ε = 0.01, p0 = −0.5, pπ = −1, M = 10, (a) the pressure p̂I vs. z, from the bottom to the top,
corresponding to δ = 0.01, 0.02, 0.03, 0.04, 0.05; (b) the flux qI vs. z, from the bottom to top at z = 1, corresponding to δ = 0.01,
0.02, 0.03, 0.04, 0.05.
Figure 8 Given the parameters β = 1, ε = 0.01, p0 = −0.5, pπ = −1, δ = 0.01, (a) the pressure p̂I vs. z, from the bottom to the top,
corresponding to M = 20, 40, 60, 80, 100; (b) the flux qI vs. z, from the bottom to the top at z = 1, corresponding to M = 20, 40, 60,
80, 100.
6 Discussion
In this article, we analyze the whole process of water uptake by a plant with a single steadily
growing slender root on the basis of hydrodynamics and thermodynamics. We take into account of
the geometric structure of plant’s interior, the characteristics of micro-tubes in plant, the interactions of soil-plant (including root-stem-leaves)-air, and establish a simplified mathematical model.
We resolve the model numerically and analytically for a simple artificial model of plant under
various conditions. The results show that the theoretical results are well consistent with the plant
physiological principle and practical experiences. The model approach presented in this paper may
be used as a basis for more realistic sophisticated mathematical models to be developed in the
future.
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197
In this article the model contains two important simplifications as follows.
(1) The model simplifies soil as the ideal, uniform, isotropic, water saturated porous medium.
The assumption is very suitable for the plant cultivated in nutrition solution. But it is a greatly
simplified assumption for the plant growing in actual soil conditions. Of course, removing such an
assumption in the model will not create essential difficulty in principle. The water flow in soil may
be still subject to Darcy’s law, but governing equation will be a more complicated elliptic equation
with variable coefficients. The problem will become more difficult and can be resolved by numerical approaches only.
(2) In this model we simplify the liquid in plant as pure water, by neglecting the variation of
water concentration of the solution along the pathway in the plant. Under such an assumption, the
osmosis pressure pπ generated by surface of root and the pulling force p0 generated by transpiration
on the leaves are set to be given constants in our model, ignored the dependences of these two
parameters on the water transportation process. A more realistic model should contain the concentration C(r, z, t) of water in the solution inside of the plant and consider it as a new unknown
quantity in the system. Thus, the problem will involve the hydrodynamic equation of water flux
and the convective diffusion equation for the concentration of water in the interior of plant. Further
study of such type of mathematical models and the comparison of theoretical results and experimental observations for a realistic individual plant will be carried out in the future papers.
Appendix
A1 The dimensionless system of water flow in the exterior of root
In terms of the dimensionless variables, the following dimensionless governing equations and
boundary conditions are as follows:
∇ 2φ = 0,
(a1)
φ = pˆ − U ∞ z .
(a2)
here,
We set
φ0 =
kpa
μ
.
The boundary conditions are:
w=
δφ
→ U ∞ , u → 0, as z → −∞ or r → ∞,
δz
(a3)
where we set
U∞ =
U 4 μ dU
=
.
V
P0 a 2
(1) On r = δ R* ( z ), 0 ≤ z ≤ ∞, from the mass conservation law, we have
J w 1 + δ 2 R*′2 ( z ) =
(2) The osmosis law:
∂φ
∂φ
− δ R*′ ( z )
.
∂r
∂z
(
(a4)
)
2πδ R* ( z ) J w = 2πδ MR* ( z ) pˆ − pˆ I − pπ .
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(a5)
(3) Normalization condition:
R* (1) = 1.
(a6)
A2 The dimensionless system for water transport in the interior of root
The dimensionless velocity profile in the interior of root is
uI ( r , z ) =
P0 a 2
∂pˆ
⎡ r 2 − r02 ( z ) ⎤ I ,
⎣
⎦
∂z
4μ dV
(a7)
here, we define the characteristic velocity:
V=
P0 a 2
.
4μ d
(a8)
We get
∂pˆ I
.
∂z
The total water flux across the section of water pathway at z is
πr 4 ( z ) ∂pˆ I
qI = − 0
.
∂z
2
The boundary conditions are as follows:
(1) the mass conservation law on the surface of root:
∂qI
= 2πδ R* ( z ) J w ;
∂z
uI (r , z ) = ⎡⎣ r 2 − r02 ( z ) ⎤⎦
(a9)
(a10)
(a11)
(2) the transpiration condition on the leaves: at z = L , the pressure
pˆ I = p0 = −1 +
ρ gL
P0
= − 1 + Ca .
(a12)
For simplicity, we omit the over bar ‘−’ hereafter.
A3
A3.1
Slender body approximations of solution in the exterior of root
Inner solution near the stem of root
In view of the boundary conditions on the interface (region (I) in Figure 1), we introduce the following inner variable:
r* = r/δ.
In terms of the inner variables, the basic equations and boundary conditions (a1)－(a4) are written
as
2
⎛ ∂ 2φ* 1 ∂φ* ⎞
2 ∂ φ*
.
⎜⎜ 2 +
⎟⎟ = −δ
r* ∂r* ⎠
∂z 2
⎝ ∂r*
The boundary condition on the interface is:
∂φ
∂φ
δ J w 1 + δ 2 R*′2 = * − δ 2 R*′ ( z ) * , at r* = R*(z).
∂r*
∂z
(a13)
(a14)
Noting that the potential function of the uniform flow at the far field is φ∞ = −U ∞ z , we may assume the following inner expansions:
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199
φ* (r* , z ) = −U ∞ z + μ0 (δ )φ*0 (r* , z ) + μ1 (δ )φ*1 (r* , z ) +
J w ( z ) = ν 0 (δ ) J w 0 ( z ) + ν 1 (δ ) J w1 ( z ) + .
,
(a15)
By substitution of the above into (a13), we may derive each order approximations of solution.
(1) Zero-th order approximate solution.
Let μ0 (δ )
δ 2 , the zero-th order approximation solution is obtained as
φ*0 (r* , z ) = φ*0 ( z ),
(a16)
which only depends on variable z.
(2) First order approximate solution.
From the boundary conditions (a14), we derive μ1(δ ) = δ 2, ν0(δ ) = δ. The first-order equation is
subject to the equation:
⎛ ∂ 2φ*1 1 ∂φ*1 ⎞
+
⎜⎜
⎟ = 0.
2
r* ∂r* ⎟⎠
⎝ ∂r*
(a17)
Its general solution is obtained as
φ*1 (r* , z ) = A1 ( z ) ln r* + B1 ( z ).
From the boundary conditions (a14), we also derive
∂φ*1
( R* , z ) + U ∞ R*′ ( z ) = J w0 .
∂r*
(a18)
By applying this condition, it follows that
1
A1 ( z ) = − U ∞ S*′ ( z ) + J w0 R* ( z ),
2
where
S* ( z ) = R*2 .
Up to now, the function B1(z) in the inner solution is still unknown, which will be determined by the
matching with the outer solution.
A3.2
Outer solution away from the root
In the outer region far away from the needle-root ((III) in Figure 1), the components of perturbed
flow velocity is of the same order in different directions. Let δ→0, the root can be treated as a line
(Figure A1). So that, the flow field can be considered as a uniform flow being perturbed by a line of
sinks extending from z = 0 to z = ∞. Therefore, we make the following outer expansion:
φ = −U ∞ z + φ (r , z ).
(a19)
Figure A1 Sketch of a line of sinks extending from z = 0 to z = ∞.
It is well-known that the fundamental solution of the flow induced by a unit singular point source is
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I (r , z ) = −
1
.
4πrP
We can write the perturbed potential function φ (r , z ) as:
φ (r , z ) = −
1 ∞ Q(ξ , δ )
dξ ,
4π ∫ 0 rP
where the point sink is located at the point P(0, ξ ) on the z-axis and Q(ξ, δ ) is its strength, whereas
rP = r 2 + ( z − ξ ) 2
(a20)
is the distance between the point sink P and any point (r, z) in space. The solution φ (r , z ) has the
following characteristics: as z → ∞ or r → ∞, solution φ (r , z ) → 0 . Hence, the far-field condition
is satisfied. Furthermore, making use of the formula
d
1
ln x + r 2 + x 2 =
dx
r 2 + x2
and in terms of integrating by parts, one may derive
)
(
∞
∞
z
Q(ξ , δ )
Q(ξ , δ )
Q(ξ , δ )
dξ = ∫
dξ + ∫
dξ
rP
rP
rP
0
0
z
I=∫
∞
z
= −Q(ξ ) ln ⎡( z − ξ ) + r 2 + ( z − ξ )2 ⎤ + Q(ξ ) ln ⎡(ξ − z ) + r 2 + (ξ − z ) 2 ⎤
⎣⎢
⎦⎥ 0
⎣⎢
⎦⎥ z
∞
z
+ ∫ Q′(ξ ) ln ⎡( z − ξ ) + r 2 + ( z − ξ ) 2 ⎤ d ξ − ∫ Q′(ξ ) ln ⎡(ξ − z ) + r 2 + (ξ − z )2 ⎤ d ξ
⎢⎣
⎥⎦
⎢⎣
⎥⎦
0
z
)
(
(a21)
= −2Q( z ) ln r + Q(0) ln z + r 2 + z 2 + G1 (r , z ),
where we have assumed Q(∞) = 0 and defined
z
∞
0
z
G1 (r , z ) = ∫ Q′(ξ ) ln ⎡( z − ξ ) + r 2 + ( z − ξ )2 ⎤ d ξ − ∫ Q′(ξ ) ln ⎡ (ξ − z ) + r 2 + (ξ − z ) 2 ⎤ d ξ .
⎣⎢
⎦⎥
⎣⎢
⎦⎥
Thus, we obtain
φ (r , z ) = −
)
(
∞
1 Q(ξ , δ )
1
1
1
dξ =
Q( z ) ln r −
Q(0) ln z + r 2 + z 2 −
G1 (r , z ) + O(r 2 )
4π ∫0 rP
2π
4π
4π
⎛ z + r2 + z2 ⎞ 1
1
1
⎟−
G1 (r , z ) + O(r 2 ).
=
[Q( z ) − Q(0)] ln r − Q(0) ln ⎜⎜
2
⎟
2π
4π
4π
r
⎝
⎠
For fixed z = O(1), let r →0, it follows that
1
1
1
φ (r , z ) ∼ Q( z ) ln r − Q(0) ln 2 z − G1 (0, z ) + O(r 2 ),
2π
4π
4π
where
z
∞
0
z
(a22)
(a23)
G10 ( z ) = G1 (0, z ) = ∫ Q′(ξ ) ln 2( z − ξ ) d ξ − ∫ Q′(ξ ) ln(ξ − z ) d ξ .
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201
A3.3 Matching condition
We expand the sink strength Q(ξ, δ ) in the asymptotic form as δ → 0:
Q(ξ , δ ) ∼ γ 0 (δ )Q0 (ξ ) + γ 1 (δ )Q1 (ξ ) + .
(a24)
Furthermore, we expand the potential function of perturbed flow velocity in the asymptotic form:
φ (r , z ) = γ 0 (δ )φ0 (r , z ) + γ 1 (δ )φ1 (r , z ) +
(a25)
,
and substitute the above two expansions into (a23). To carry out matching procedure for the inner
solution with outer solution, we let r = δ r* in outer solution (a23) and let δ → 0. From the matching
conditions, it is derived that
μ0 = δ 2 ln δ , μ1 (δ ) = γ 0 (δ ) = δ 2 , φ*0 =
φ*1 =
1
Q0 ( z ),
2π
1
1
1
Q0 ( z ) ln rˆ* − Q0 (0) ln 2 z −
G10 ( z ),
2π
2π
4π
where
z
∞
0
z
G10 ( z ) = ∫ Q0′ (ξ ) ln 2( z − ξ ) d ξ − ∫ Q0′ (ξ ) ln 2(ξ − z ) d ξ .
Furthermore, one may obtain
Q0 ( z )
1
= A1 ( z ) = − U ∞ S*′ ( z ) + J w0 R* ( z ),
2π
2
or
Q0 ( z ) = 2πJ w0 R* ( z ) − πU ∞ S*′′( z ),
and
1
1
Q0 (0) ln 2 z −
G10 ( z ).
4π
4π
Thus, the inner solution can be written in the form:
B( z ) = −
δ 2 ln δ
δ2
[ 2Q0 ( z ) ln rˆ* − Q0 (0) ln 2 z − G10 ( z )] +
2π
4π
Especially, on the interface r* = R*(z), the potential function is
φ* (r* , z ) = −U ∞ z +
δ 2 ln δ
Q0 ( z ) +
(a26)
.
(a27)
δ2
[ 2Q0 ( z ) ln R* ( z ) − Q0 (0) ln 2 z − G10 ( z )] + . (a28)
2π
4π
It is seen that the inner solution (a27) has a logarithmic singularity at the tip area of the dendrite
(r*→ 0 or z → 0), when Q0(0) ≠ 0. Hence, one needs to find a different asymptotic expansion solution in the tip region.
φ* ( R* , z ) = −U ∞ z +
A3.4
Q0 ( z ) +
Tip solution in the tip region
In the tip region (region (II) in Figure 1), the tip radius
b
is the proper characteristic length
(Figure A2). Thus, we introduce the tip variables as follows:
r
z
rˆ = 2 , zˆ = 2 .
δ
δ
(a29)
By use of these variables, we can transform the equations in the following Laplace equation:
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2
2
ˆ 2φˆ = ⎧⎪⎨ ∂ + 1 ∂ + ∂ ⎫⎪⎬ φˆ = 0.
∇
2
rˆ ∂rˆ ∂zˆ 2 ⎭⎪
⎩⎪ ∂rˆ
(a30)
The boundary conditions on the interface are:
∂φˆ ˆ
∂φˆ
δ 2 Jˆw ( zˆ) 1 + Rˆ ′2 =
− R′( zˆ ) , at rˆ = Rˆ ( zˆ ).
(a31)
∂rˆ
∂zˆ
Suppose that the root has a rounded tip. Then as z→0, one may assume the root interface shape has
the following expansion:
{
R* ( z ) = S*′ (0) z 1 + a1 z + a2 z 2 +
},
( z → 0),
where a1, a2, … are known coefficients. With the tip inner variables, we can rewrite the shape
function in the form:
r = δ 2 rˆ = δ R* ( zˆ ) = δ 2 S*′ (0) zˆ ⎡⎣1 + a1δ 2 zˆ + a2δ 4 zˆ 2 +
⎤,
⎦
or
rˆ = Rˆ ( zˆ ) = Rˆ0 ( zˆ ) + δ 2 Rˆ1 ( zˆ ) +
,
(a32)
where
Rˆ0 ( zˆ ) = S*′ (0) zˆ.
Figure A2 Sketch of the tip region.
We now make the following asymptotic expansion for the inner solution in the tip region:
φˆ(rˆ, zˆ ) ∼ −δ 2φˆ (rˆ, zˆ ) + , δ → 0.
0
(a33)
Note that the Laplace formula (a30) has the following solutions:
Uˆ 0 (rˆ, zˆ ) = − ln ⎡ −( zˆ + aˆ ) + rˆ 2 + ( zˆ + aˆ )2 ⎤ , Vˆ0 (rˆ, zˆ ) = Azˆ + B.
⎣⎢
⎦⎥
It is easy to find that zero-th order approximation solution of the tip region has the following form:
Q (0)
1
φˆ0 (rˆ, zˆ ) = U ∞ zˆ − 0 ln ⎡⎢ −( zˆ + aˆ ) + rˆ 2 + ( zˆ + aˆ )2 ⎤⎥ + G10 (0).
(a34)
⎣
⎦ 4π
4π
In fact, we may argue that: (1) The above function satisfy eq. (a30). (2) The above function can
match with the outer solution. To demonstrate this point, we exam the outer solution (a23). It is
seen that as one approaches the tip region, namely as r → 0 and z → 0, the outer solution becomes
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203
φ (r , z ) ∼ −U ∞ z
⎛ z + r2 + z2
1
⎪⎧ 1
+ δ 2 ⎨ [Q0 ( z ) − Q0 (0)] ln r −
Q0 (0) ln ⎜
⎜
2π
4π
r2
⎝
⎩⎪
⎞ 1
⎪⎫
⎟−
G1 (r , z ) + O(r 2 ) ⎬ +
⎟ 4π
⎠
⎭⎪
.
On the other hand, as rˆ → ∞, zˆ → ∞, the function δ 2φ 0 becomes
⎧⎪
⎫⎪
Q0 (0) ⎛ zˆ + rˆ 2 + zˆ 2 ⎞ 1
⎟
+
ln ⎜
G
(0)
⎬+ .
10
⎜
⎟ 4π
4π
rˆ 2
⎪⎩
⎪⎭
⎝
⎠
Obviously, both are matched. (3) Finally, because that there is no water flux crossing the surface of
root at the tip z = 0, namely Jw(0) = 0, it can be proved that solution (a34) also satisfies the interface
condition (a31).
In terms of the results obtained, we gain the uniformly valid expansion in the whole region
composing the outer sub-region, inner sub-region and the tip sub-region as follows.
In the inner sub-region (II):
δ 2φˆ0 (r , z ) ∼ −δ 2 ⎨U ∞ zˆ +
φ* (r* , z ) = −U ∞ z −
δ 2 | ln δ |
2π
Q0 ( z )
δ 2 ⎧⎪
⎡
r*2
⎨Q0 ( z ) ln ⎢
4π ⎪
⎢ ( z + δ 2 aˆ ) + ( z + δ 2 aˆ )2 + δ 2 r*2
⎣
⎩
The potential function at the interface r* = R*(z) is
−
φ*[ R* ( z ), z ] = −U ∞ z −
−
δ 2 ⎧⎪
δ 2 | ln δ |
2π
⎫
⎤
⎥ + [Q0 ( z ) − Q0 (0)] ln 2 z + G10 ( z ) ⎪⎬ +
⎥
⎪⎭
⎦
. (a35)
⎫
⎤
⎥ + [Q0 ( z ) − Q0 (0) ] ln 2 z + G10 ( z ) ⎪⎬ +
⎥
⎪⎭
⎦
, (a36)
Q0 ( z )
⎡
R*2
⎢
Q
(
z
)
ln
⎨ 0
4π ⎪
⎢ ( z + δ 2 aˆ ) + ( z + δ 2 aˆ )2 + δ 2 R*2
⎣
⎩
and
pˆ ( z ) = φ* ( R* , z ) + U ∞ z ,
where aˆ = −
A4
(a37)
S*′ (0)
.
4
Governing equation of water transport in the interior of plant
Combining eqs. (a5), (a10) and (a11), one may derive the governing equation for the water pressure
pI(z) inside root:
1 d ⎧ 4 d pˆ I ⎫
MR* ( z ) [ pˆ ( z ) − pˆ I ( z ) − pπ ] = −
(a38)
⎨r0 ( z )
⎬,
4δ d z ⎩
dz ⎭
and
Q0 ( z ) = 2πδ −1MR* ( z ) ( pˆ − pˆ I − pπ ) − πU ∞ S*′ ( z ).
(a39)
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