MATHEMATICAL MODELING FOR WATER UPTAKE …...MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT...

Antarctica J Math 11(3)(2014) 265-277

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MATHEMATICAL MODELING FOR WATER UPTAKE

BY PLANT ROOT IN SATURATED AND

UNSATURATED ZONE

P S AVHALE [Email avhalepsyahoocom]

Department of Mathematics Shivaji Arts and Science College Kannad Aurangabad

[District] Maharastra India

amp

S B KIWNE Department of Mathematics Deogiri College Aurangabad Maharastra India

(Received on 18th

August 2014)

The root system of the plants uptake the water and nutrients from the soil The

present paper explains systematic description of root functioning for water uptake

from the saturated and unsaturated soil and transport through the stem to the leaves

We use basic principles of physics and fluid-dynamics tiny xylems tubes as water

transport channel in the stem and develop simple mathematical model for water

transport In this paper we write mathematical model for water profile at root

surface at the xylem and leaves We solve the mathematical model using root surface

boundary conditions by analytical method

1 INTRODUCTION

The primary physiological function of root is up taking the water as well as

nutrients and transport to leaves for photosynthesis Investigations and observation of

the uptake of water and nutrient in plant root and stem can be traced back to many

years ago it possesses importance in point of view of agricultural production and

economical development In traditional farming like planting and agricultural the -----------------------------------------------------------------------------------------------------------------

Subject classification (AMS) 35XX-35K xx 76XX-76Rxx 76XX-7605 76XX-76Zxx-

76Z05 and 92XX-92B 265

266 P S AVHALE and S B KIWNE

Key words amp PhrasesPlant root Root water uptake Poiseuille flow Mathematical modelXX

mechanism of water and nutrients is invaluable for utilizing water and fertilizer for

increasing production Now a new trend of planting inedible plant emerges on

industrial basis The view of planting inedible plant are prevent the salinization

desertification of soil to clean pollution of heavy metals radioelement and plants

mining To collect the valuable metals like gold in soil by planting some plants

whose roots possess a special capability of absorbing the valuable metals The plants

of genus Bauhina have many species out of which Bauhinia variegata plant extract is

analyzed and found it contain micro-particles of gold [1] Since ancient times

Bauhinia racemosa Lam family Caesalpinaceae has been an integral part of life in

India Leaves of Bauhinia racemosa are traditionally used on occasion of Dashera

festival as symbol of gold in India [2] Recently proved that Bauhina racemosa extract

also contain micro particles of gold Water enters into root through root surface from

the soil Then it flow through micro-tube of xylem in the stem to rich leaves and being

vaporized into air from the pores on leaves The transport of water in plant can affect

the water flow in soil and vice versa Then study of Mathematical model for plant

water uptake divides into the parts (1) the flow of water in the porous soil (2) water

flow into root through root surface (3) flow of water through micro-tube of xylem

(4) vaporization of water through leaves In recent years a number of researchers

from various fields such as physics applied mathematics and plant physiology paid

more attention to develop mathematical model for water and nutrient uptake The

outstanding work in this field is done by T Roose and proposed a mathematical

model for uptake of water and nutrient Roose work is development of Nye Tinker

and Barber model for water and nutrient uptake assuming that the root is an infinitely

long cylinder [7][15] Before this model many root water uptake model like a linear

root water uptake model (Prasad Rama 1988) [3] CERES (Ritche 1985) [4] and

SWAP (Van Dam et al 1997) [5] was developed

2 FLOW OF WATER IN POROUS MEDIUM

In 1856 Henry Darcy make experiment and concluded that rate of flow ( i e

volume of water per unit time) 119876 is (a) proportional to the cross-sectional area 119860 (b)

proportional to (1198931 minus 1198932) the lengths 1198931 and 1198932are measured with respect to some

arbitrary datum level (c) inversely proportional to the length 119871 [6]

119876 = 119870119860(1198931minus1198932)

119871 (2 1)

where 119870 is a coefficient of proportionality which is known as soil hydraulic

conductivity Equation (2 1) is known as Darcy formula Specific discharge 119958 as the

volume of water owing per unit time through a unit cross-sectional area normal to the

direction of flow i e 119958 =119928

119912 we obtain

119958 = 119870119869 (2 2)

where 119869 = (1198931minus1198932)

119871 is in one dimension In three dimension the generalization

MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT 267 IN SATURATED AND UNSATURATED ZONE

of equation (2 2) is given by

119958 = 119870119869 = minus 1048576119870119892119903119886119899119889 = 1048576minus nablaempty 2 3

The average areal porosity is equal to the volumetric porosity 119899 the portion of

the area 119860 available to flow is 119899119860 The average velocity 119881 of flow is given by

119881 =119876

119899119860=

119958

119899 (2 4)

Another form of Darcy formula is in term of specific discharge i e Darcy flux is

written as

119958 = minus119948

120641 119892119903119886119899119889 119901 minus 120588119892 (2 5)

where =119896120588119892

120583 120583 is dynamic viscosity of fluid 120588 =

120574

119892 is the density of the fluid k is the

mediums permeability and 119944 = minus119944119949119963 is the vector gravity acceleration directed

downward If fluid is not compressible then nabla119906 = 0 The equation for the

conservation of water in the soil with no water sink is given by [7]

120597empty119897

120597119905+ nabla 119958 = 0 (2 6)

where 119958 is given by

119958 = minus119948

120641 nabla119901 minus 120588119892119896 (2 7)

The 119896 soil permeability is just differ from hydraulic conductivity K by scaling

factor which is given by

119870 =119896119901119892

120583 (2 8)

where 120588119892 present the density of water and gravity If soil is saturated then we have

120588119892119896 = 0 therefore the Darcy flux is given by = minus119948

120641[nablap] However the

conductivity of unsaturated soil is determined from experiments in terms of effective

water saturation where effective water saturation 119878 is defined as

119878 = empty119897 minus empty119897 119903

empty119897 119904 minus empty119897 119903 (2 9)

where empty119897 is the moisture content of the soil empty119897 119903 is the residual (minimum) moisture

content of the soil and empty119897 119904 is the saturated moisture content of the soil i e the soil


porosity Van Genuchten drive a formula for the soil hydraulic conductivity as a

function of soil moisture content and porosity which is given by [14]

119870 = 11987011990411987812[1 1048576minus (1 1048576minus 119878119898 )119898 ]2 119891119900119903 0 lt 119898 lt 1 (2 10)

where 119870119904 is the hydraulic conductivity of fully saturated soil In addition to deriving

the soil hydraulic conductivity formula (2 10) he also presents a formula for the

suction characteristic He uses the pressure head 119893 suction minus119901 gt 0 can be written in

terms of pressure head i e minus119901 = minus120588119892119893 Hence we have

= [1

1 + 120572119893 119899 ]119898 =

1

1 + 120572120588119892

minus119901 119899

119898

119898 = 1 minus1

119899 0 lt 119898 lt 1 (2 11)

where 119870119904 is a fitting parameter that can be inversely linked to the bubbling pressure

We write equation (2 6) of conservation of water in the soil in terms of relative

moisture content S using the full van Genuchten formulas and then we get the

following nonlinear diffusion-convection equation

empty119897 119904 minus empty119897 119903 120597119878

120597119905= nabla 119863 119878 nabla119878 minus

120588119892

120583

119889119896

119889119878

120597119878

120597119911 2 12

where D(S) is called the soil water diffusivity and it is defined as

119863 119878 =119896 119878

120583 120597119901

120597119878

=119896119904120588119892

120583

times 1 minus 119898

prop 11989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 (2 13)

and

119889119896(119878)

119889119878= 119896119904

1

2119878minus

12 1 minus 1 minus 119878

1119898

119898

2

+ 2119878 minus12

+1119898

1 minus 1 minus 1198781119898

119898

1 minus 1198781119898

119898minus1

(2 14)

RANGE OF VALIDITY

In the flow through conduits the Reynolds number 119877119890 which is

dimensionless number expressing the ratio of inertial to viscous forces acting on the

fluid is used as a criterion to distinguish between laminar flow occurring at low

velocities and turbulent flow occurring at higher velocities By analogy a Reynolds

number is defined also for flow through porous media [6]


119877119890 =119958119889

119907 (2 15)

where 119889 is some representative length of the porous matrix and 119907 is the kinematic

viscosity of the fluid Practically all evidence indicates that Darcys law is valid as

long as the Reynolds number does not exceed some value between 1 and 10 There

the equation (2 12) valued as long as Darcys law obeys i e for Reynolds number

value less than flow will be at very low pressure gradients through low-permeability

sediments also for Reynolds number value greater than 10 there will be large flows

through high permeability gradients

3 WATER FLOW THROUGH ROOT SURFACE INSIDE THE

ROOT

Water flow through root surface then there will be change in water pressure

which serves as boundary condition in equation of water conservation in the root The

root surface membrane work as semi permeable membrane represents the water

movement path across the cortex to the xylem The rate of water uptake by pant roots

per unit surface area 119869119908[1198881198983119888119898minus210485761199041048576minus1] is given by [7][11]

119869119908 = 119896119903 nabla119875 minus 120589Δprod (3 1)

where 119896119903 is the speed of water flow through the root surface membrane per unit

difference in water pressure on either side of the membrane nabla119875 is the hydraulic water

pressure across the membrane 120589 is the reflection which give relation between osmotic

pressure to hydrostatic and Δprod is the osmotic pressure difference across the

membrane Neglecting the osmotic pressure effects on water uptake by plant roots and

119863 is the water diffusivity in cortical tissues then for a root surface membrane with the

inner boundary at 119886119909 and an outer boundary at the root surface 119886 the water pressure

distribution inside this membrane is given by a solution to the equation

120597119901119898

120597119905=

119863

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 2)

with boundary conditions given for example by 119901119898 = 119901 on 119903 = 119886 and 119901119898 = 119901119903 at

119903 = 119886119909 where 119901119898 is the water pressure in the root cortex tissues 119901 is the water

pressure at the root surface and 119901119903 is the water pressure in the xylem Non-

dimensionalising the above equation with typical timescale [119905] and root radius a we

find that the dimensionless problem becomes

1198862

119863 119905

120597119901119898

120597119905=

1

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 3)

With

119901119898 = 119901 on 119903 = 1 and 119901119898 = 119901119903 at 119903 = 119886119909119886 = 119903119909 (3 4)


For small order radius water pressure distribution profile is at pseudo steady

state i e the time derivative term in the equation (3 3) is small and the steady state

solution is given by [8]

119901119898 =119901 minus 119901119903

minus119897119899 119903119909 119897119899 119903 + 119901 (3 5)

asymp119901 minus 119901119903

1 minus 119903119909 119903 minus 1 + 119901 119891119900119903 119903119909 lt 119903 lt 1

Hence in dimensional terms the ux of water through the unit of root surface

area is given approximately by

119863120597119901119898

120597119903|119903=119886 =

119863

119886minus119886119909 119901 minus 119901119903 (3 6)

where 119896119903 =119863

119886minus119886119909 is generally known as a root surface membrane hydraulic

conductivity Using boundary condition equation (3 6) at the root surface we can be

written as

119896 119878

120583

120597119901

120597119903= 119896119903 119901 minus 119901119903 119886119905 119903 = 119886 (3 7)

where 119896(119878) is the soil water permeability 119896119903 is the radial conductivity of root surface

membrane 119901 is the water pressure in the soil 119901119909 is the water pressure in the root

xylem and 119886 is the radius of the root

4 WATER FLOW IN THE XYLEM LONGITUDINALLY UP-

WARDS INTERIOR OF ROOT

Interior parts of root are made of macro tubes of xylem with average radius

119903119898 Assume that the number of the micro-tubes is N The total area cross-section of

the micro tubes can be written as 119873empty1199031198982 = empty1198772 We assume that the water path

effectively by a single tiny circular tube with the radius 119877 Water motion in the

interior of root is described by one dimensional Poiseuille flow along the pipe with

variable cross section area of radius r [7][9]

120583nabla2119906 = nabla119901 minus 120588119892119948 (4 1)

where 119906 is water velocity profile with non-slip boundary condition which is given by

119906 119903 119911 =1199032minus1198772

4120583 120597119901119903

120597119911minus 120588119892 (4 2)

The total flux of water passing upwards through the cross section at the level 119911

is given by


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774

8120583 then equation (4 3) becomes

119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)

5 MODEL FOR CONSERVATION OF WATER INSIDE THE

ROOT

We consider a unit length of the root and assume that there is balance between

water flow through the root surface membrane and upward in the xylem tubes Then

mass conservation law yields [7][9][16]

2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)

where 119886 is the radius of the root and 119902 is the longitudinal flux of water Substituting

the value of (4 4) in equation (5 1) then we get an ordinary differential equation for

119901119903 as a function of 119911 i e

2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)

which we can solve subject to boundary conditions

119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)

where 119879 is the average pressure at the top of the root i e at the root-shoot boundary

and L is the length of the root

6 MODEL FOR CONSERVATION OF WATER INSIDE OF THE

ROOT AT CONSTANT SOIL PRESSURE

Suppose the root surrounded by water in porous medium are at constant

pressure i e 119901 = 119875 then water pressure equation in the root system is

2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)

Non-dimensionalise the equation (6 1) with

119911 = 119871119911lowast and 119901119903 = 119879119901119903lowast (6 2)


where 119911lowast and 119901119903lowast are the dimensionless length and pressure respectively The

dimensionless equation (6 2) becomes with dropping lowast becomes

1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)

with dimensionless boundary condition (5 3) becomes

119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712

119896119909 are the dimensionless water uptake coefficients

Solution of equation (6 3) with boundary conditions (6 4) is given by

119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1

2 minus 1198750 minus 1 tan 119893 119896 minus 1198750 +

1 119890=119896119911 (6 5)

7 MODEL FOR WATER FLOW IN THE SOIL WITH ROOT

SURFACE BOUNDARY CONDITION SUBJECTED TO

CONSTANT ROOT INTERNAL PRESSURE

Water flow in soil given by equation (2 12) can be write in radial co-ordinates

form with boundary condition (3 7) is given by [10]


120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)

with boundary condition

119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)

where 119875 119903 is the constant pressure inside the root and

119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4

We non-dimensionalise this model with choice of time radius and pressure in

new scales as follows

119905 = 119905 119905lowast = empty119897 119904 minus empty119897 119903 1205721198981198862120583

1 minus 119898 119896119904120588119892119905lowast 119903 = 119886119903lowast 120583 =

120588119892

120572119901lowast (7 5)

and the dimensionless model with boundary condition becomes by dropping


120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 119901 119878 = minus 119878minus1119898 minus 1 1119899

(7 8)

and dimensionless parameters are

120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)

Equation (7 1) can also written in term of relative water saturation in the soil

[12] The equation for conservation of water in the soil is thus

120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)

where empty is the (constant) porosity of the soil 119878 is the relative water saturation in the

soil i e 119878 = empty119897empty where empty119897 is the volumetric water fraction 119958 is the volume flux of

water i e Darcy flux and 119865119908 is the uptake of water by plant roots Consider value of

u given in equation (2 7) with equation (2 10) Also the water pressure in the soil

pores can be linked to the relative water saturation via suction characteristic [14]

119901119886 minus 119901 = 119901119888119891 119878 119891 119878 = 119878minus1119898 minus 1 1minus119898

(7 11)

where 119901119886 is atmospheric pressure and 119901119888 is a characteristic suction pressure It is

common to take 119901119886 = 0

Then the Darcy-Richards equation in terms of relative water saturation only it

is written as

empty120597119878

120597119905= nabla 1198630119863 119878 nabla119878 minus 119870119904119870 119878 119948 minus 119865119960 (7 12)

where the water diffusivity in the soil is 1198630119863 119878 =119896

120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)

Also equation (2 8) can written as follows


119870119904 =120588119892119896119904

120583 (7 14)

Equation (7 12) is studied with boundary condition at soil surface and at the

base of soil layer If amount of water is flowing due to rainfall then we take boundary

minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)

where 119902119904 is the volume flux of water per unit soil surface area per unit time If surface

pounding occur then the boundary condition will be 119878 = 1 at 119911 = 0 The boundary

condition at base of soil layer is written as

minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)

The model for water uptake of a single cylindrical root in cylindrical radial co-

ordinates is written as

empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)

with boundary conditions

1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)

119886 is the radius of the root 119886119894119899119905 is a measure of inter branch distance 119896119903 is root radial

conductivity and 119901119903 is the root internal pressure 119875 le 119901119903 le 119901 119878 We take the far-field

soil water content i e effective water saturation is 119878infin which is constant Then the

far-field boundary condition is given by

119878 rarr infin 119886119904 119903 rarr infin

Non-dimensionalizing this model with 119903 = 119886 119905 = empty11988621198630 and 119901119903 = 119875

120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875

The far-filed boundary condition is still

119878 rarr infin as 119903 rarr infin

Solution of equation (7 21) is given by [7] for outer region far away from the root

119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)

where S(119903 119905) = 119878infin + 119904(119903 119905) For inner region near the root solution is given by

119878 = 119878infin + 1199041 +120582119908 119901 119878infin minus 119901119903

119863 119878infin 119897119899119903 (7 23)

where 1199041 is 1199041 = minus120582119908 119901 119878infinminus119901119903

2119863 119878infin 119897119899 4119890minus120574119863 119878infin 119905 + 1

Also we have the dimensional expression for water uptake by root from soil

given in equation(7 10) which is written as[8]

119865119908 = 2empty119886119897119889119896119903 minus119901119888119891 119878 minus 119901119903 (7 24)

where 119897119889 is the root length density

8 VAPORIZATION OF WATER THROUGH PLANT LEAVES

Leaf of plant considered as plane at 119911 = 119871 from tip of the root Let 119873119886 is the

number of total opened micro pores of water pathway on the leaves with effective

radius 119903119894 hence effective area of water pathway in plant is 1205871199032 which is actually the

total areas of all micro pores on leaves namely 1205871199031198942119873119886

119894=1 [9][13] The liquid pressure

of the opened pore on the leaves 119901119894119900

is negative It is caused by the transpiration on

the leaves There are two well known theories about 119901119894119900

(1) The liquid in micro pores and air are separated by curved surface due to capillarity

negative 119901119894119900 is generated Suppose 119901119886 is air pressure and 120589 is surface tension

coefficient of liquid then we have 119901119886 minus 119901119894119900 =

120589

119903119894 where 119903119894 is radius of the vessel


(2) We start with assumption that liquid and air are separated by a semi permeable

membrane and the fractional pressure of air 119901119886depends on the humidity Suppose the

concentration of water in micro pores is C according 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 119901119894119900 = 119901119886 + 119901120587 where 119901120587

is the osmotic pressure which depends on the difference of the concentrations

(119862 minus 119901119894119900 119901119886) of the water on the both sides of the membrane

The above condition is applicable for the system that the solutions on the both

sides of the membrane are in the thermodynamics equilibrium If the solution on the

both sides of the membrane are not in the thermodynamics equilibrium then there

exists a difference of pressures ∆119901 ne 0 then for first case we have ∆119901 = 119901119894119900 minus 119901119886 minus

120589

119903119894 and in second case ∆119901 = 119901119894119900

minus 119901119886 + 119901120587 When ∆119901 gt 0 the water will vaporize

into air otherwise water in air will enter leaves The flux vaporizing into air can be

expressed as 119902119894 = 119896119894∆119901 where 119896119894 is the osmosis coefficient

9 DISCUSSION

In this paper we try to elaborate the idea of water uptake by plant root which

based on the model discussed by Roose T in his dissertation of doctorate and idea

are extended by different angle using many research paper related to this field We

discussed the mathematical model for plant which is in normal circumstances the

amount of water within a plant is set by a balance between the water uptake through

the roots and the rate of transpiration through the leaves

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water uptake of steadily growing plant root Sci China Ser G-Phy Mech Astron

vol 51 no 2 2008 184-205

[10] T Roose A C Fowler A mathematical model for water and nutrient uptake by

plant root systems Journal of Theoretical Biology 228 2004 173-184

[11] Melvin T Tyree Shudong Yang Pierre Cruiziat and Bronwen Sinclair Novel

Methods of Measuring Hydralic conductivity of tree root Systems

and Interpretation using AMAIZED Plant Physiol 104 1994 189-199



[13] Vant Hoff J H The role of osmotic pressure in the analogy between solution

and gases Z Phy Chem 1887 481-508

[14] M TH Van Genuchten A closed-form equation for predicting the hydraulic

conductivity of unsaturated soils Soil Science Society of America Journal 44

1980 892-898

[15] F J Molz Models of water transport in the soil-plant system Water Resources

Research 17(5) 1981 1245-1260

[16] J J Landsberg and N D Fowkes Water movement through plant roots Annals

of Botany 42 1978 493-508

ajm608


Key words amp PhrasesPlant root Root water uptake Poiseuille flow Mathematical modelXX

mechanism of water and nutrients is invaluable for utilizing water and fertilizer for

increasing production Now a new trend of planting inedible plant emerges on

industrial basis The view of planting inedible plant are prevent the salinization

desertification of soil to clean pollution of heavy metals radioelement and plants

mining To collect the valuable metals like gold in soil by planting some plants

whose roots possess a special capability of absorbing the valuable metals The plants

of genus Bauhina have many species out of which Bauhinia variegata plant extract is

analyzed and found it contain micro-particles of gold [1] Since ancient times

Bauhinia racemosa Lam family Caesalpinaceae has been an integral part of life in

India Leaves of Bauhinia racemosa are traditionally used on occasion of Dashera

festival as symbol of gold in India [2] Recently proved that Bauhina racemosa extract

also contain micro particles of gold Water enters into root through root surface from

the soil Then it flow through micro-tube of xylem in the stem to rich leaves and being

vaporized into air from the pores on leaves The transport of water in plant can affect

the water flow in soil and vice versa Then study of Mathematical model for plant

water uptake divides into the parts (1) the flow of water in the porous soil (2) water

flow into root through root surface (3) flow of water through micro-tube of xylem

(4) vaporization of water through leaves In recent years a number of researchers

from various fields such as physics applied mathematics and plant physiology paid

more attention to develop mathematical model for water and nutrient uptake The

outstanding work in this field is done by T Roose and proposed a mathematical

model for uptake of water and nutrient Roose work is development of Nye Tinker

and Barber model for water and nutrient uptake assuming that the root is an infinitely

long cylinder [7][15] Before this model many root water uptake model like a linear

root water uptake model (Prasad Rama 1988) [3] CERES (Ritche 1985) [4] and

SWAP (Van Dam et al 1997) [5] was developed

2 FLOW OF WATER IN POROUS MEDIUM

In 1856 Henry Darcy make experiment and concluded that rate of flow ( i e

volume of water per unit time) 119876 is (a) proportional to the cross-sectional area 119860 (b)

proportional to (1198931 minus 1198932) the lengths 1198931 and 1198932are measured with respect to some

arbitrary datum level (c) inversely proportional to the length 119871 [6]

119876 = 119870119860(1198931minus1198932)

119871 (2 1)

where 119870 is a coefficient of proportionality which is known as soil hydraulic

conductivity Equation (2 1) is known as Darcy formula Specific discharge 119958 as the

volume of water owing per unit time through a unit cross-sectional area normal to the

direction of flow i e 119958 =119928

119912 we obtain

119958 = 119870119869 (2 2)

where 119869 = (1198931minus1198932)

119871 is in one dimension In three dimension the generalization






119881 =119876

119899119860=

119958

119899 (2 4)


written as

119958 = minus119948

120641 119892119903119886119899119889 119901 minus 120588119892 (2 5)

where =119896120588119892


120574





120597empty119897

120597119905+ nabla 119958 = 0 (2 6)


119958 = minus119948

120641 nabla119901 minus 120588119892119896 (2 7)



119870 =119896119901119892

120583 (2 8)













119870 = 11987011990411987812[1 1048576minus (1 1048576minus 119878119898 )119898 ]2 119891119900119903 0 lt 119898 lt 1 (2 10)





= [1

1 + 120572119893 119899 ]119898 =

1

1 + 120572120588119892

minus119901 119899

119898

119898 = 1 minus1

119899 0 lt 119898 lt 1 (2 11)







120588119892

120583

119889119896

119889119878

120597119878

120597119911 2 12


119863 119878 =119896 119878

120583 120597119901

120597119878

=119896119904120588119892

120583


prop 11989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 (2 13)

and

119889119896(119878)

119889119878= 119896119904

1

2119878minus


1119898

119898

2

+ 2119878 minus12

+1119898

1 minus 1 minus 1198781119898

119898

1 minus 1198781119898

119898minus1

(2 14)

RANGE OF VALIDITY







119877119890 =119958119889

119907 (2 15)









ROOT















120597119901119898

120597119905=

119863

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 2)






1198862

119863 119905

120597119901119898

120597119905=

1

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 3)

With

119901119898 = 119901 on 119903 = 1 and 119901119898 = 119901119903 at 119903 = 119886119909119886 = 119903119909 (3 4)





119901119898 =119901 minus 119901119903

minus119897119899 119903119909 119897119899 119903 + 119901 (3 5)

asymp119901 minus 119901119903

1 minus 119903119909 119903 minus 1 + 119901 119891119900119903 119903119909 lt 119903 lt 1



119863120597119901119898

120597119903|119903=119886 =

119863

119886minus119886119909 119901 minus 119901119903 (3 6)

where 119896119903 =119863



written as

119896 119878

120583

120597119901

120597119903= 119896119903 119901 minus 119901119903 119886119905 119903 = 119886 (3 7)














119906 119903 119911 =1199032minus1198772

4120583 120597119901119903

120597119911minus 120588119892 (4 2)


is given by


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774


119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)


ROOT




2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)




2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)


119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)







2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)






1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608






119881 =119876

119899119860=

119958

119899 (2 4)


written as

119958 = minus119948

120641 119892119903119886119899119889 119901 minus 120588119892 (2 5)

where =119896120588119892


120574





120597empty119897

120597119905+ nabla 119958 = 0 (2 6)


119958 = minus119948

120641 nabla119901 minus 120588119892119896 (2 7)



119870 =119896119901119892

120583 (2 8)













119870 = 11987011990411987812[1 1048576minus (1 1048576minus 119878119898 )119898 ]2 119891119900119903 0 lt 119898 lt 1 (2 10)





= [1

1 + 120572119893 119899 ]119898 =

1

1 + 120572120588119892

minus119901 119899

119898

119898 = 1 minus1

119899 0 lt 119898 lt 1 (2 11)







120588119892

120583

119889119896

119889119878

120597119878

120597119911 2 12


119863 119878 =119896 119878

120583 120597119901

120597119878

=119896119904120588119892

120583


prop 11989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 (2 13)

and

119889119896(119878)

119889119878= 119896119904

1

2119878minus


1119898

119898

2

+ 2119878 minus12

+1119898

1 minus 1 minus 1198781119898

119898

1 minus 1198781119898

119898minus1

(2 14)

RANGE OF VALIDITY







119877119890 =119958119889

119907 (2 15)









ROOT















120597119901119898

120597119905=

119863

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 2)






1198862

119863 119905

120597119901119898

120597119905=

1

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 3)

With

119901119898 = 119901 on 119903 = 1 and 119901119898 = 119901119903 at 119903 = 119886119909119886 = 119903119909 (3 4)





119901119898 =119901 minus 119901119903

minus119897119899 119903119909 119897119899 119903 + 119901 (3 5)

asymp119901 minus 119901119903

1 minus 119903119909 119903 minus 1 + 119901 119891119900119903 119903119909 lt 119903 lt 1



119863120597119901119898

120597119903|119903=119886 =

119863

119886minus119886119909 119901 minus 119901119903 (3 6)

where 119896119903 =119863



written as

119896 119878

120583

120597119901

120597119903= 119896119903 119901 minus 119901119903 119886119905 119903 = 119886 (3 7)














119906 119903 119911 =1199032minus1198772

4120583 120597119901119903

120597119911minus 120588119892 (4 2)


is given by


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774


119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)


ROOT




2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)




2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)


119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)







2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)






1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608




119870 = 11987011990411987812[1 1048576minus (1 1048576minus 119878119898 )119898 ]2 119891119900119903 0 lt 119898 lt 1 (2 10)





= [1

1 + 120572119893 119899 ]119898 =

1

1 + 120572120588119892

minus119901 119899

119898

119898 = 1 minus1

119899 0 lt 119898 lt 1 (2 11)







120588119892

120583

119889119896

119889119878

120597119878

120597119911 2 12


119863 119878 =119896 119878

120583 120597119901

120597119878

=119896119904120588119892

120583


prop 11989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 (2 13)

and

119889119896(119878)

119889119878= 119896119904

1

2119878minus


1119898

119898

2

+ 2119878 minus12

+1119898

1 minus 1 minus 1198781119898

119898

1 minus 1198781119898

119898minus1

(2 14)

RANGE OF VALIDITY







119877119890 =119958119889

119907 (2 15)









ROOT















120597119901119898

120597119905=

119863

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 2)






1198862

119863 119905

120597119901119898

120597119905=

1

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 3)

With

119901119898 = 119901 on 119903 = 1 and 119901119898 = 119901119903 at 119903 = 119886119909119886 = 119903119909 (3 4)





119901119898 =119901 minus 119901119903

minus119897119899 119903119909 119897119899 119903 + 119901 (3 5)

asymp119901 minus 119901119903

1 minus 119903119909 119903 minus 1 + 119901 119891119900119903 119903119909 lt 119903 lt 1



119863120597119901119898

120597119903|119903=119886 =

119863

119886minus119886119909 119901 minus 119901119903 (3 6)

where 119896119903 =119863



written as

119896 119878

120583

120597119901

120597119903= 119896119903 119901 minus 119901119903 119886119905 119903 = 119886 (3 7)














119906 119903 119911 =1199032minus1198772

4120583 120597119901119903

120597119911minus 120588119892 (4 2)


is given by


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774


119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)


ROOT




2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)




2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)


119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)







2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)






1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608


119877119890 =119958119889

119907 (2 15)









ROOT















120597119901119898

120597119905=

119863

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 2)






1198862

119863 119905

120597119901119898

120597119905=

1

119903

120597

120597119903 119903

120597119901119898

120597119903 (3 3)

With

119901119898 = 119901 on 119903 = 1 and 119901119898 = 119901119903 at 119903 = 119886119909119886 = 119903119909 (3 4)





119901119898 =119901 minus 119901119903

minus119897119899 119903119909 119897119899 119903 + 119901 (3 5)

asymp119901 minus 119901119903

1 minus 119903119909 119903 minus 1 + 119901 119891119900119903 119903119909 lt 119903 lt 1



119863120597119901119898

120597119903|119903=119886 =

119863

119886minus119886119909 119901 minus 119901119903 (3 6)

where 119896119903 =119863



written as

119896 119878

120583

120597119901

120597119903= 119896119903 119901 minus 119901119903 119886119905 119903 = 119886 (3 7)














119906 119903 119911 =1199032minus1198772

4120583 120597119901119903

120597119911minus 120588119892 (4 2)


is given by


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774


119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)


ROOT




2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)




2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)


119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)







2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)






1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608





119901119898 =119901 minus 119901119903

minus119897119899 119903119909 119897119899 119903 + 119901 (3 5)

asymp119901 minus 119901119903

1 minus 119903119909 119903 minus 1 + 119901 119891119900119903 119903119909 lt 119903 lt 1



119863120597119901119898

120597119903|119903=119886 =

119863

119886minus119886119909 119901 minus 119901119903 (3 6)

where 119896119903 =119863



written as

119896 119878

120583

120597119901

120597119903= 119896119903 119901 minus 119901119903 119886119905 119903 = 119886 (3 7)














119906 119903 119911 =1199032minus1198772

4120583 120597119901119903

120597119911minus 120588119892 (4 2)


is given by


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774


119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)


ROOT




2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)




2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)


119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)







2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)






1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608


119902 = minusempty1198774

8120583 120597119901119903

120597119911minus 120588119892 (4 3)

let 119896119909 =empty1198774


119902 = minus119896119909 120597119901119903

120597119911minus 120588119892 (4 4)


ROOT




2120587119886119896119903 119901 minus 119901119903 =120597119902

120597119911 (5 1)




2120587119886119896119903 119901 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (5 2)


119901119903 = 119879 119886119905 119911 = 0 and 120597119901119903

120597119911= 0 at 119911 = 119871 (5 3)







2120587119886119896119903 119875 minus 119901119903 = minus119896119909

1205972119901119903

1205971199112 (6 1)






1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608




1198962 1198750 minus 119901119903 = minus1205972119901119903

1205971199112 (6 3)


119901119903 = 1 at 119911 = 119900 and 120597119901119903

120597119911= 0 at 119911 = 1 (6 4)

where 1198750 =119875

119879 and 1198962 =

21205871198861198961199031198712



119901119903 = 1198750 +1

2 1198750 minus 1 119905119886119899119893 119896 minus 1198750+ + 1 119890119896119911 +

1


1 119890=119896119911 (6 5)







120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 1)


119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119875119903 119886119905 119903 = 119886 (7 2)


119863 119878 =119896119904120588119892

120583times

1 minus 119898

12057211989811987812minus1119898 1 minus 1198781119898

minus119898+ 1 minus 1198781119898

119898minus 2 7 3

119901 119878 = minus120588119892

120572 119878minus1119898 minus 1

1119899 7 4





120588119892

120572119901lowast (7 5)



120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608


120597119878

120597119905=

1

119903

120597

120597119903 119863 119878 119903

120597119878

120597119903 (7 6)

119863 119878 120597119878

120597119903 = 120582119908 119901 119878 minus 119875119903 119886119905 119903 = 1 (7 7)

where

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898


(7 8)


120582119908 =119896119903120588119892119898119886

1 minus 119898 119870119904 119908119894119905119893 119870119904 =

119896119904120588119892

120583 119886119899119889 119901119903 =

119875119903120572

120588119892 (7 9)



120593120597119878

120597119905+ nabla 119958 = minus119865119960 (7 10)







(7 11)


common to take 119901119886 = 0


is written as

empty120597119878



120583

120597119901

120597119878 1198630 =

119901119888119896119904

120583

1minus119898

119898 and

119863 119878 = 11987812minus

1119898 1 minus 119878

1119898

minus119898

+ 1 minus 1198781119898

119898

minus 2 (7 13)



119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608


119870119904 =120588119892119896119904

120583 (7 14)



minus1198630119863 119878 120597119878

120597119911= 119902119904 119886119905 119911 = 0 (7 15)




minus1198630119863 119878 120597119878

120597119911+ 119870119904119870 119878 = 0 119886119905 119911 = 119897119908 (7 16)



empty120597119878

120597119903=

1

119903

120597

120597119903 1198630119863 119878 119903

120597119878

120597119903 (7 17)


1198630119863 119878 120597119878

120597119903 = 119896119903 119901 119878 minus 119901119903 119886119905 119903 = 119886

120597119878

120597119903= 0 119886119905 119903 = 119886119894119899119905 (7 18)

where

p(S) = -119901119888 119878minus1119898 minus 1

1minus119898 (7 19)







120597119878

120597119905=

1

119903

120597

120597119903 119903119863 119878

120597119878

120597119903 (7 20)



119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608


119863 119878 120597119878

120597119903= 120582 minus1205760119891 119878 minus 119901119903 119886119905 119903 = 1

120597119878

120597119903= 0 119886119905 119903 =

119886119894119899119905

119886 (7 21)

where 120582 =119896119903119886 119875

1198630 and 120576 = 119901119888 119875




119878 = 119878infin + 119904 = 119878infin + 1198611198641 1199032

4119905119863(119878infin) (7 22)



119863 119878infin 119897119899119903 (7 23)



















120589














120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608













120589





9 DISCUSSION







REFERENCES






2012


(3-4) 1988 297-306







Wageningen 1997








vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608




vol 51 no 2 2008 184-205












1980 892-898


Research 17(5) 1981 1245-1260


of Botany 42 1978 493-508

ajm608

MATHEMATICAL MODELING FOR WATER UPTAKE …...MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT...

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Transcript of MATHEMATICAL MODELING FOR WATER UPTAKE …...MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT...