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Antarctica J. Math., 11(3)(2014), 265-277 www.domainsmoon.com MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT IN SATURATED AND UNSATURATED ZONE P. S. AVHALE [Email : [email protected]] Department of Mathematics, Shivaji Arts and Science College, Kannad, Aurangabad [District], Maharastra, India. & 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-76Zxx76Z05 and 92XX-92B 265 266 P. S. AVHALE and S. B. KIWNE Key words & Phrases:Plant 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 plant's 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), π is (a) proportional to the cross-sectional area π΄, (b) proportional to (β1 − β2 ), the lengths β1 and β2 are measured with respect to some arbitrary datum level (c) inversely proportional to the length πΏ [6]. π=πΎ π΄(β1 −β2 ) πΏ (2. 1) where πΎ is a coefficient of proportionality, which is known as soil hydraulic conductivity. Equation (2. 1) is known as Darcy formula. Specific discharge, π as the volume of water owing per unit time through a unit cross-sectional area normal to the πΈ direction of flow, i. e. , π = π¨, we obtain π = πΎπ½ (2. 2) where π½ = (β1 −β2 ) πΏ is in one dimension. In three dimension, the generalization MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT IN SATURATED AND UNSATURATED ZONE 267 of equation (2. 2) is given by π = πΎπ½ = − τπΎπππππ = τ − ∇∅ (2. 3) The average areal porosity is equal to the volumetric porosity π, the portion of the area π΄ available to flow is ππ΄. The average velocity π of flow is given by π= π π = (2. 4) ππ΄ π Another form of Darcy formula is in term of specific discharge, i. e. , Darcy flux is written as π π = − (πππππ π − ππ) (2. 5) π where = πππ π πΎ , π is dynamic viscosity of fluid, π = π is the density of the fluid, k is the medium's permeability and π = −πππ is the vector gravity acceleration directed downward. If fluid is not compressible then ∇π’ = 0 . The equation for the conservation of water in the soil with no water sink is given by [7], π∅π + ∇. π = 0 (2. 6) ππ‘ where π is given by, π π = − [∇π − πππΜ] (2. 7) π The π soil permeability is just differ from hydraulic conductivity K by scaling factor which is given by, πΎ= πππ , (2. 8) π where ππ present the density of water and gravity. If soil is saturated then we have π πππ = 0 therefore the Darcy flux is given by = − π [∇p] . However, the conductivity of unsaturated soil is determined from experiments in terms of effective water saturation, where effective water saturation π is defined as π = ∅π − ∅π,π (2. 9) ∅π,π − ∅π,π where ∅π is the moisture content of the soil, ∅π,π is the residual (minimum) moisture content of the soil and ∅π,π is the saturated moisture content of the soil, i. e. the soil 268 P. S. AVHALE and S. B. KIWNE 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] πΎ = πΎπ π 1/2 [1 τ − (1 τ − π π )π ]2 πππ 0 < π < 1; (2. 10) where πΎπ 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 β suction −π > 0 can be written in terms of pressure head, . i. e. , −π = −ππβ. Hence we have π = [ 1 πΌ 1+( ) β π] π 1 1 =[ , 0 < π < 1; (2. 11) π] , π = 1 − π πΌ 1 + [ππ (−π)] where πΎπ 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 (∅π,π − ∅π,π ) ππ ππ ππ ππ (2. 12) = ∇. (π·(π)∇π) − ππ‘ π ππ ππ§ where D(S) is called the soil water diffusivity and it is defined as π·(π) = π(π) ππ | | π ππ ππ ππ π (1 − π) 1/2−1/π −π π × π [(1 − π 1/π ) + (1 − π 1/π ) − 2] (2. 13) ∝π = and 1 π 2 ππ(π) 1 −1 = ππ [ π 2 [1 − (1 − π π ) ] ππ 2 1 1 1 π 1 (π−1) + 2π (−2+π) [1 − (1 − π π ) ] (1 − π π ) ] (2. 14) RANGE OF VALIDITY In the flow through conduits, the Reynolds number, π π , 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] MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT IN SATURATED AND UNSATURATED ZONE π π = 269 ππ (2. 15) π£ where π is some representative length of the porous matrix and π£ is the kinematic viscosity of the fluid. Practically all evidence indicates that Darcy's 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 Darcy's 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 π½π€[ππ3 ππ−2 τπ τ−1 ] is given by [7][11] π½π€ = ππ (∇π − πΔ∏) (3. 1) where ππ is the speed of water flow through the root surface membrane per unit difference in water pressure on either side of the membrane, ∇π is the hydraulic water pressure across the membrane, π is the reflection which give relation between osmotic pressure to hydrostatic and Δ∏ is the osmotic pressure difference across the membrane. Neglecting the osmotic pressure effects on water uptake by plant roots and π· is the water diffusivity in cortical tissues, then for a root surface membrane with the inner boundary at ππ₯ and an outer boundary at the root surface π, the water pressure distribution inside this membrane is given by a solution to the equation πππ π· π πππ = (π ) (3. 2) ππ‘ π ππ ππ with boundary conditions given for example by ππ = π on π = π and ππ = ππ at π = ππ₯ , where ππ is the water pressure in the root cortex tissues, π is the water pressure at the root surface and ππ is the water pressure in the xylem. Nondimensionalising the above equation with typical timescale [π‘] and root radius a, we find that the dimensionless problem becomes π2 πππ 1 π πππ = (π ) (3. 3) π·[π‘] ππ‘ π ππ ππ With ππ = π on π = 1 and ππ = ππ at π = ππ₯ /π = ππ₯ (3. 4) 270 P. S. AVHALE and S. B. KIWNE 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] ππ = ≈ π − ππ ππ(π) + π (3. 5) −ππ(ππ₯ ) π − ππ (π − 1) + π πππ ππ₯ < π < 1 1 − ππ₯ Hence, in dimensional terms the ux of water through the unit of root surface area is given approximately by π· where ππ = π· (π−ππ₯ ) πππ ππ π· |π=π = π−π (π − ππ ) (3. 6) π₯ 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 π(π) ππ π ππ = ππ (π − ππ ) ππ‘ π = π (3. 7) where π(π) is the soil water permeability, ππ is the radial conductivity of root surface membrane, π is the water pressure in the soil, ππ₯ is the water pressure in the root xylem and π is the radius of the root. 4. WATER FLOW IN THE XYLEM LONGITUDINALLY UPWARDS INTERIOR OF ROOT Interior parts of root are made of macro tubes of xylem with average radius ππ . Assume that the number of the micro-tubes is N. The total area cross-section of the micro tubes can be written as π∅ππ2 = ∅π 2 . We assume that the water path effectively by a single tiny circular tube with the radius π . 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]. π∇2 π’ = ∇π − πππ (4. 1) where π’ is water velocity profile with non-slip boundary condition which is given by π’(π, π§) = π 2 −π 2 πππ 4π [ ππ§ − ππ] (4. 2) The total flux of water passing upwards through the cross section at the level π§ is given by MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT IN SATURATED AND UNSATURATED ZONE π=− let ππ₯ = ∅π 4 8π 271 ∅π 4 πππ 8π [ ππ§ − ππ] (4. 3) then equation (4. 3) becomes ππ π = −ππ₯ [ ππ§π − ππ] (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], 2ππππ (π − ππ ) = ππ (5. 1) ππ§ where π is the radius of the root and π is the longitudinal flux of water. Substituting the value of (4. 4) in equation (5. 1), then we get an ordinary differential equation for ππ as a function of π§, i. e. , 2ππππ (π − ππ ) = −ππ₯ π 2 ππ (5. 2) ππ§ 2 which we can solve, subject to boundary conditions ππ = π ππ‘ π§ = 0, and πππ ππ§ = 0 at π§ = πΏ (5. 3) where π 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. , π = π then water pressure equation in the root system is 2ππππ (π − ππ ) = −ππ₯ π 2 ππ (6. 1) ππ§ 2 Non-dimensionalise the equation (6. 1) with π§ = πΏπ§ ∗ and ππ = πππ∗ (6. 2) 272 P. S. AVHALE and S. B. KIWNE where π§ ∗ and ππ∗ are the dimensionless length and pressure respectively. The dimensionless equation (6. 2) becomes with dropping ∗ becomes π 2 (π0 π 2 ππ − ππ ) = − 2 (6. 3) ππ§ with dimensionless boundary condition (5. 3) becomes ππ = 1 at π§ = π and π where π0 = π and π 2 = 2ππππ πΏ2 ππ₯ πππ ππ§ = 0 at π§ = 1 (6. 4) are the dimensionless water uptake coefficients. Solution of equation (6. 3) with boundary conditions (6. 4) is given by 1 1 ππ = π0 + 2 [(π0 − 1)π‘ππβ(π) − π0+ + 1]π ππ§ + 2 [−(π0 − 1) tan β(π) − π0 + 1]π =ππ§ (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] (∅π,π − ∅π,π ) ππ 1 π ππ = (π·(π)π ) (7. 1) ππ‘ π ππ ππ with boundary condition π·(π) ππ = ππ [π(π) − ππ ] ππ‘ π = π (7. 2) ππ where π π is the constant pressure inside the root and π·(π) = ππ ππ (1 − π) 1/2−1/π −π π × π [(1 − π 1/π ) + (1 − π 1/π ) − 2] (7. 3) π πΌπ ππ −1/π 1/π π(π) = − (π − 1) (7. 4) πΌ We non-dimensionalise this model with choice of time, radius and pressure in new scales as follows π‘ = [π‘]π‘ ∗ = (∅π,π − ∅π,π )πΌππ2 π ∗ ππ ∗ π‘ , π = ππ ∗ , π = π (7. 5) (1 − π)ππ ππ πΌ and the dimensionless model with boundary condition becomes by dropping 273 MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT IN SATURATED AND UNSATURATED ZONE ππ 1 π ππ = (π·(π)π ) (7. 6) ππ‘ π ππ ππ π·(π) where π·(π) = 1 1 π 2−π [(1 − ππ = ππ€ [π(π) − ππ ] ππ‘ π = 1 (7. 7) ππ 1 −π π π ) + (1 − 1 π π π ) 1/π − 2] , π(π) = −(π −1/π − 1) (7. 8) and dimensionless parameters are ππ€ = ππ ππππ ππ ππ ππ πΌ π€ππ‘β πΎπ = πππ ππ = (7. 9) (1 − π)πΎπ π ππ 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 π ππ + ∇. π = −πΉπ (7. 10) ππ‘ where ∅ is the (constant) porosity of the soil, π is the relative water saturation in the soil i. e. , π = ∅π /∅ where ∅π is the volumetric water fraction, π is the volume flux of water, i. e. , Darcy flux and πΉπ€ 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], 1−π ππ − π = ππ π(π), π(π) = (π −1/π − 1) (7. 11) where ππ is atmospheric pressure, and ππ is a characteristic suction pressure. It is common to take ππ = 0. Then the Darcy-Richards equation in terms of relative water saturation only, it is written as ∅ ππ = ∇. [π·0 π·(π)∇π − πΎπ πΎ(π)π − πΉπ ] (7. 12) ππ‘ π ππ where the water diffusivity in the soil is π·0 π·(π) = π ππ , π·0 = 1 1 1 −π π·(π) = π 2−π [(1 − π π ) Also equation (2. 8) can written as follows 1 π ππ ππ 1−π π ( π + (1 − π π ) − 2] (7. 13) ) and 274 P. S. AVHALE and S. B. KIWNE πΎπ = ππππ (7. 14) π 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 −π·0 π·(π) ππ = ππ , ππ‘ π§ = 0, (7. 15) ππ§ where ππ is the volume flux of water per unit soil surface area per unit time. If surface pounding occur, then the boundary condition will be π = 1 at π§ = 0. The boundary condition at base of soil layer is written as −π·0 π·(π) ππ + πΎπ πΎ(π) = 0, ππ‘ π§ = ππ€ , (7. 16) ππ§ The model for water uptake of a single cylindrical root in cylindrical radial coordinates is written as ∅ ππ 1 π ππ = (π·0 π·(π)π ) (7. 17) ππ π ππ ππ with boundary conditions ππ π·0 π·(π) ππ = ππ [π(π) − ππ ] ππ‘ π = π ππ = 0 ππ‘ π = ππππ‘ (7. 18) ππ where 1−π p(S) = -ππ (π −1/π − 1) (7. 19) π is the radius of the root, ππππ‘ is a measure of inter branch distance, ππ is root radial conductivity and ππ is the root internal pressure, π ≤ ππ ≤ π(π). We take the far-field soil water content, i. e. , effective water saturation is π∞ which is constant. Then the far-field boundary condition is given by π → ∞ ππ π → ∞ Non-dimensionalizing this model with π = π; π‘ = ∅π2 /π·0 and ππ = |π| ππ ππ‘ with boundary conditions 1 π ππ = π ππ (ππ·(π) ππ) (7. 20) MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT IN SATURATED AND UNSATURATED ZONE π·(π) 275 ππ = π[−π0 π(π) − ππ ]ππ‘ π = 1 ππ ππ ππππ‘ = 0 ππ‘ π = (7. 21) ππ π where π = ππ π|π| π·0 and π = ππ /|π| The far-filed boundary condition is still π → ∞ as π → ∞. Solution of equation (7. 21) is given by [7] for outer region far away from the root π2 π = π∞ + π = π∞ + π΅πΈ1 (4π‘π·(π )) (7. 22) ∞ where S(π; π‘) = π∞ + π (π, π‘), For inner region near the root solution is given by π = π∞ + π 1 + where π 1 is π 1 = − ππ€[π(π∞ −ππ )] 2π·(π∞ ) ππ€ [π(π∞ − ππ )] πππ (7. 23) π·(π∞ ) ππ[4π −πΎ π·(π∞ )π‘ + 1] Also we have the dimensional expression for water uptake by root from soil given in equation(7. 10) which is written as[8], πΉπ€ = 2∅πππ ππ [−ππ π(π) − ππ ] (7. 24) where ππ is the root length density. 8. VAPORIZATION OF WATER THROUGH PLANT LEAVES Leaf of plant considered as plane at π§ = πΏ from tip of the root. Let ππ is the number of total opened micro pores of water pathway on the leaves with effective radius ππ , hence effective area of water pathway in plant is ππ 2 which is actually the ππ total areas of all micro pores on leaves, namely ∑π=1 πππ2 [9][13]. The liquid pressure , of the opened pore on the leaves πππ is negative. It is caused by the transpiration on , the leaves. There are two well known theories about πππ . (1) The liquid in micro pores and air are separated by curved surface due to capillarity , negative πππ is generated. Suppose ππ is air pressure and π is surface tension π , coefficient of liquid, then we have ππ − πππ = π , where ππ is radius of the vessel. π 276 P. S. AVHALE and S. B. KIWNE (2) We start with assumption that liquid and air are separated by a semi permeable membrane and the fractional pressure of air ππ, depends 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 πππ = ππ + ππ where ππ is the osmotic pressure, which depends on the difference of the concentrations (πΆ − , πππ /ππ ) 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 βπ ≠ 0 then for first case we have, βπ = πππ − ππ − π , and in second case βπ = πππ − ππ + ππ . When βπ > 0 , the water will vaporize π π into air, otherwise water in air will enter leaves. The flux vaporizing into air can be expressed as ππ = ππ βπ, where ππ 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. 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