Laplace Adomian Decomposition

Method to study Chemical ion transport

through soil

Abstract: The paper deals with a

theoretical study of chemical ion transport in soil under a uniform external

force in the transverse direction where the soil is taken as porous medium. The

problem is formulated in terms of boundary value problem that consists of a set

of partial differential equations, which is subsequently converted to a system

of ordinary differential equations through the use of a similarity transformation

along with boundary layer approximation. The equations hence obtained by utilizing

Laplace Adomian Decomposition Method (LADM). The merit of this method lies in

the fact that much of simplifying assumptions need not be made to solve the

non-linear problem. The decomposition parameter is used only for

grouping the terms, therefore, the nonlinearities is

handled easily in the operator equation and accurate approximate solution are obtained for the said physical problem. The computational outcomes are introduced

graphically. By utilizing parametric variety, it has been demonstrated that the

intensity of the external pressure extensively influences the flow behaviour.

Keywords: Porous

medium; Adomian’s decomposition method; Laplace transformation, Reynolds number

1. Introduction

Most of the phenomena

occurring in nature are non-linear. In the biological world, non-linearity is a

common problem. Modelling different problems of soil science, for example,

fluid flow in soil involves nonlinear partial differential equations.

Khuri 1 proposed a

numerical Laplace decomposition algorithm to solve a class of nonlinear

differential equations. Agadjanov 2 applied this method for the solution of

Duffing equation. Nasser and Elgazery 3 used this method to solve

Falkner-Skan equation. The numerical system essentially outlines how the

Laplace Transform might be utilized to solve the nonlinear differential

equations by utilizing the decomposition technique.

For solving a certain

class of problems, it is found that application of a combination of Laplace

transform method and Adomian decomposition method namely Laplace Adomian

decomposition method (LADM) is very useful. Some further discussion on this

method has been made by Babolian et al. 4 and Biazar et al. 5. The method

was used by Wazwaz 6 in handling Volterra integro-differential equations and

by Dogan 7 for solving a system of ordinary differential equations.

In the present paper,

the effect of an external pressure / force on flow through a porous medium has

been analyzed by assuming the flow to be Newtonian. The analysis is carried out

by employing Laplace Adomian Decomposition Method and some important

predictions have been made on the basis of the present study. Since the study

has been carried out for a situation when the soil is subject to an external

force that may be due to gravity.

2.

Mathematical

Modelling of the Problem

Nomenclature: (x, y) : Cartesian coordinates of a point,

(u, v) : Velocity components along x- and y- directions, U0: Characteristic velocity, F0: Applied external force, k: Permeability of porous matrix, h: Half-width of

channel, ?:Non-dimensional distance, ?: Coefficient of viscosity, ?:Kinematic viscosity

of solute, ?:Density of solute, Re: Reynolds number

A transport system

mainly consists of three-dimensional (3D) vessels. However, in some cases, such

as in micro-vessels of soil it is approximately 2D and it can be considered as

channel flow. A physical sketch of the geometry is shown in Fig. 1. The x-axis is taken along the centre line of the channel, parallel to the

channel surface and y-axis in the

transverse direction. The flow is taken to be symmetric about x-axis. Although solutions of

multidimensional transient flow of any solute can be obtained by numerical

modeling, their applications are limited in the field. The one-dimensional

uptake model is based on the model given by Nye-Tinker and Barbar, but is

extended with a radial component and water uptake parameters. The developed

salt transport model is extremely flexible and allows spatial variations of

salt movement as influenced by non-uniform and uniform water application

patterns.

Fig. 1 Physical Sketch of the Problem

From a hydrological

perspective, solute movement in soil and its spatial distribution can largely

control by the water fluxes of the groundwater 14. For an improved

understanding of the magnitude of these fluxes, accurate estimates of the

temporal and spatial water uptake patterns are needed. Let

and

be the velocity components along

– axis and

-axis respectively and

be the applied

external pressure. In the absence of pressure gradient, the equation for

boundary layer flow of an incompressible fluid is

, (1)

and the continuity equation

(2)

where

the density of solute, v is the kinetic coefficient of

viscosity and k denotes the permeability of the porous medium. Assuming

that the flow is symmetric about the central line

of the channel, we

focus our attention to the flow in the region

only. Then the

boundary conditions to be taken care of are as follows:

(3)

(4)

We now introduce the non-dimensional

quantities defined by

(5)

Where U0 is the

characteristic velocity

It may be noted that the continuity

equation (2) is automatically satisfied.

In terms of the non-dimensional

variables, Equation (1) reads

(6)

Where Re and K are the Reynolds

number and porosity permeability parameter respectively, defined by

(7)

With the use of transformation (5),

the boundary conditions (3) and (4)

become

(8)

(9)

3.

Analysis

of the Model

In this section, in order to analyze the

model, we first solve equation (6) subjected to the boundary conditions (8) and

(9). For this purpose, we use the Laplace Adomian Decomposition Method (LADM).

In the first step, we consider the Laplace transformation of equation (6),

whereby we get.

(10)

Here and in the sequel, LF stands

for the Laplace transform of the function F.

Using the property of the Laplace

transform, we have

(11)

Using the boundary condition (9),

from equation (11) we obtain

(12)

Writing

(where b is a

constant), equation (12) assumes the form

(13)

Following Adomian Decomposition

Method. We assume the solution for

in the form of an

infinite series:

(14)

To write it in the form

;

(15)

Where

are the so-called

Adomian polynomials 15. To Find An, we introduce a scalar

such that

(16)

In which the parameter

used in (16) is

not a perturbation parameter; it is used only for grouping the terms of

different orders. Thus the parameterized form of (15) is given by

(17)

From the definition of Adomian

polynomials, it follows that

(18)

Now substituting (17) into (18), we

get

(19)

And so on. Substitution of equations

(14) and (15) into the equation (13), further yields

(20)

Matching both sides of equation (20) yields the

iterative algorithm:

(21)

(22)

(23)

Fig. 2 Distribution of Vm

with

for different values of

Reynolds number Re

(24)

and so on. Now considering the

inverse Laplace transform of equation (21) the following value

is obtained for

:

(25)

The first Adomian polynomial A0 calculated from eqns.

(19) and (25) is found in the form

(26)

Since

by applying Laplace

inversion, we obtain

(27)

Proceeding in a similar manner, using

(19) and (27), we calculate the second Adomian polynomial A1 given

by

(28)

Next we find the Laplace

transformation of A1 given by (28), substitute it in (23) and then

consider Laplace inversion. Thus we have found the expression of

given below.

(29)

If we consider three-term

approximation of the solution

(30)

By taking

Thus the calculated expression for

reads

(31)

The first derivative of

is given by

(32)

Now using the boundary condition

, we can obtain the expression for b in the form

(33)

The volumetric flow rate is then

given by

(34)

can be obtained by differentiating (32) and

then considering

4.

Results and Discussion

We now present here

the important results from our work in terms of pertinent dimensionless

parameters. However, for practical considerations, we also mention some typical

values of

Fig. 3 Distribution of f’ with

for different values of

Reynolds number Re

Fig. 4 Distribution of f with

for different values of

Reynolds number Re

the corresponding

dimensional parameters, as appropriate to the results subsequently obtained.

In finding the

estimates, we have taken density, ? = 1440kg.m?3, viscosity of the solute ? = 10?3kg.m?1.s?1. Fig. 2 illustrates the extent of variation in the

volumetric flow rate corresponding to different values of Reynolds number Re = 1.0, 2.0, 3.0, 4.0. The plots presented in this

figure reveal that volumetric flow rate increases with a rise in the value of

the Reynolds Re. The variation of f’

and f with

are shown in Fig. 3

and Fig. 4 respectively.

5.

Concluding Remarks

The present study deals with a theoretical investigation

of solute flow through a porous soil under the action of an external force. The

study is quite suitable for the application to the hydro-dynamical flow when it

is subjected to the influence of an externally applied force. The solution to

the nonlinear equations that govern the flow is obtained by using Adomain’s

decomposition method.