Laplace medium. The problem is formulated in terms of

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.

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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.

x

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