Solution of Porous Medium Equation Arising in Fluid ...

International Journal of Futuristic Trends in Engineering and Technology ISSN: 2348-5264 (Print), ISSN: 2348-4071 (Online) Vol. 1 (10), 2014

Solution of Porous Medium Equation Arising in Fluid Flow through Porous Media by Homotopy Perturbation Method Using Elzaki Transform (Paper ID: 10ET30112014010)

Shraddha S. Chavan

Mihir M. Panchal

Department of Applied Mathematics and Humanities SardarVallabhbhai National Institute of Technology Surat. Gujarat. India [email protected]

Department of Applied Sciences and Humanities C. G. Patel Institute of Technology Bardoli, Gujarat. India [email protected]

Abstract: In this Paper, We have discussed the Application of Homotopy Perturbation Method using Elzaki transform for finding Solution of Porous Medium Equation arising in fluid flow through porous media. In this paper we have discussed Ground water infiltration problem in Fluid Flow through Homogeneous Porous Media.

storage capacity of a basin. Thus first an equation is derived for mean water table height on the basis of hydraulic theory of ground water by means of Boussinesq’s equation; second is to solve that equation by using Elzaki Transform[1],[2] using Homotopy perturbation method[3]. Then with the help of height of free surface, the atmospheric pressure in dry region and velocity of filtered ground water is calculated by using Darcy’s law.

I.

INTRODUCTION

Infiltration is the process by which water on the ground surface enters the soil. Infiltration rate in soil science is a measure of the rate at which soil is able to absorb rainfall or irrigation. It is measured in inches per hour or millimetre per hour. The rate decreases as the soil becomes saturated. If the precipitation rate exceeds the infiltration rate, round off will usually occur unless; there is some physical barrier. It is related to the saturated hydraulic conductivity of the near surface soil. The rate of infiltration can be measured using an infiltrometer. Infiltration is measured by two forces: Gravity and capillary action, while smaller pores offer greater resistance to gravity, very small pores pull water through capillary action in addition to and even against the force of gravity. The groundwater flow plays an important role in fluid mechanics; fluid flow through porous media, chemical engineering, ground water hydrology and other fields of fluids. It deals with the filtration of an incompressible fluid flow through the porous medium. The infiltration of incompressible fluid is useful in control salinity of water, contamination of water including agricultural purpose and it is also useful in nuclear waste deposable problems.Such problems are also useful to measure moisture content of water in vertical one dimensional ground water recharge and dispersion of any fluid in porous media. It has been discussed by Mehta & Meher [6] from different point of view. II.

III.

MATHEMATICAL FORMULATION AND SOLUTION OF THE PROBLEM

Infiltration is a process by which water on the ground surface enter the soil. The purpose of this chapter is to present physically meaningful technique to determine effective height of the water table as a measure of initial storage capacity of a basin. Impose the following simplifying assumptions: (i) The stratum has height H and lies on top of a horizontal impervious bed, which is labelled as z = 0; (ii) Ignore the transversal variable y; and (iii)The water mass which infiltrates the soil occupies a region described as Ω = {(x, z)∈ R: z ≤ h(x, t)} Here it is assumed that there is no region of partial saturation. This is an evolution model. Clearly, 0 ≤ h(x, t) ≤ H, H maximum height of free surface and the free boundary function h is also an unknown of the problem. In this situation, we arrive at a system of three equations with unknowns the two velocity components u, w and the pressure P in a variable domain: one equation of mass conservation for an incompressible fluid and two equations for the conservation of momentum of the Navier–Stokes type. The resulting system is too complicated and can be simplified for the practical computation after introducing a suitable assumption, the hypothesis of almost horizontal flow, i.e., assume that the flow has an almost horizontal speed u  (u, 0), so that h has small gradients. It follows that in the vertical component of the momentum equation ( ) + u.∇ )=− – (3.1)

STATEMENT OF THE PROBLEM

Infiltration is the process by which water on the ground surface enters the unsaturated soil. The purpose of this chapter is to present physically meaningful technique to determine effective height of the water table as a measure of initial Akshar Publication © 2014

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Now neglect the inertial term (the left-hand side).Integration in z gives for this first approximation P + ρgz = constant.

As Height of water table decreases exponentially when increases for any time t > 0, which is physically consistent with the real Phenomena [11]. − ℎ − =0 (3.11) Taking Elzaki transform of equation (3.11) subject to the initial condition, it gives [ℎ( , )] =

(

)+

ℎ

+

(3.12)

The inverse of the Elzaki transform of above equation (3.12) gives, ℎ( , ) = Fig. 1.

Now calculate the constant on the free surface z = h(x, t). If we impose continuity of the pressure across the interface, we have P = 0 (assuming constant atmospheric pressure in the air that fills the pores of the dry region z > h(x, t)) then it gives, P = ρg (h – z) (3.2) In other words, the pressure is determined by means of the hydrostatic approximation. Now go to the mass conservation law which will give the equation. Now proceed as follows: Take a section S = (x, x + a) × (0, C). Then =∫ . (3.3) ∫ ∫ Where is the porosity of the medium, i.e., the fraction of volume available for the flow circulation, and u is the velocity, which obeys Darcy’s law[4] in the form that includes gravity effects u= − ∇( + ℎ) (3.4)

Thus the Boussinesq’s equation is obtained ℎ = (ℎ ) (3.6) With constant = ρg /2 μ. This is the fundamental equation in ground water infiltration which represents porous media equation [5]. So finally the derivation of Boussinesq’s equation in one dimension is obtained for simplicity, but it generalizes immediately to several dimensions and gives ht = ∆(h2 ) (3.7) =

=

(h )

+

(3.8)

(3.9)

For C=2c=Constant, equation (3.9) becomes =

ℎ

+

(ℎ)=ℎ

+(

(ℎ)=ℎ

+ℎ

) +2

176 + …… (3.15) ! Equation (3.15) is the approximate solution of the equation (3.10). Now calculating pressure by equation (3.2) i.e. P= ρg (h – z) Putting the value of h from (3.15) in (3.2) it gives the pressure P P= ( +2 +9 − ) P= ( +2 +9 )− (3.16) P+ = ( +2 +9 ) Calculating Velocity u by using Darcy Law i.e. } u = − ∇{P +

2ℎ

ℎ

(3.13)

[ 176 ] = 176 = ! Proceeding in similar manner 4 :ℎ ( , ) … … … … … … … … …. 5 :ℎ ( , ) … … … … … … … … So that the solution ℎ( , ) = ℎ + ℎ + ℎ + ℎ … … …. = +2 +9 +

=2 ℎ Finally it gives, =2

+

(ℎ)= ℎ +ℎ +ℎ +( ) +2 Comparing the coefficient of like powers of it gives 0 :ℎ ( , )= , (ℎ) = 2 1 [ (ℎ)] : ℎ ( , )= [2 ] =2 = (ℎ) = 18 2 [ (ℎ)] :ℎ ( , )= [ 18 ] =9 = (ℎ) = 176 3 [ (ℎ)] :ℎ ( , )=

On the right-hand lateral surface it haves u · n ≈ (u, 0) · (1, 0) = u, i.e., − (k/μ)px, while on the left-hand side it have −u. Using the formula for P and differentiating in x, it gives = ℎ (3.5) ∫

The equation becomes

ℎ

Now, we apply Homotopy Perturbation Method to equation (3.13) it gives, ∑∞ [∑∞ ℎ ( , )= + (ℎ)] (3.14) Where Hn(h) are He’s polynomials. The first few components of He’s polynomials are given by

Groundwater Infiltration

2

+

(3.10)

Having initial condition ℎ( , 0) = Akshar Publication © 2014


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Therefore the velocity is obtained, u= IV.

n=8.90*(10)^-4 q=0.5 g=9.8 c=(g*q*k)/(2*n) s0={Exp[-x]+c*(2*Exp[-2*x])*t+(9*Exp[-3*x])*(t^2)*c} s=Expand[s0] p=Expand[s/.{t->0.1}] u=Expand[s/.{t->0.2}] v=Expand[s/.{t->0.3}] w=Expand[s/.{t->0.4}] x1=Expand[s/.{t->0.5}] Table[{p,u,v,w,r,x1},{x,0.1,1,.1}] Plot[{p,u,v,w,r,x1},{x,0.1,1}]

(3.17)

NUMERICAL AND GRAPHICAL PRESENTATIONS:

Numerical and graphical presentations of equation (3.15), (3.16) and (3.17) have been obtained by using MATHEMATICA Coding which is as follows: Programm :A k=1.256*(10)^-6 Table. 1.

Height of water table h(x,t) at different distance

keeping time fixed

Distance t=0.2

0.1

0.9056

0.2

0.8253

0.3

0.7443

0.7458

0.4

0.6730

0.6742

0.5

0.6086

0.6095

0.6

0.5504

0.5511

0.7

0.4978

0.4983

0.4503

0.4507

0.4073

0.4076

0.3685

0.3687

0.8 0.9

0.4071

1

Height of water table different distance when

c=(g*q*k)/(0.99*n) z=0.1 s0=(q)*(g)*{Exp[-x]+c*(2*Exp[-2*x])*t+(9*Exp[3*x])*(t^2)*c}-(q)*(g)*(z) s=Expand[s0] p=Expand[s/.{t->0.1}] u=Expand[s/.{t->0.2}] v=Expand[s/.{t->0.3}]

at 0.1, 0.2, 0.3, 0.4, 0.5

Program:B k=1.256*(10)^-6 n=8.90*(10)^-4 q=0.5 g=9.8 Akshar Publication © 2014


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w=Expand[s/.{t->0.4}] r=Expand[s/.{t->0.5}] Table[{p,u,v,w,r},{x,0.1,0.5,.1}] Plot[{p,u,v,w,r},{x,0.1,0.5}] Table. 2.

Distan ce

0.2 0.3 0.4 0.5

Pressure P of filtered water at different distance x when time is fixed

Pressure

Pressure

Pressure

Pressure

Pressure

0.1 0.2 0.3

3.9640 3.5377 3.1525

3.9810 3.5507 3.1625

4.0026 3.5671 3.1750

4.0281 3.5869 3.1901

0.4 0.5

2.8044 2.4897

2.8121 2.4957

2.8217 2.5030

2.8331 2.5117

2.4852

Velocity U at different distance when =0.1,0.2,0.3,0.4,0.5 V.

Equation 3.15 represents height of free surface of infiltrated water in unsaturated porous media. The solution is in the exponential term in ‘t’ and ‘x’ which shows that from graphical and tabular value that the height of free surface or water mound in unsaturated porous media is decreasing as distance x increases for given time t positive and which is physical fact for the phenomenon of infiltration form deduction. We have calculated pressure p of filtered water in unsaturated porous media. Here pressure and velocity of filtered water after infiltration is decreasing as distance x increases for a given time t>0 which is physically consistent with the real phenomena.

Pressure P at different distance when 0.1, 0.2, 0.3, 0.4, 0.5 Program:C k=1.256*(10)^-6 n=8.90*(10)^-4 q=0.5 g=9.8 c=(g*q*k)/(0.99*n) s0={(k)*(q)*(g)}/n*{Exp[-x]+c*(4*Exp[2*x])*t+(27*Exp[-3*x])*(t^2)*c} s=Expand[s0] p=Expand[s/.{t->0.1}] u=Expand[s/.{t->0.2}] v=Expand[s/.{t->0.3}] w=Expand[s/.{t->0.4}] r=Expand[s/.{t->0.5}] Table[{p,u,v,w,r},{x,0.1,0.5,0.1}] Plot[{p,u,v,w,r},{x,0.1,0.5}] Table. 3.

CONCLUSION

REFERENCES [1] [2]

[3] [4] [5] [6]

Elzaki, Tarig M (2011), The New Integral Transform “Elzaki Transform” Global Journal of Pure and Applied Mathematics, ISSN 0973-1768. Elzaki, Tarig M & Eman M. A.(2012), Homotopy Perturbation and Elzaki Transform for Solving Nonlinear Partial Differential Equations , Mathematical Theory and Modeling ,ISSN 2224-5804 Vol.2. Hemeda A. A.(2012).Homotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations. Applied Mathematical Sciences, Vol. 6 Henry Darcy (1856). Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris. pp.305–401 JUAN LUIS V´AZQUEZ. The Porous Medium Equation. Oxford Scienc Publications. Meher, R .(2011), Solution of Nonlinear equation of single and double phase flow phenomena arising in oil recovery process, Ph.D Thesis.

Table-3 Velocity of filtered water at different distance when time is fixed

Distanc Velocity U e t=0.1 0.1

Velocity U t=0.2

Velocity U t=0.3

Velocity U t=0.4

Velocity U t=0.5

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