The main aim of the paper is to examine the concentration of the longitudinal dispersion phenomenon arising in fluid flow through porous media. These phenomenon yields a partial differential equation namely Burger’s equation, which is solved by mixture of the new integral transform and the homotopy perturbation method under suitable conditions and the standard assumption. This method provides an analytical approximation in a rapidly convergent sequence with in exclusive manner computed terms. Its rapid convergence shows that the method is trustworthy and introduces a significant improvement in solving nonlinear partial differential equations over existing methods. It is concluded that the behaviour of concentration in longitudinal dispersion phenomenon is decreases as distance x is increasing with fixed time t > 0 and slightly increases with time t.
The present paper discusses the solution of longitudinal dispersion phenomenon arising in the miscible fluid flow through homogenous porous media. The problem of solute dispersion during underground water movement has attracted interest from the early days of this century [
The problem is to find the concentration as a function of time t and position x, as two miscible fluids flow through porous media on either sides of the mixed region, the single fluid equation describes the motion of the fluid. The problem becomes more complicated in one dimension with fluids of equal properties. Hence, the mixing takes place longitudinally as well as transversely at time
Most of the works reveal common assumption of homogenous porous media with constant porosity, steady seepage flow velocity and constant dispersion coefficient. For such assumption, Ebach and White [
A new integral transform is derived from the classical Fourier integral. A new integral transform [
the exact or approximate solutions of nonlinear partial differential equations in physics and mathematics is still a significant problem that needs new methods to discover exact or approximate solutions. Also a new integral transform and some of its fundamental properties are used to solve general nonlinear partial differential equation with appropriate initial conditions.
A new integral transform is defined for functions of exponential order. We consider functions in the set defined as
For a given function in the set F, the constant M must be finite number,
A new integral transform denoted by the operator
In this paper, we have combined a new integral transform and homotopy perturbation method (HPM) to solve Burger’s equation arising in the dispersion phenomenon. The purpose of this study is to show the applicability and the efficiency of this mixture method.
According to Darcy’s law, the equation of continuity for the mixture, in the case of compressible fluids is given by Bear [
where
The equation of diffusion for a fluid flow through a homogeneous porous medium, without increasing or decreasing the dispersing material is given by,
where C is the concentration of the fluids,
In a laminar flow through homogeneous porous medium at a constant temperature,
Then
Therefore Equation (4) becomes,
When the seepage velocity is the along x-axis, the nonzero components are
In this case, Equation (6) becomes
where is the component of velocity along the
where
Hence Equation (7) becomes
This is the non linear Burger’s equation for longitudinal dispersion of miscible fluid flow through porous media. The theory that follows is confined to dispersion in unidirectional seepage flow through semi-infinite homogeneous porous media. The seepage flow velocity is assumed unsteady. The dispersion systems to be considered are subject to an input concentration of contaminants
In this section, the effectiveness and the usefulness of mixture of new integral transform and homotopy perturbation method is demonstrated by finding the solution of non-linear Burger’s equation for longitudinal dispersion arising in fluid flow through porous media.
The initial and boundary conditions for the Problem (9) are
Since concentration is decreasing as x with distance x. Therefore for the sake of convenience
By applying a new integral transform of Equation (9) subject to the boundary and initial conditions (12) we have
The inverse new integral transform implies that,
Now applying homotopy perturbation method in Equation (12), we get
where,
Comparing the coefficients of the same power of
Thus the solution
The solution (14) represents the concentration of the longitudinal dispersion phenomenon for any value of
Expression (14) represents the solution of Burger’s equation arising in longitudinal dispersion phenomenon in fluid flow through porous media which is the concentration for any time t = 0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009 and 0.01 (
The main goal of this paper is to solve Burger’s equation arising in longitudinal dispersion phenomenon in fluid flow through porous media by the mixture of the new integral transform with the homotopy perturbation method. The analytical expressions obtained here are useful to the study of salinity intrusion in groundwater, helpful in making quantitative predictions on the possible contamination of groundwater supplies resulting from groundwater movement through buried wastes. Numerical and graphical representation of solution presents possible
Concentration C(x, t) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
t x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
0.001 | 0.906565 | 0.820873 | 0.74211 | 0.671442 | 0.607507 | 0.549663 | 0.49733 | 0.443981 | 0.407142 | 0.368303 |
0.002 | 0.908301 | 0.821722 | 0.743408 | 0.672568 | 0.608486 | 0.550517 | 0.498076 | 0.450635 | 0.407717 | 0.368889 |
0.003 | 0.910045 | 0.823227 | 0.744711 | 0.673698 | 0.60947 | 0.551375 | 0.498825 | 0.451291 | 0.408293 | 0.369395 |
0.004 | 0.911797 | 0.824739 | 0.746019 | 0.674833 | 0.610456 | 0.552235 | 0.499877 | 0.451949 | 0.40887 | 0.369903 |
0.005 | 0.913557 | 0.826258 | 0.747333 | 0.675972 | 0.611447 | 0.553098 | 0.500331 | 0.452609 | 0.409449 | 0.370412 |
0.006 | 0.915326 | 0.827784 | 0.748652 | 0.677116 | 0.612441 | 0.553961 | 0.501087 | 0.453271 | 0.41003 | 0.370923 |
0.007 | 0.917104 | 0.829316 | 0.749977 | 0.678264 | 0.613438 | 0.554833 | 0.501846 | 0.453936 | 0.410612 | 0.371435 |
0.008 | 0.918889 | 0.830855 | 0.751307 | 0.679417 | 0.614439 | 0.585705 | 0.502607 | 0.454602 | 0.411197 | 0.371948 |
0.009 | 0.920684 | 0.832402 | 0.752643 | 0.680574 | 0.615444 | 0.586579 | 0.503371 | 0.45827 | 0.411782 | 0.372463 |
0.010 | 0.922481 | 0.833955 | 0.753984 | 0.681735 | 0.616453 | 0.587457 | 0.504137 | 0.45594 | 0.41237 | 0.372979 |
concentration of a given dissolved substance in unsteady unidirectional seepage flows through semi-infinite, homogeneous, isotropic porous media subject to the source concentrations that vary negative exponentially with distance and slightly increases with time.