Solution of a Mathematical Model on Pollutant Concentration Using Adomian Decomposition Method
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Solution of a Mathematical Model of
Pollutant Concentration in a Channel Flow using Adomian Decomposition Method
U. Isip, T. Aboiyar and E. S. Onah
Department of Mathematics/Statistics/Computer Science,
University of Agriculture, Makurdi, Nigeria
Abstract
This paper focuses on the solution of a model for nonlinear dispersion of a pollutant ejected by an external source into a laminar flow of an incompressible fluid in a channel. The model equations are solved using the Adomian Decomposition Method, which is a semi-analytical method. The Adomian Decomposition Method (ADM) can be used to obtain exact solutions of nonlinear functional equations of various kinds without discretizing the equations or approximating the operators. The solution, when it exists, is found in a rapidly converging series form, and time and space are not discretized. Solutions of the mathematical model are presented in graphical form and given in terms of fluid velocity and pollutant concentration, for various parameter values. The results agree with results in literature obtained by high order finite difference methods.
1 Introduction
Water pollution, resulting from industrial waste discharge into water bodies such as rivers, lakes, streams among others, is a serious environmental concern, having large scale impact on both people and other living organisms in both small and large communities, especially in the riverine areas. An example of this is the accidental spillage of crude oil into water channels in the Niger Delta area of Nigeria, destroying both marine and land animals. In this work we are particularly concerned with the blockage of river channels as a result of excessive pollutant discharge. According to Taylor (1954), spread of pollutants in a fluid flow depends largely on concentration coefficients. These can be determined empirically for each type of pollutant. Investigations such as those by Shulka (2002) and Chinyoka & Makinde (2010) can help identify the pollutant physical properties and the related mathematical parameters likely to cause the greatest harm in spreading the pollutant downstream. Chinyoka & Makinde (2010), under the assumption that a pollutant is introduced nonlinearly into a channel flow via an external source presented the mathematical model described in this paper.
The Adomian Decomposition Method (ADM) allows exact solutions of nonlinear functional equations of various kinds without discretizing the equations or approximating the operators. The solution, when it exists, is found in a rapidly converging series form, and time and space are not discretized. The decomposition method yields rapidly convergent series solutions by using a few iterations for both linear and nonlinear deterministic and stochastic equations. The advantage of this method is that it provides a direct scheme for solving the problem, i.e., without the need for linearization, perturbation, massive computation and any transformation. This can be seen for instance in the work of Abbaoui and Cherrualt (1994a,1994b), Lesnic (2002), Al-Khaled and Allan (2005), and Wazwaz (2000)
In this paper, we will use the ADM to find a reliable and accurate solution to the model of pollutant concentration proposed Chinyoka & Makinde (2010). This yields an approximation to the exact solution in series form.
2 Methods 3.1 The Mathematical Model
We will first of all describe the model of Chinyoka & Makinde (2010). They considered a transient problem of fluid flow and nonlinear dispersion of pollutant in a rectangular channel, and provided the dimensionless model of the form:
where ϕ and y are the dimensionless equivalents for the pollutant concentration and flow velocity, respectively, is the pollutant external source parameter, Gc is the solutal Grashof number (property of the pollutant), K is the axial pressure gradient, Sc is the Schmidt number, is the viscosity variation parameter, is the mass diffusivity variation parameter, and n0 is the pollutant external source variation parameter. The dimensionless shear stress (Cf) and the mass transfer rate (Sh) at the channel wall are given by (2)
2.2 The Adomian Decomposition Method for Systems of Partial Differential Equations Following Wazwaz (2000), we consider the system of partial differential equations
∂u∂t+∂v∂x+N1u,v=g1
∂v∂t+∂u∂x+N2u,v=g2 (3) with initial data ux,0=f1x vx,0=f2x (4)
Rewriting in operator form we have, Ltu+Lxv+N1u,v=g1
Ltv+Lxu+N2u,v=g2 (5) with initial data ux,0=f1x vx,0=f2x (6) where Lt and Lx are the first order partial differential operators, ∂∂t and ∂∂x, respectively. N1 and N2 are nonlinear operators, and g1and g2 are non-homogenous terms. Applying the inverse operator Lt-1 to the system (5) and using the initial data (6) yields ux,t=f1x+Lt-1g1-Lt-1Lxv-Lt-1N1u,v, vx,t=f2x+Lt-1g2-Lt-1Lxu-Lt-1N1u,v. (7) where Lt-1=0t.dt
The linear terms u(x,t) and v(x,t) are thereafter decomposed by an infinite series of components ux,t=n=0∞unx,t, vx,t=n=0∞vnx,t, (8) which are obtained systematically as the terms of the infinite series progress. The nonlinear operators N1(u,v) and N2(u,v) are defined by the infinite series of Adomian polynomials N1u,v=n=0∞An,
N2u,v=n=0∞Bn, (9) which can be uniquely determined using algorithms derived by Wazwaz (2000b) for each nonlinear expression.
Substituting (8) and (9) into (7) gives n=0∞unx,t=f1x+Lt-1g1-Lt-1Lxn=0∞vn-Lt-1n=0∞An, n=0∞vnx,t=f2x+Lt-1g2-Lt-1Lxn=0∞un-Lt-1n=0∞Bn. (10)
Following Adomian analysis, the nonlinear system (7) is transformed into a set of recursive relations given by uox,t=f1x+Lt-1g1, uk+1x,t=-Lt-1Lxvk-Lt-1Ak, k≥0, (11) and vox,t=f2x+Lt-1g2, vk+1x,t=-Lt-1Lxuk-Lt-1Bk, k≥0, (12)
The solution is thus obtained in series form as
ux,t=n=0∞unx,t, vx,t=n=0∞vnx,t, (13)
3 Results
We solve the model equations above using the Adomian Decomposition Method by first simplifying them and then rewriting them in operator form. Simplifying using the product rule for differentiation we have,
We now rewrite the system of PDEs (14) in operator form as follows (15) with initial data
(16) where the operator and the inverse operator
The linear term is
,
and nonlinear terms: (17)
Applying the inverse operator to the system (15) and using the initial data (16) yields (18) which we simplify to obtain (19)
The Adomian decomposition method suggests that the linear terms and be decomposed by an infinite series of components: (20) and the nonlinear terms (21)
Substituting (20) and (21) into (19) gives (22)
Following Adomian analysis, the nonlinear system (21) is transformed into a set of recursive relations given by
It is an essential feature of the decomposition method that the zeroth components and are defined always by all terms that arise from initial data and from integrating the inhomogeneous terms. Having defined the zeroth pair the pair can be determined recurrently by using (23) and (24). The remaining pairs ; k2; can be easily determined in a parallel manner. Wazwaz (2000) notes that additional pairs for the decomposition series normally account for higher accuracy. Having determined the components of and , the solution of the system follows immediately in the form of a power series expansion upon using (22). The series obtained can be summed up in many cases to give a closed form solution. For concrete problems, the nth term approximants can be used for numerical purposes.
Following Wazwaz (2000), and the algorithm designed in MAPLE, we obtain the Adomian polynomials for the nonlinear terms as follows:
A0=eαϕ0y∂w0∂y
A1=eαϕ0y∂w1∂y+αϕ1eαϕ0y∂w0∂y
A2=eαϕ0y∂w2∂y+αϕ1eαϕ0y∂w1∂y+αϕ2eαϕ0+12ϕ12α2eαϕ0y∂w0∂y
A3=eαϕ0y∂w3∂y+αϕ1eαϕ0y∂w2∂y+αϕ2eαϕ0+12ϕ12α2eαϕ0y∂w1∂y+αϕ3eαϕ0+ α2ϕ1ϕ2eαϕ0+16α3ϕ13eαϕ0∂w0∂y (25) and, B0=eαϕ0y∂2w0∂y2 B1=eαϕ0y∂2w1∂y2+αϕ1eαϕ0y∂w0∂y B2=eαϕ0y∂2w2∂y2+αϕ1eαϕ0y∂2w1∂y2+αϕ2eαϕ0+12ϕ12α2eαϕ0y∂2w0∂y2 B3=eαϕ0y∂2w3∂y2+αϕ1eαϕ0y∂2w2∂y2+αϕ2eαϕ0+12ϕ12α2eαϕ0y∂2w1∂y2+ αϕ3eαϕ0+α2ϕ1ϕ2eαϕ0+16α3ϕ13eαϕ0∂2w0∂y2 (26) and,
and,
D0=1Sceγϕ0y∂2ϕ0∂y2
D1=eγϕ0y∂2ϕ1∂y2+γϕ1eγϕ0y∂2ϕ0∂y2
D2=eγϕ0y∂2ϕ2∂y2+γϕ1eγϕ0y∂ϕ1∂y+γϕ2eγϕ0+12ϕ12γ2eγϕ0y∂ϕ0∂y
D3=eγϕ0y∂2ϕ3∂y2+γϕ1eγϕ0y∂2ϕ2∂y2+γϕ2eγϕ0+12ϕ12γ2eγϕ0y∂2ϕ1∂y2+γϕ3eγϕ0+ γ2ϕ1ϕ2eγϕ0+16γ3ϕ13eγϕ0∂2ϕ0∂y2 (28) and E0=λen0ϕ0
E1=λn0ϕ1en0ϕ0
E2= λn0ϕ2en0ϕ0+12ϕ12n02en0ϕ0 E3=λn0ϕ3en0ϕ0+ϕ1ϕ2n02en0ϕ0+16ϕ13n03en0ϕ0 (29)
Computing the polynomials using MAPLE, we have the solutions:
(38)
wy,t=m1-y2+K-4αe2αymy-4αe2αym+2Gcyt+12-2α8γ2e2γySc+2λn0e2n0ye2αy+2α24γe2γySc+λe2n0ye2αymy+4αe2αy-8α2e2αymy-4αe2αym-8α2e2αym+2Gc-2α8γ2e2γySc+2λn0e2n0ye2αy+2α24γe2γySc+λe2n0ye2αym+Gc4γe2γySc+λe2n0yt2+…and
(30)
and,
(31)
ϕy,t=2y+4γe2γySc+λe2noyt+12Sc2γ8γ2e2γySc+2λn0e2n0ye2γy+2γ24γe2γySc+λe2n0ye2γy+2γe2γy8γ2e2γySc+2λn0e2n0y+2γe2γy16γ3e2γySc+4λn02e2n0y+λn0Sc4γe2γySc+λe2n0ye2n0yt2+…
4 Numerical Experiments
Here we obtain the solutions of the model graphically, using various parameter values.
4.1 Transient Solutions
Figures 2 and 3 illustrate the changes in velocity and concentration as time increases.
Figure 2: Graph of Velocity (w) versus ‘y’ at t = 0.1, 10, 30, 50 using K=1, no=0.1, =0.5, Gc=0.1, Sc=0.6, γ=0.1
Figure 3: Graph of Concentration (ϕ) versus ‘y’ at t = 0.1, 10, 25, 50, using K=1, no=0.1, =0.5, Gc=0.1, Sc=0.6, γ=0.1
Figure 2 shows the time development of the velocity profile. As expected, the velocity increases, following a parabolic path. This increase is due to increased momentum, as the flow progresses. This is consistent with the results obtained by Chinyoka and Makinde (2010). Figure 3 shows increase in concentration profiles as time progresses. This increase is sustained till some future time when there is no further injection of pollutants.
4.2 Dependence on Flow Parameters
Figures 4 to 7 show the dependence of velocity and concentration on the flow parameters. Two key parameters are modified in this analysis: λ and Sc.
Figure 4: Graph of Velocity (w) versus ‘y’ at λ = 0, 1, 5, 10, using K=1, no=0.1, Gc=0.1, Sc=0.6, γ=0.1
In the figure above, we observe the change in velocity using different values of , a property of the pollutant, which depends inversely on the pollutant viscosity. Figure 4 shows higher velocities for higher values of . This observation is consistent with those obtained by Chinyoka and Makinde (2010).
Figure 5: Graph of Concentration (ϕ) versus ‘y’ at λ = 0, 1, 5, 10 using K=1, no=0.1, Gc=0.1, Sc=0.6, γ=0.1
Similar to the result obtained in Figure 4, Figure 5 shows that the concentration increases significantly as increases. Very high values of may however lead to a blow up in the concentration values.
Figure 6: Graph of Velocity (w) versus ‘y’ at Sc = 0.24, 0.6, 0.78, 2.62 using K=1, no=0.1, =0.5, Gc=0.1, γ=0.1
Sc is the Schmidt number, which depends on the chemical composition of the pollutant. Usually chemical compounds with higher Sc values are more viscous. As observed in Figure 6, lower values of Sc lead to higher velocity. This is expected, as a result of reduced viscous drag. Pollutants with lower Sc values are less likely to cause blockages in water channels.
Figure 7: Graph of Concentration (ϕ) versus ‘y’ at Sc = 0.24, 0.6, 0.78, 2.62 using K=1, no=0.1, =0.5, Gc=0.1, γ=0.1
In this case, we observe the change in concentration as we vary the Schmidt values. The reduced viscosity leads to a drastic increase in concentration with lower Sc values. This phenomenon is observed in Figure 7.
5 Conclusion
In this paper, we studied the model developed by Chinyoka and Makinde (2010), and obtained solutions for the pollutant concentration in a channel flow, as well as the flow velocity, using the Adomian Decomposition Method. Graphical illustration of the solution shows a clear similarity with results obtained by Chinyoka and Makinde (2010) using high order finite difference methods. The results obtained by the Adomian Decomposition Method shows clearer demarcations between the graphs for different parameter values, making observation and analysis easier.
6 References
Abbaoui, K. and Cherrualt, Y. (1994a) “Convergence of Adomians method applied to differential equations”, Comp. Math Appl., Vol. 28, pp. 103-109.
Abbaoui, K. and Cherrualt Y.(1994b) “Convergence of Adomians method applied to nonlinear equations”, Mathematical and Computer Modelling, Vol. 20, pp. 69-73.
Al-Khaled, K. and Allan, F. (2005) “Construction of solution for the shallow water equations by the decomposition method”, Mathematics and Computers in simulation, Vol. 66, pp. 479-486.
Chinyoka, T. and Makinde, O. D. (2010) “Analysis of Nonlinear Dispersion of a Pollutant Ejected by an External Source into a Channel Flow”. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2010.
Lesnic, D. (2002) “The decomposition method for forward and backward time-dependent problems”, Journal of Computational and Applied Mathematics, 147, 27-38.
Shulka, P. (2002). “Analytical solutions for steady transport dispersion of nonconservative pollutant with time-dependent periodic waste discharge concentration,” Journal of Hydraulic Engineering, vol. 129, no. 9, pp. 866–869.
Taylor, G. I. (1954) “The dispersion of matter in turbulent flow through pipe,” Proceedings of the Royal Society of London, vol. 233, pp. 446–448.
Wazwaz, A. (2000). “A new algorithm for calculating Adomian polynomials for nonlinear operators”, Appl. Math, Comput., Vol. 111 (2000), pp. 53-69.
