Analysis of Nonlinear Dispersion of a Pollutant Ejected by an External Source into a Channel Flow

This paper focuses on the transient analysis of nonlinear dispersion of a pollutant ejected by an external source into a laminar flow of an incompressible fluid in a channel. The influence of density variation with pollutant concentration is approximated according to the Boussinesq approximation, and the nonlinear governing equations of momentum and pollutant concentration are obtained. The problem is solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, pollutant concentration, skin friction, and wall mass transfer rate for various parametric values. The model can be a useful tool for understanding the polluting situations of an improper discharge incident and evaluating the effects of decontaminating measures for the water body.


Introduction
Pollution of water sources like rivers and lakes, say resulting from industrial waste discharge, is a serious socioecological hazard. If the problem is not carefully controlled and monitored, large communities can be exposed to extensive health risks. Early detection of such pollution accidents, both in terms of extent and impact, is thus of major primary importance, and the subsequent requirement to take immediate corrective measure to redress the pollution problem and mitigate against its impact is equally so. In line with prevention better than cure adage, the development of accurate methods to predict the spread of a pollutant once a discharge has been detected and hence also the development of equally reliable preventive/corrective techniques is thus of paramount importance 1 . Spread of pollutants in a fluid flow depends largely on concentration coefficients 2 . These can be determined empirically for each type of pollutant. Investigations such as 3 and also the current one can help identify the pollutant physical properties and the related mathematical parameters likely to cause the greatest harm in the spreading the pollutant downstream. The importance of these kinds of investigations as well as the complimentary experimental works, say in large-scale water treatment and redistribution networks, thus makes them of great relevance 4-9 .
It should however be remarked that to date, the literature on the transient analysis of problems consisting of buoyancy effects and nonlinear pollutant injection is still quite sparse. This then forms the crux of the current investigation. We suppose that a pollutant is introduced nonlinearly into a channel flow via an external source and proceed, under Boussinesq approximations, to investigate the transient diffusion and resultant spatialtemporal concentration.
In Section 2, the model problem is formulated and in particular the mathematical governing equations, initial and boundary conditions are outlined. We outline, in Section 3, the finite difference schemes that will be employed in the solution process, and graphical results and discussions follow in Section 4.

Formulation of the Problem
We consider a transient problem of fluid flow and nonlinear dispersion of pollutant in a rectangular channel as shown in Figure 1. In order to derive the governing equations, the following assumptions are made: i the fluid is viscous and incompressible; ii initially, the flow is fully developed through a rectangular channel; iii at time t > 0, a given pollutant is injected into the flow from an external source; the viscosity of the fluid and the pollutant mass diffusivity then vary with its concentration; iv the influence of density variation with pollutant concentration has been considered only in the body-force term of the momentum equation and is approximated according to the Boussinesq approximation.
Under laminar flow conditions, the problem is reduced mathematically to a transient coupled fluid flow and mass transfer problem given in one dimension as 1, 3, 8, 9 ρ ∂u subject to the following initial and boundary conditions: where u is the fluid axial velocity, r, z are the radial and axial coordinates, respectively, C is the pollutant concentration, C 0 is the pollutant reference concentration, C w is the pollutant concentration at the walls, S is the pollutant external source function, g is the gravitational acceleration, ρ is the density, β is the concentration expansion coefficient, a is the channel half width, and P is the fluid pressure. As shown in 2.3 , we employed a fully developed Poiseuille parabolic velocity profile as the initial condition for the flow. The pollutant concentration-dependent fluid dynamic viscosity μ, mass diffusivity D, and external source are prescribed as follows: where μ 0 , D 0 , b 1 , b 2 , and b 3 are the viscosity coefficient, mass diffusivity coefficient, viscosity variation parameter, mass diffusivity variation parameter, and the pollutant external source variation parameter, respectively. The following dimensionless variables are introduced: where λ is the pollutant external source parameter, Gc is the solutal Grashof number, K is the axial pressure gradient parameter, Sc is the Schmidt number, α is the viscosity variation parameter, γ is the mass diffusivity variation parameter, and n is the pollutant external source variation parameter. The dimensionless shear stress C f and the mass transfer rate Sh at the channel wall are given by In the following section, 2.8 is solved numerically and the skin-friction together with the wall mass transfer rate given in 2.9 are computed.

Numerical Solution
Our numerical algorithm is based on the semi-implicit finite difference scheme given in 10 for the isothermal viscoelastic case. As in 11 , we extend the algorithm to the temperature equation and take the implicit terms at the intermediate time level The algorithm employed in 11 uses ξ 1/2; we will however follow the formulation in 10 and thus take ξ 1 in this article so that we can use larger time steps. The discretization of the governing equations is based on a linear Cartesian mesh and uniform grid on which finite-differences are taken. We approximate both the second and first spatial derivatives with second-order central differences. The equations corresponding to the first and last grid point are modified to incorporate the boundary conditions. The semi-implicit scheme for the velocity component reads Gc φ e αφ y w y N .

3.1
Mathematical Problems in Engineering 5 The equation for w N 1 then becomes where r 1 exp αφ N Δt/Δy 2 . The solution procedure for w N 1 thus reduces to inversion of tri-diagonal matrices which is an advantage over a full implicit scheme. The semiimplicit integration scheme for the concentration equation is similar to that for the velocity component. Unmixed second partial derivatives of the concentration are treated implicitly: The equation for φ N 1 thus becomes where r 2 exp γφ N Δt/Δy 2 . The solution procedure again reduces to inversion of tridiagonal matrices. The schemes 3.2 and 3.4 were checked for consistency. For ξ 1, these are first-order accurate in time but second order in space. The schemes in 11 have ξ 1/2 which improves the accuracy in time to second order. We use ξ 1 here so that we are free to choose larger time steps and still converge to the steady solutions. The algorithm was also tested for both spatial and temporal convergences and shown to be independent of both mesh size and time step size.

Results and Discussion
Unless otherwise stated, we employ the parameter values: Gc 0.1, Sc 0.6, α 0.1, γ 0.1, λ 0.5, n 0.1, m 0.1, K 1, Δr 0.02, Δt 0.005, and t 50. Figures 2 and 3 show the time development of the velocity and pollutant concentration profiles. The velocity profile eventually settles to a steady parabolic flow which is qualitatively similar to the initial profiles but quantitatively higher than the initial. This is expected since the injection of pollutant into the flow acts as a momentum source and hence naturally increases the flow velocity at subsequent times after the injection. Since the injection ceases at some point, we also naturally expect the velocities to eventually settle to steady state as illustrated in Figure 2.

Transient Solutions
The observed increase in the pollutant concentration with time shown in Figure 3 is a trivial consequence of the fact that pollutant is injected into the flow at time > 0. The cessation of the injection process at some future time leads to the eventual steady concentration profiles displayed.

Dependence on Flow Parameters
As already noted, the pollutant injection is connected to momentum source terms and hence leads to increased steady flow velocities. Increase of the pollutant injection parameter, λ, thus expectedly indirectly increases flow velocity as illustrated in Figure 4. The increases pollutant concentration however also increases fluid viscosity and hence the competing effects of higher viscosity and higher pollutant injection result in the marginal velocity increases shown in We should however remark as will be illustrated shortly that large values of λ may not allow for the attainment of steady solutions but rather lead to finite time blowup phenomena.
For large values of λ, the possibility of blowup phenomena is clearly illustrated in Figure 6. Physically, this results primarily from flow blockage due to excessive discharge of pollutant into the channel flow. In Figure 6, we compute and plot for each value of λ the corresponding pollutant maximum concentration φ max in steady state t 10 until a value of λ is reached around λ < 0.4 for the current parameter values at which no steady solutions exist but instead blowup phenomena is observed.
We next look at the effects of the solutal Grashof number Gc on both the flow velocity and pollutant concentration. Clearly Gc is connected to momentum source terms and thus flow velocities should directly increase with increasing Gc. This is illustrated in Figure 7.
On  Figure 9 are, as explained before, due to competing effects of increased Boussinesq source terms and increased fluid viscosity, both due to increased Sc and hence increased φ.  Figure 9 shows that pollutant substances with higher Schmidt numbers lead to correspondingly higher pollutant concentrations within the flow.
Increasing the parameter α leads to corresponding increases in the fluid viscosity and hence leads to reduced flow velocities. This is well illustrated in Figure 11.
As with the Grashof number, the fluid viscosity only plays a role in the momentum equations and since the velocity is uncoupled from the concentration equation, any changes in the viscosity even though they result in changes in the velocity field will have no effect on the pollutant concentration as shown in Figure 12. The diffusion parameter γ manifests itself appreciably in the pollutant concentration but barely plays a role in the flow velocity as illustrated in Figures 13 and 14.
Due to limited fluid motion close to the wall, due to viscous effects, increased diffusion values would not be able to redistribute pollutant concentration around the flow hence the observed increases in pollutant concentration close to the walls. On the other hand, higher

Shear Stress and Mass Transfer Rate
According to Figures 4,5,13, and 14, the velocity increases with λ and hence the wall shear stress should also correspondingly increase with λ as shown in each graph of Figure 17. Similarly, the velocity decreases with increasing γ and hence the observed decrease in C f with increasing γ also shown in Figure 17. A similar reasoning is applied to explain the features observed in Figure 18. According to Figures 7,8,9, and 10, the velocity increases with Gc and hence the wall shear stress should correspondingly increase with Gc as shown in each graph of Figure 19. Similarly, the velocity increases even though marginally with increasing Sc and hence the  observed increase in C f with increasing Sc also shown in Figure 19. A similar reasoning is applied to explain the features observed in Figure 20. According to Figures 2, 3, 15, and 16, the velocity increases with t until a steady state is reached and hence the wall shear stress should correspondingly increase with t until also a steady state is reached when the time dependence ceases. This is shown in Figure 21. Similarly, the velocity increases even though marginally with increasing n and hence the observed marginal increases in C f with increasing n also shown in Figure 21. A similar reasoning is applied to explain the features observed in Figure 22 except in this case, based on the scalings on the vertical axis, the increase in Sh with increasing n is more significant as compared to that for C f .

Conclusion
We computationally investigate the transient dynamics of pollutant dispersion in a channel with a nonlinear waste discharge concentration subjected to one-dimensional gravity driven flow under Boussinesq approximations. We observe that there is a transient increase in the flow velocity with an increase in the pollutant injection. Larger increases in the injection are also shown to potentially lead to blockage due to excessive discharge of pollutant into the channel flow. We also similarly investigated the effects of the various other physical parameters on the flow velocity, the pollutant concentration, and hence also on the wall shear stress and wall mass transfer rates.