Abstract
Some analytical solutions of one-dimensional advection–diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green’s function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant’s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.
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Acknowledgements
The first author expresses his gratitude to University Grants Commission, Government of India, for financial and academic assistance in the form of Senior Research Fellowship. The authors express their gratitude to the reviewers for their valuable comments and suggestions which improved the present work to a great extent.
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List of symbols
c: Solute concentration in domain x
C: Solute concentration in new domain X
C i : Initial concentration
C 0: Reference concentration
D 0: Dispersion coefficient in homogeneousmedium
u 0: Velocity in homogeneous medium
K: Solute concentration in the new domain η
M: Injected pollutant mass
k: Asymptotically varying parameter
ν , t 0,
χ , σ , ξ: Dummy variables
m: Temporal dependence parameter
m 1: Another temporal dependence parameter
x: Position variable
T: Another new time variable
t: Time variable
t′: Time in new domain
X: Position in new domain
η: A new space variable
α: Coefficient in the decay term
δ(⋅): Dirac delta function
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SANSKRITYAYN, A., KUMAR, N. Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method. J Earth Syst Sci 125, 1713–1723 (2016). https://doi.org/10.1007/s12040-016-0756-0
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DOI: https://doi.org/10.1007/s12040-016-0756-0