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Abstract
A mathematical model for the diffusion of dust particles emitted from a fixed source is investigated using the atmospheric diffusion equation. This model poses an initial boundary value problem with a second order linear partial differential equation. The steady state case of this problem when the uniform source is situated at ground level was examined by Sharan et al. [1]. The solution of the unsteady case in closed form for a time dependent source is derived. Two special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken, are examined in detail. In the case when diffusion is present only in the vertical direction, it is shown that for small times, the particles spread with a front that travels with the speed of the wind. When diffusion is present only in the direction of the wind, there is no discontinuity front and the particles diffuse slowly into the direction of the wind. The solutions for the special cases considered are examined for large values of time. It is found that the solution approaches that of the corresponding steady state solution of the equation.
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References
- Sharan, M., Singh, M. and Yadav, A. Mathematical model for atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmospheric Environment, 1996, 30, 1137-1145.
- Taylor, G. Diffusion by continuous movements. Proceedings of the London Mathematical Society A, 1921, 20, 196 - 212.
- Abbott, J. and Francis, J. Saltation and suspension trajectories of solid grains in a water stream. Philosophical Transactions of the Royal Society of London, 1977, A284, 225-253.
- Eltayeb, I. and Hassan, M. On the non-linear evolution of sand dunes. Geophysical Journal of the Royal Astronomical Society, 1981, 65, 31-45.
- Koch, W. A solution of the two-dimensional atmospheric diffusion equation with height-dependent diffusion coefficient and height-dependent wind speed and eddy diffusivities. Atmospheric Environment, 1989, 23, 1729-1732.
- Hill, J. Similarity solutions for non-linear diffusion a new integration procedure. Journal of Engineering Mathematics, 1989, 23,141-155.
- Eltayeb, I. and Hassan, M. Two-dimensional transport of dust from an infinite line source at ground level. Geophysical Journal International, 1992, 110, 571-576.
- Lin, J. and Hildemann, L. Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities. Atmospheric Environment, 1996, 30, 239-254.
- Lin, J. and Hildemann, L. A generalized mathematical scheme to analytically solve the atmospheric diffusion equation with dry deposition. Atmospheric Environment, 1997, 31, 59-71.
- AL-Mashrafi, K. Diffusion of dust particles emitted from a fixed source. International Journal of Applied Mathematical Research, 2015, 4(4), 454-465.
- AL-Mashrafi, K. The influence of longitudinal diffusion on the transport of dust particles emitted from a fixed source. International Journal of Applied Mathematical Research, 2016, 5(1), 11-17.
- Roberts, O. 1923. The theoretical scattering of smoke in a turbulent atmosphere. Proceedings of the Royal Society of London.,1923, A 104, 640-654.
- Smith, F. The diffusion of smoke from a continuous elevated point source into a turbulent atmosphere. Journal of Fluid Mechanics, 1957, 2, 49-76.
- Gillette, D. Threshold friction velocities for dust production for agricultural soils. Journal of Geophysical Research, 1988, 93, 12645-12662.
- Eltayeb, I. and Hassan, M. Diffusion of dust particles from a point source above ground level and a line source at ground level. Geophysical Journal International, 2000, 142, 426-438.
- Hassan, M. and Eltayeb, I. Time-dependent transport of dust. Journal of Geophysical Research, 1991, 96, 9337-9339.
- Gillette, D. and Passi, R. Modelling dust emission caused by wind erosion. Journal of Geophysical Research, 1988, 93, 14233-14242.
- Pasquill, F. Atmospheric Diffusion, Van Nostrand, Princeton. 1962
- Abramowitz, M. and Stegun, I. 1965 Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, New York, Dover.
References
Sharan, M., Singh, M. and Yadav, A. Mathematical model for atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmospheric Environment, 1996, 30, 1137-1145.
Taylor, G. Diffusion by continuous movements. Proceedings of the London Mathematical Society A, 1921, 20, 196 - 212.
Abbott, J. and Francis, J. Saltation and suspension trajectories of solid grains in a water stream. Philosophical Transactions of the Royal Society of London, 1977, A284, 225-253.
Eltayeb, I. and Hassan, M. On the non-linear evolution of sand dunes. Geophysical Journal of the Royal Astronomical Society, 1981, 65, 31-45.
Koch, W. A solution of the two-dimensional atmospheric diffusion equation with height-dependent diffusion coefficient and height-dependent wind speed and eddy diffusivities. Atmospheric Environment, 1989, 23, 1729-1732.
Hill, J. Similarity solutions for non-linear diffusion a new integration procedure. Journal of Engineering Mathematics, 1989, 23,141-155.
Eltayeb, I. and Hassan, M. Two-dimensional transport of dust from an infinite line source at ground level. Geophysical Journal International, 1992, 110, 571-576.
Lin, J. and Hildemann, L. Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities. Atmospheric Environment, 1996, 30, 239-254.
Lin, J. and Hildemann, L. A generalized mathematical scheme to analytically solve the atmospheric diffusion equation with dry deposition. Atmospheric Environment, 1997, 31, 59-71.
AL-Mashrafi, K. Diffusion of dust particles emitted from a fixed source. International Journal of Applied Mathematical Research, 2015, 4(4), 454-465.
AL-Mashrafi, K. The influence of longitudinal diffusion on the transport of dust particles emitted from a fixed source. International Journal of Applied Mathematical Research, 2016, 5(1), 11-17.
Roberts, O. 1923. The theoretical scattering of smoke in a turbulent atmosphere. Proceedings of the Royal Society of London.,1923, A 104, 640-654.
Smith, F. The diffusion of smoke from a continuous elevated point source into a turbulent atmosphere. Journal of Fluid Mechanics, 1957, 2, 49-76.
Gillette, D. Threshold friction velocities for dust production for agricultural soils. Journal of Geophysical Research, 1988, 93, 12645-12662.
Eltayeb, I. and Hassan, M. Diffusion of dust particles from a point source above ground level and a line source at ground level. Geophysical Journal International, 2000, 142, 426-438.
Hassan, M. and Eltayeb, I. Time-dependent transport of dust. Journal of Geophysical Research, 1991, 96, 9337-9339.
Gillette, D. and Passi, R. Modelling dust emission caused by wind erosion. Journal of Geophysical Research, 1988, 93, 14233-14242.
Pasquill, F. Atmospheric Diffusion, Van Nostrand, Princeton. 1962
Abramowitz, M. and Stegun, I. 1965 Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, New York, Dover.