# The influence of longitudinal diffusion on the transport of dust particles emitted from a fixed source

## DOI:

https://doi.org/10.14419/ijamr.v5i1.5367## Published:

2016-01-09## Keywords:

Dust Diffusion, Atmospheric Diffusion Equation, Longitudinal Diffusion, Wind Speed, Unsteady State.## Abstract

The mathematical model for the diffusion of dust particles emitted from a fixed source in the presence of the longitudinal diffusion and absence of latitudinal and vertical diffusions, is investigated. The diffusion of dust particles in the atmosphere is governed by the atmospheric diffusion equation. In the previous paper [1], the general case of the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time, was solved. The calculations showed that the diffusion parameters play an important role in the spread of the dust particles in the atmosphere. In the previous paper, we examined the model in the presence of vertical diffusion and absence of other diffusions to show that for small times, the dust spreads with a front that travels with the speed of the wind. In the current paper, the vertical and latitudinal diffusions are absent while the longitudinal diffusion is present. It is found that the solution depends on the source of time dependence. To study the nature of the solution, two special cases of the source are specified. In the both cases, it is found that there is no discontinuity front, and the dust particles spread slowly into the direction of the wind.

## References

[1] Al-Mashrafi, K.S., â€œDiffusion of dust particles emitted from a fixed sourceâ€, *International Journal of Applied Mathematical Research*, 4 (4), (2015), pp: 454-465. http://dx.doi.org/10.14419/ijamr.v4i4.5058.

[2] Roberts, O.F.T., â€œThe theoretical scattering of smoke in a turbulent atmosphereâ€, *Proc. Roy. Soc. London Ser. A*, 104, (1923), pp: 640-654. http://dx.doi.org/10.1098/rspa.1923.0132.

[3] Llewelyn, R.P., â€œAn analytical model for transport, dispersion and elimination of air pollutants emitted from a point sourceâ€, *Atmospheric Environment*, 17, (1983), pp: 249-256. http://dx.doi.org/10.1016/0004-6981(83)90040-9.

[4] Skidmore, E.L*., *â€œSoil erosion by wind: an overview, in physics of desertificationâ€* *(F. EL-Baz and M. H. Hassan, eds.),* Martinus Nijhoff Publishers*, Dordrecht, (1986), pp: 261-273.

[5] Kevin B.M., Howard R.B. and Ronald G.R., â€œNumerical simulation of smoke plumes from large oil firesâ€, *Atmospheric Environment*, 30, (1996), pp: 4125-4136. http://dx.doi.org/10.1016/1352-2310(96)00151-3.

[6] Bagnold, R.A., â€œThe physics of Blown sand and Desert Dunesâ€, Methuen, London, (1954).

[7] Kennedy, J.F., â€œThe formation of sediment ripples, dunes and anti-dunesâ€, *Ann. Rev. Fluid Mech*., 1, (1969), pp: 147-168. http://dx.doi.org/10.1146/annurev.fl.01.010169.001051.

[8] Eltayeb, I.A. and Hassan, M.H.A., â€œTwo-dimensional transport of dust from an infinite line source at ground levelâ€, *Geophys. J. Int.,* .110, (1992), pp: 571-576. http://dx.doi.org/10.1111/j.1365-246X.1992.tb02092.x.

[9] Eltayeb, I.A. and Hassan, M.H.A., â€œThe Evolution of dust emitted by a uniform source above ground levelâ€, *International Journal of Mathematics and Mathematical Sciences*, (2003), pp: 3327-3343. http://dx.doi.org/10.1155/S0161171203106163.

[10] Gillette, D.A., â€œThreshold Friction Velocities for Dust production for Agricultural Soilsâ€. *J. Geophys. Res.,* 93, (1988), pp: 12645 â€“ 12662. http://dx.doi.org/10.1029/JD093iD10p12645.

[11] Gillette, D. and Goodwin, P., â€œMicroscale transport of sand-sized soil aggregates eroded by windâ€, *J. Geophys. Res*, 79, (1974), pp: 4080-4084. http://dx.doi.org/10.1029/JC079i027p04080.

[12] Sharan M., Singh M.P., and Yadav, A.K., â€œMathematical model for atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distanceâ€, *Atmospheric Environment*, 30, (1996), pp: 1137-1145. http://dx.doi.org/10.1016/1352-2310(95)00368-1.

[13] Sharan M., and Gupta S., â€œTwo-dimensional analytical model for estimating crosswind-integrated concentration in a capping inversion: eddy diffusivity as a function of downwind distance from the sourceâ€, *Atmospheric Environment*, 36, (2002), pp: 97-105. http://dx.doi.org/10.1016/S1352-2310(01)00449-6.

[14] Eltayeb, I.A. and Hassan, M.H.A., â€œDiffusion of dust particles from a point source above ground level and a line source at ground levelâ€, *Geophys. J. Int.,* 142, (2000), pp: 426-438. http://dx.doi.org/10.1046/j.1365-246x.2000.00172.x.

[15] Hassan, M.H.A. and Eltayeb, I.A., â€œSuspension transport of wind eroded sand particlesâ€, *Geophys. J. Internet.* , (1991b), 104, pp: 147-152. http://dx.doi.org/10.1111/j.1365-246X.1991.tb02500.x.

[16] Pasquill, F., â€œAtmospheric Diffusionâ€, *D. Van Nostrand, *Princeton, (1962).

[17] Taylor G.I., â€œDiffusion by continuous movementsâ€. *Proc. London Math. Soc., 20*, (1921), pp: 196â€“212.

[18] Zwillinger, D., â€œ*Handbook of Differential Equations*â€, Mass, Academic Press, Boston, (1989).

[19] Abramowitz, M. and Stegun, I., â€œ*Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables*â€, Dover, New York, (1965).

## License

Authors who publish with this journal agree to the following terms:

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [1]Â Â Â Â Â Â Â Â Â Â Â **Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under aÂ Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [2]Â Â Â Â Â Â Â Â Â Â Â **Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [3]Â Â Â Â Â Â Â Â Â Â Â **Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (SeeÂ The Effect of Open Access).