# American Institute of Mathematical Sciences

September  2011, 6(3): 465-483. doi: 10.3934/nhm.2011.6.465

## Towards a mathematical model for stability in pedestrian flows

 1 Department of Infrastructure Engineering, University of Melbourne, Parkville, Victoria 3010, Australia, Australia

Received  December 2010 Revised  June 2011 Published  August 2011

It is suggested that flows of pedestrians on curved paths, such as the recirculating flow that occur around the Kaaba in Mecca, continually stratify themselves according to tolerance of crowd density with those pedestrians who are more tolerant of high densities taking a path of shorter length. Such stratification occurs over a distance, referred to here as the stratification distance scale" and is generally of the order of the radius of curvature of the flow. Once stratified a pedestrian crowd flows smoothly around an obstacle with a distance scale greater than the stratification distance scale. However, flow past a smaller obstacle with a distance scale less than the stratification distance scale, leads to some temporary breakdown in this stratification, with the flow developing patches of turbulent-like behavior with different pedestrian types responding differently to the obstacle. The flow between the nearby Safa and Marwa Hills is poorly stratified because of the lack of curvature of the flow between the Hills even though the flow is recirculating. At the start of each turning point by each Hill, the change in curvature leads to turbulence-like behavior as the stratification reforms, after its breakdown between the Hills, for the flow around the Hills.
Citation: Abdul M. Kamareddine, Roger L. Hughes. Towards a mathematical model for stability in pedestrian flows. Networks and Heterogeneous Media, 2011, 6 (3) : 465-483. doi: 10.3934/nhm.2011.6.465
##### References:
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show all references

##### References:
 [1] A. T. Chahly, "Descriptive Geometry," The Higher School Pub, Moscow, Undated. [2] J. W. Daily and D. R. F. Harleman, "Fluid Dynamics," Addison-Wesley, USA, 1966. [3] B. D. Hankin and R. A. Wright, Passenger flow in subways, Journal of the Operational Research Society, 9 (1958), 81-88. doi: 10.1057/jors.1958.9. [4] B. Hofmann-Wellenhof, H. Lichtenegger and J. Collins, "GPS: Theory and Practice," Springer-Verlag, Vienna, 1992. [5] R. L. Hughes, A continuum theory for the flow of pedestrians, Trans. Res. Part B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7. [6] N. A. Koshak and A. Fouda, "Analyzing Pedestrian Movement in Mataf Using GPS and GIS to Support Space Redesign," Proceedings of the Ninth International Conference on Design and Decision Support Systems (DDSS) in Architecture and Urban Planning, International Conference, 2008. [7] R. S. C. Lee and R. L. Hughes, Exploring trampling and crushing in a crowd, Journal of Transportation Engineering, 131 (2005), 575-582. doi: 10.1061/(ASCE)0733-947X(2005)131:8(575). [8] R. S. C. Lee, "The Danger in a Crowd," Ph.D thesis, The University of Melbourne, 2005. [9] A. Leick, "GPS Satellite Surveying," 2nd edition, John Wiley & Sons, New York, 1995. [10] B. W. Parkinson, Introduction and heritage of NAVSTAR, Global Positioning System: Theory and Applications, 1 (1996), 3-28. [11] B. W. Parkinson and J. J. Spilker, Global Positioning System: Theory and applications, American Institute of Aeronautics and Astronautics, 2 (1996), 3-142. [12] S. Sarmady, F. Haron, M. M. Mohd Salahudin and A. Z. H. Talib, Evaluation of existing software for simulating of crowd at masjid Al-Haram, Jurnal Pengurusan JWZH, 1 (2007), 83-95. [13] B. W. Parkinson and J. J. Spilker, Overview of GPS operation and design, Global Positioning System: Theory and applications, (1996), 29-55. [14] D. Stewart, "Mecca," Newsweek, New York, 1980. [15] D. E. Wells, N. Beck, D. Delikaraoglou, A. Kleusberg, E. J. Krakiwsky, G. Lachapelle, R. B. Langley, M. Nakiboglu, K. P. Schwarz, J. M. Tranquilla and P. Vanícek, "Guide to GPS Positioning," University of New Brunswick, New Brunswick, 1987. [16] , http://www.fathersez.com/wp-content/uploads/2008/09/kaabah-tawafpreview.jpg,, Accessed on April 2010., (2010). [17] , http://image62.webshots.com/662/4/10/80/2938410800103225830Msadrn_ph.jpg,, Accessed on April 2010., (2010). [18] , http://www.watchingamerica.com/images/kaaba_pic.jpeg,, Accessed on May 2010., (2010). [19] , http://farm3.static.flickr.com/2097/2119107921_a2739375f9_b.jpg,, Accessed on May 2010., (2010). [20] , http://navedz.files.wordpress.com/2009/03/kaaba-1429-hijri.jpg,, Accessed on May 2010., (2010). [21] , http://islamfrance.free.fr/photo/mosquee/normale/kaaba03.jpg,, Accessed on May 2010., (2010). [22] , http://www.princeton.edu/~humcomp/tawaf.jpg,, Accessed on May 2010., (2010). [23] , http://www.muslimherald.com/Files/k/Masjed-Alharam_Tawaf1a.jpg,, Accessed on May 2010., (2010). [24] Z. Zainuddin, K. Thinakaran and I. M. Abu-Sulyman, Simulating the circumambulation of the Ka'aba using Sim Walk, European Journal of Scientific Research, 38 (2009), 454-464.
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