By Joaquim P. Marques de Sá
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Extra info for Applied Partial Differential Equations A Visual Approach
The thermal ﬂuctuations are not strong enough to take the granular sand arrangement back into a global energy minimizing state, instead it settles into (one of many possible) local equilibria. html 3 Granular Material Flows 38 Fig. 1. Dune 45 in Sossus Vlei, Namibia 3 Granular Material Flows 39 Here are a few more examples of granular ﬂows: grains such as corn or wheat ﬂowing from a silo; landslides of boulders and debris; rock and ice collisions in planetary rings; transport and handling of coal or certain chemicals in industrial plants; powder metallurgy; powder spray coating and lava ﬂow in volcanic eruptions.
Also we refer to the webpage of Dr. H. Momiji5 for more information on dune dynamics and its mathematical modeling. gov/gallery/sanddunes/. uk/∼hmomiji/ 3 Granular Material Flows 48 3 Granular Material Flows 49 Fig. 6. Sunset in a dune ﬁeld, Sossus Vlei, Namibia Fig. 7. A sand dune, Atacama desert, Chile 3 Granular Material Flows 50 Fig. 8. Dunes, Death Valley, California 3 Granular Material Flows 51 Fig. 9. Footprints in a sand dune, Death Valley, California. Just one stable conﬁguration, out of many possible ones… Fig.
Due to chemical reactions). oelz/ 4 Chemotactic Cell Motion and Biological Pattern Formation 56 leads to the reaction-diffusion equation: St = div(D1 grad S) + g(r, S) . e. g > 0 describes production of the chemical and g < 0 its degradation. g. the Petri dish boundaries) and by initial conditions for r and S. 3) where d and e are positive constants. This classical Keller-Segel model, with appropriately ﬁtted parameters, is often sufﬁcient to describe real chemotactic processes with good qualitative and reasonable quantitative agreement.
Applied Partial Differential Equations A Visual Approach by Joaquim P. Marques de Sá