By Joaquim P. Marques de Sá
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Additional resources for Applied Partial Differential Equations A Visual Approach
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. 10. Pattern of wind ripples, Death Valley California 3 Granular Material Flows 52 3 Granular Material Flows 53 Fig. 11. A granular (pattern) equilibrium state in a Zen garden in Kyoto, Japan Fig. 12. A stable pile of small pebbles in a Zen garden in Kyoto, Japan. For the modeling of the growth, collapse and stability of piles of granular materials, in the context of the Monge– Kantorovich mass transportation theory, using p-Laplace equations we refer to the survey of L.
A typical boundary condition is the so-called no-slip condition which reads u=0 on the boundary of the ﬂuid domain. The constraint div u = 0 enforces the incompressibility of the ﬂuid and serves to determine the pressure p from the evolution equation for the ﬂuid velocity u. If ν = 0 then the so called incompressible Euler8 equations, valid for very small viscosity ﬂows (ideal ﬂuids), are obtained. Note that the viscous Navier– Stokes equations form a parabolic system while the Euler equations (inviscid case) are hyperbolic.
The signiﬁcance of this is apparent when one considers the following data: – In the chemical industry approximately one-half of the products and at least three-quarters of the raw materials are in granular form. – Landslides cause more than one billion dollars of property damage and at least 25 fatalities in the United States annually (FEMA). – In Mexico 5 million tons of corn are handled each year, 30% of which is lost due to poor handling systems. Even small increases in efﬁciency can make a signiﬁcant economic impact.