Markowich P. Applied Partial Differential Equation: A Visual Approach

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3 Granular Material Flows

Fig. 3.4. Ridge of a Dune in Sossus Vlei, Namibia

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Fig. 3.5. Wind ripples in Sossus Vlei, Namibia

The obtained ﬂow equations have the form of a compressible Euler system

with a temperature relaxation term. More precisely, the temperature T(x, t)re-

laxes according to the so-called Haff’s law. This implies that – in the case of

vanishing bulk velocity and time-independent position density – the tempera-

ture relaxes to 0 with the algebraic rate t

−2

For details and for a collection of references on granular ﬂows we refer to [5]

and [6].

Comments on the Images 3.1– 3.10 The formation of sand dunes involves vari-

ous complicated geophysical mechanisms: sediment transport, avalanches, wind

ﬁeld driven aeolian transport of course taking into account that sand is a typical

granular material, with transport being dominated by localised inelastic colli-

sions between sand grains andbysaltation (jumpingmovementofgrainsoverthe

surface), driven by turbulent wind ﬂow. Clearly, these two types of sand grain

movement require different mathematical modeling: the former is described

by short range interactions modeled by the granular material Boltzmann-type

equation as stated in Chapter 3, the latter by convection representing the wind

ﬁeld in conjunction with inelastic collisions when grains hit the sand surface

after saltation. Also, we remark that the microscopic properties (shape, size etc.)

of sand grains may vary substantially! We refer to the Ph.D. thesis [4] for various

3 Granular Material Flows

ﬂuid type modeling approaches, with interesting simulations of barchan dune

evolution. Also we refer to the webpage of Dr. H. Momiji

for more information

on dune dynamics and its mathematical modeling. For fascinating images of

sand dunes on Mars see: http://mars.jpl.nasa.gov/gallery/sanddunes/.

http://www.geog.ucl.ac.uk/∼hmomiji/

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Fig. 3.6. Sunset in a dune ﬁeld, Sossus Vlei, Namibia

Fig. 3.7. A sand dune,

Atacama desert, Chile

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Fig. 3.8. Dunes, Death Valley, California

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Fig. 3.9. Footprints in a sand dune, Death Valley, California. Just one stable conﬁguration, out of

many possible ones…

Fig. 3.10. Pattern of wind ripples, Death Valley California

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Fig. 3.11. A granular (pattern) equilibrium state in a Zen garden in Kyoto, Japan



Fig. 3.12. 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. C.

Evans [2] and, for more mathematical detail, to [3]