2 D. AMADORI AND W. SHEN
1. The model of granular flow
The following model was proposed in [20] to describe granular flows
(1.1)
ht = div(h∇u) (1 |∇u|)h ,
ut =
(
1 |∇u|)h .
These equations describe conservation of masses. The material is divided in two
parts: a moving layer with height h on top and a standing layer with height u at the
bottom. The moving layer slides downhill, in the direction of steepest descent, with
speed proportional to the slope of the standing layer. If the slope |∇u| 1 then
grains initially at rest are hit by rolling grains of the moving layer and start moving
as well. Hence the moving layer gets bigger. On the other hand, if |∇u| 1, grains
which are rolling can be deposited on the bed. Hence the moving layer becomes
smaller.
Figure 1. A mass of “snow” flowing down. The thick/thin line
corresponds to the profile before/after the “snow” has flown.
This model is studied in one space dimension in the rest of the paper. Define
p
.
= ux, and assume p 0, one can rewrite (1.1) into the following 2 × 2 system of
balance laws
(1.2)
(hp)x = (p 1)h ,
pt +
(ht
(p 1)h
)
x
= 0 .
Writing the system of balance laws (1.2) in quasilinear form, the corresponding
Jacobian matrix is computed as
A(h, p) =
−p −h
p 1 h
.
For h 0 and p 0, one finds two real distinct eigenvalues λ1 0 λ2, as
λ1,2 =
1
2
h p ± (p h)2 + 4h .
When h is small, i.e., with h 0, we have
λ1 = −p +
p 1
p
h +
O(h2)
, λ2 =
h
p
+
O(h2)
.
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