Environmental Engineering Reference
In-Depth Information
with
K
=
ρg
sin
θ/μ
. The governing equation takes the form of a nonlinear advection
equation, which can be solved using the method of characteristics [LEV 02].
Using the chain rule for interpreting this partial differential equation [1.26], we
can show that it is equivalent to the following ordinary equation
d
d
t
=0
,
[1.27]
along the characteristic curve
d
d
t
=
λ
(
h
)
,
[1.28]
in the (
x, t
) plane, with
λ
(
h
)=
Kh
(
h − h
p
). Equation [1.27] shows that the flow
depth is constant along the characteristic curve, hence the characteristic curves are
straight lines, the slope of which are given by the right-hand side term
λ
(
h
) in
equation [1.28]. These characteristic curves can be used to solve an initial value
problem, where the initial value of
h
is known over a given interval:
h
=
h
i
(
x
i
)
(at
t
=0). The value of
h
along each characteristic curve is the value of
h
at the initial
point
x
(0) =
x
i
. We can thus write
h
(
x, t
)=
h
i
(
x
i
)=
h
i
(
x − λ
(
h
i
(
x
i
))
t
)
.
It is worth noting that because of the nonlinearity of equations [1.26], a smooth initial
condition can generate a discontinuous solution (shock) if the characteristic curves
intersect, since at the point of intersection
h
takes (at least) two values [LEV 02].
1.5. Bibliography
[ANC 97] ANCEY C., Rhéologie des écoulements granulaires en cisaillement simple, Ph.D.
thesis, École Centrale de Paris, 1997.
[ANC 07] ANCEY C., “Plasticity and geophysical flows: a review”.
J. Non-Newtonian Fluid
Mech
., vol. 142, p. 4-35, 2007.
[ANC 99] ANCEY C., COUSSOT P., “Transition from frictional to viscous regime for
granular suspensions”.
C. R. Acad. Sci. Paris sér. I
, vol. 327, p. 515-522, 1999.
[ANC 00] ANCEY C., EVESQUE P., “Frictional-collisional regime for granular suspension
flows down an inclined channel”.
Phys. Rev. E
, vol. 62, p. 8349-8360, 2000.
[ANC 08] ANCEY C., IVERSON R. M., RENTSCHLER M., DENLINGER R. P., “An exact
solution for ideal dam-break floods on steep slopes”.
Water Resour. Res.
, vol. 44, W01430,
2008.
[ANC 01] ANCEY C., JORROT H., “Yield stress for particle suspensions within a clay
dispersion”.
J. Rheol
., vol. 45, p. 297-319, 2001.
