Environmental Engineering Reference
In-Depth Information
Substituting
q
with its expression [1.11] and the yield surface elevation
h
0
with
equation [1.10] into equation [1.12], we obtain a governing equation for
h
, which
takes the form of a nonlinear diffusion equation
F
(
h, h
0
)
∂h
∂h
∂t
=
∂
∂x
∂x
−
tan
θ
,
[1.13]
with
F
=
ρgh
0
(3
h − h
0
)cos
θ/
(6
μ
).
A typical application of this analysis is the derivation of the shape of a viscoplastic
deposit. Contrary to a Newtonian fluid, the flow depth of a viscoplastic fluid cannot
decrease indefinitely when the fluid spreads out along an infinite plane. Because of
the finite yield stress, when it comes to rest, the fluid exhibits a non-uniform flow-
depth profile, where the pressure gradient is exactly balanced by the yield stress. On
an infinite horizontal plane, the bottom shear stress must equal the yield stress. Using
equation [1.8] with
θ
=0and
y
=0, we eventually obtain [LIU 90a]
−ρgh
∂h
σ
xy
|
y
=0
=
τ
c
=
∂x
,
[1.14]
which, on integrating, provides
− h
i
=
2
τ
c
ρg
h
(
x
)
(
x
i
− x
)
,
where
h
=
h
i
at
x
=
x
i
is a boundary condition. This equation shows that the deposit-
thickness profile depends on the square root of the distance. When the slope is non-
zero, an implicit solution for
h
(
x
) to equation [1.8] is found [LIU 90a]
log
τ
c
− ρgh
sin
θ
τ
c
− ρgh
i
sin
θ
=tan
2
θ
(
x − x
i
)
.
τ
c
ρg
cos
θ
tan
θ
(
h
(
x
)
− h
i
)+
[1.15]
The shape of a static two-dimensional pile of viscoplastic fluid was investigated
by Coussot
et al
. [COU 96b], Mei and Yuhi [MEI 01b], Osmond and Griffiths
[OSM 01], and Balmforth
et al
. [BAL 02]; the latter derived an exact solution, while
the former authors used numerical methods or
ad hoc
approximations to solve the
two-dimensional equivalent to equation [1.8]. Similarity solutions to equation [1.13]
have also been provided by Balmforth
et al.
[BAL 02] in the case of a viscoplastic flow
down a gently inclined, unconfined surface with a time-varying source at the inlet.
1.4.5.
Fast motion
The most common method for solving fast-motion free-surface problems is to
depth-average the local equations of motion. In the literature, this method is referred
to as the Saint-Venant approach, the boundary-layer approximation, the lubrication
