Geoscience Reference
In-Depth Information
pressure of the mixture;
are the stresses of the mixture; and
F
i
is the
i
-component of the external force on the mixture, such as gravity.
Note that the spatial direction indices
i
and
j
are subject to Einstein's summation
convention: Subscript (or superscript) repeated twice in any product or quotient of
terms is summed over the entire range of values of that subscript (or superscript). For
example,
a
ij
b
j
τ
(
i
,
j
=
1, 2, 3
)
ij
=
j
=
1
a
ij
b
j
a
i
3
b
3
in the 3-D system.
According to numerous experimental studies, the water-sediment mixture with low
sediment concentration (less than about 200 kg
=
a
i
1
b
1
+
a
i
2
b
2
+
m
−
3
) is a kind of Newtonian fluid,
the constitutive relation of which is given by the Navier-Poisson law:
·
2
3
µ
τ
=
2
µ
m
D
ij
−
m
D
kk
δ
(2.27)
ij
ij
µ
m
is the dynamic viscosity of the mixture;
D
ij
is the tensor of deformation rate,
defined as
D
ij
where
=
(∂
u
i
/∂
x
j
+
∂
u
j
/∂
x
i
)/
2; and
δ
ij
is the Kronecker delta, with
δ
=
1
ij
when
i
j
.
When the sediment concentration is high, the mixture becomes a non-Newtonian
fluid, such as Bingham fluid. The relation between shear stress and deformation rate
for the one-directional shear flow of Bingham fluid is written as
=
j
and
δ
=
0 when
i
=
ij
du
dz
τ
13
=
τ
B
+
η
(2.28)
where
is the plastic viscosity.
Extending Eq. (2.28) to the multi-directional shear flow yields the general constitu-
tive relation of Bingham fluid (Prager, 1961; Wu and Wang, 2000):
τ
13
is the shear stress,
τ
B
is the yield stress, and
η
2
D
ij
ij
+
τ
B
I
1
/
2
2
1
3
D
kk
δ
τ
=
µ
−
(2.29)
ij
m
1
2
1
3
D
kk
)
where
I
2
.
The single-directional shear flow field can be divided into two zones by
=
(
D
ij
D
ij
−
, and
µ
m
is
η
τ
≥
τ
B
and
13
τ
13
<τ
B
. Eq. (2.28) is only applicable to the zone of
τ
12
≥
τ
B
. Similarly, Eq. (2.29) is
2
valid in the zone of
τ
τ
≥
2
τ
B
for the multi-directional shear flow.
ij
ij
2.2.2 Sediment transport equation
Sediment transport is governed by the following mass balance equation:
∂
+
∂(
)
c
u
si
c
=
0
(2.30)
∂
t
∂
x
i
Because the sediment velocity
u
si
is not a dependent variable in the diffusion model,
Eq. (2.30) is rewritten as
∂
c
+
∂(
u
i
c
)
=−
∂
∂
x
i
[
(
u
si
−
u
i
)
c
]
(2.31)
∂
t
∂
x
i
