Environmental Engineering Reference
In-Depth Information
Proust [COU 96a], Balmforth and Craster [BAL 99, BAL 02], Matson and Hogg
[MAT 07], and Hogg and Matson [HOG 09]. Taking the two dominant contributions
in equations [1.6] and [1.7] and returning to the physical variables, we deduce
σ
xy
=
ρg
cos
θ
(
h − y
)
tan
θ −
∂h
∂x
,
[1.8]
p
=
ρg
(
h − y
)cos
θ.
[1.9]
The bottom shear stress is then found to be
τ
b
=
σ
xy
|
y
=0
. For bottom shear stresses
in excess of the yield stress
τ
c
, flow is possible. When this condition is satisfied, there
is a yield surface at depth
y
=
h
0
within the bulk, along which the shear stress matches
the yield stress
σ
xy
|
y
=
h
0
=
ρg
cos
θ
(
h − h
0
)
tan
θ −
=
τ
c
.
∂h
∂x
[1.10]
The yield surface separates the flow into two layers [BAL 99, LIU 90a]: the bottom
layer, which is sheared, and the upper layer or plug layer, where the shear rate is nearly
zero. Indeed, using an asymptotic analysis, Balmforth and Craster demonstrated that
in the so-called plug layer, the shear rate is close to zero, but non-zero [BAL 99]. This
result may be seen as anecdotic, but it is in fact of great importance since it resolves a
number of paradoxes raised about viscoplastic solutions.
On integrating the shear-stress distribution, we can derive a governing equation for
the flow depth
h
(
x, t
). For this purpose, we must specify the constitutive equation. For
the sake of simplicity, we consider a Bingham fluid in one-dimensional flows as Liu
and Mei [LIU 90a] did; the extension to Herschel-Bulkley and/or two-dimensional
flows can be found in [BAL 99, BAL 02, MEI 01b]. In the sheared zone, the velocity
profile is parabolic
tan
θ −
h
0
y −
2
y
2
for
y ≤ h
0
,
u
(
y
)=
ρg
cos
θ
μ
∂h
∂x
1
while the velocity is constant to leading order within the plug
tan
θ −
for
y ≥ h
0
.
u
(
y
)=
u
0
=
ρgh
0
cos
θ
μ
∂h
∂x
The flow rate is then
q
=
h
0
tan
θ −
u
(
y
)d
y
=
ρgh
0
(3
h − h
0
)cos
θ
6
μ
∂h
∂x
.
[1.11]
Integrating the mass balance equation over the flow depth provides
∂h
∂t
+
∂q
∂x
=0
.
[1.12]
