Environmental Engineering Reference
In-Depth Information
y
y = h
(
x, t
)
y = h
0
x
θ
Figure 1.12.
The configuration of the flow
equation, most examples will be based on this model, but we will also refer to other
papers dealing with alternative viscoplastic models or Coulomb friction.
We consider a shallow layer of fluid flowing over a rigid impermeable plane
inclined at an angle
θ
(see Figure 1.12). The fluid is viscoplastic and incompressible;
its density is denoted by
ρ
and its bulk viscosity by
η
=
τ/γ
. The ratio
=
H
∗
/L
∗
between the typical vertical and horizontal lengthscales,
H
∗
and
L
∗
, respectively, is
assumed to be small. The streamwise and vertical coordinates are denoted by
x
and
y
,
respectively.
A two-dimensional flow regime is assumed, i.e. any cross-stream variation is
neglected. The depth of the layer is given by
h
(
x, t
). The horizontal and vertical
velocity components of the velocity
u
are denoted by
u
and
v
, respectively. The fluid
pressure is referred to as
p
(
x, y, t
), where
t
denotes time. The surrounding fluid
(assumed to be air) is assumed to be dynamically passive (i.e. inviscid and low density
compared with the moving fluid) and surface tension is neglected, which implies that
the stress state at the free surface is zero.
The governing equations are given by the mass and momentum balance equations
∇·
u
=0
,
[1.2]
d
d
t
=
ρ
∂
u
∂t
ρ
+
ρ
(
u
·∇
)
u
[1.3]
=
ρ
g
−∇p
+
∇·
σ
,
supplemented by the following boundary conditions at the free surface
v
(
x,h,t
)=
d
d
t
=
∂h
∂t
+
u
(
x,h, t
)
∂h
∂x
,
(
x,
0
,t
)=0
.
[1.4]
There are many ways of transforming these governing equations into dimensionless
expressions [BAL 99, KEL 03, LIU 90a]. Here, we depart slightly from the
presentation given by Liu and Mei [LIU 90a]. The characteristic streamwise and
