Debris Flows - Environmental Geomechanics - page 22

- Viscous regime , where the pressure gradient is balanced by viscous stresses
within the bulk. In that case, we have
U ∗ = ρg cos θH ∗
η ∗ L ∗

.

Inertial terms must be low compared with the pressured gradient and the slope must
be gentle (tan θ ). This imposes the following constraint: Re

1. We deduced

that Fr 2 = O ( Re) 1.

- Visco-inertial regime , where inertial and viscous contributions are nearly equal.
In that case, we have

η ∗
ρH ∗ .
The pressure gradient must be low com pared with the viscous stress, which entails

U ∗ = 1

the follo wing condition η ∗ ρ gH ∗ . We obtain Re

∼ 1 and Fr =

η ∗ / ( ρ gH ∗

1.
- Nearly steady uniform regime , where the viscous contribution matches gravity
acceleration. In that case, we have
U ∗ = ρg sin θH ∗
η ∗

)

.

Inertia must be negligible, which means 1 (stretched flows). We obtain Re =
O (Fr 2 ) and tan θ (mild slopes).

In the inertial regime, the rheological effects are so low that they can be neglected
and the final governing equations are the Euler equations. The visco-inertial regime
is more spurious and has no specific interest in geophysics, notably because the flows
are rapidly unstable. More interesting is the viscous regime that maybe achieved for
very slow flows on gentle slopes ( θ 1), typically when flows come to rest. We will
further describe this regime in section 1.4.4. When there is no balance between two
contributions, we have to solve the full governing equations. This is usually a difficult
task, even numerically. To simplify the problem, one can use flow-depth averaged
equations (see sect 1.4.5). The nearly steady regime will be exemplified in sect. 1.4.5
within the framework of the kinematic-wave approximation. Finally, it should be kept
in mind that the partitioning into four regimes holds for viscous (Newtonian) fluids
and non-Newtonian materials for which the bulk viscosity does not vary significantly
with shear rate over a sufficiently wide range of shear rates. In the converse case,
further dimensionless groups (e.g. the Bingham number Bi = τ c H ∗ / ( μU ∗ )) must be
introduced, which makes this classification more complicated.

1.4.4. Slow motion

Slow motion of a viscoplastic material has been investigated by Liu and Mei
[LIU 90a, LIU 90c], Mei et al . [MEI 01a], Coussot et al . [COU 96b], Coussot and