Geology Reference
In-Depth Information
isotropic viscous fluid, analogous to Hooke's law for an isotropic elastic solid, we
write the viscous stress as
λ
e
kk
δ
i
j
2μ
e
ij
,
+
(6.108)
where λ
and μ
are two constants of proportionality and
∂v
1
∂
x
1
+
∂v
2
∂
x
2
+
∂v
3
∂
x
3
=∇·
e
kk
=
v
(6.109)
is the flow divergence rate. The mean normal viscous stress is
3
μ
3
3λ
∇·
v
1
2
2μ
∇·
λ
+
v
+
=
∇·
v
.
(6.110)
The viscous stress over and above the mean normal stress is
+
μ
∂v
j
3
μ
=
μ
∂v
j
i
j
∂
x
i
+
∂v
i
2
∂
x
i
+
∂v
i
2
3
∇·
λ
∇·
i
j
λ
+
i
j
v
δ
−
∇·
v
δ
∂
x
j
−
v
δ
,
∂
x
j
(6.111)
which is called the stress deviator.
Similarly, the mean normal strain rate is
1
3
∇·
v
,
(6.112)
while the rate of strain over and above the mean normal rate of strain is
∂v
j
∂
x
i
+
∂v
j
∂
x
i
+
1
2
∂v
i
∂
x
j
1
3
∇·
1
2
∂v
i
∂
x
j
−
2
3
∇·
i
j
i
j
−
v
δ
=
v
δ
,
(6.113)
which is called the strain rate deviator. The total viscous stress can be written as
the sum of the stress deviator plus the mean normal stress, or
μ
∂v
j
i
j
3
μ
∂
x
i
+
∂v
i
2
3
∇·
2
λ
+
i
j
.
∂
x
j
−
v
δ
+
∇·
v
δ
(6.114)
Hence, the viscous stress deviator is proportional to the rate of strain deviator,
and the mean normal stress is proportional to the mean normal rate of strain. The
constants of proportionality define the first and second coe
cients of viscosity, η
and ζ. The total viscous stress is then
∂v
j
∂
x
i
+
j
∂v
i
∂
x
j
−
2
3
∇·
i
i
η
v
δ
+
ζ
∇·
v
δ
j
.
(6.115)
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