Geology Reference
In-Depth Information
From the equation (6.90) for the conservation of mass, the quantity in square brack-
ets vanishes, and after dividing by ρ the
equation of motion
becomes
∂τ
ji
∂
x
j
.
∂v
i
∂
t
+
v
j
∂v
i
1
ρ
∂
x
j
=
f
i
+
(6.103)
The stress field thus far has been left unspecified. In a fluid, the stress field arises
from pressure and the resistance of the fluid to deformation. While a fluid, by
definition, can support no permanent shear stress, it does in general resist a finite
deformation rate.
Consideration of the surface forces and moments on an arbitrary volume ele-
ment, as in Section 1.4.1, as the volume element becomes vanishingly small, shows
the stress τ
ji
to be symmetric. The diagonal components of the stress, τ
11
,τ
22
,τ
33
,
as before are the
normal stresses
, while τ
12
,τ
13
,τ
21
,τ
23
,τ
31
,τ
32
are
shear stresses
.
In a static fluid, the shear stresses vanish by definition of a fluid and the normal
stresses are all equal to
p
, where
p
is the scalar pressure field.
The analysis of the rate of deformation of a fluid closely follows that for a solid
in Section 1.4.3, with velocities replacing displacements. The velocity of a fluid
particle at
P
(
x
k
−
+
dx
k
) relative to a neighbouring fluid particle at
P
(
x
k
)is
d
v
i
=
∂v
i
∂
x
j
dx
j
.
(6.104)
Once again, the vector gradient can be split into its symmetric and antisymmetric
parts as
∂v
i
∂
x
j
=
e
ji
+
ω
ji
,
(6.105)
where
∂v
j
∂
x
i
+
∂v
j
∂
x
i
−
1
2
∂v
i
∂
x
j
1
2
∂v
i
∂
x
j
e
ij
=
, ω
ij
=
(6.106)
are both second-order tensors.
e
ij
is the
strain rate tensor
. As in the case of the
deformation of a solid,
e
ij
represents the true rate of deformation of the fluid, while
ω
ij
can be shown to represent a rigid-body rotation of
P
around
P
at the angular
velocity
1
2
(
ξ
=
∇×
v
)
;
(6.107)
ξ
is called the
vorticity
of the flow.
We now require a law relating the stresses produced in resistance to the deform-
ation rate, as measured by the strain rate tensor. The most widely used assumption
is that these quantities are linearly related, the law of Newtonian viscosity. For an
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