Geology Reference
In-Depth Information
compress significantly under the extreme pressures at its base, we don't need to
consider that here. The main effect of compression can be taken into account in
a fairly simple way later if it is needed. So here we can consider our fluid to be
incompressible and just focus on deformation.
The box defined by ABDC becomes deformed into the parallelogram AB D C.
We can use this change in shape of the box to measure the deformation of the fluid.
One way to measure the deformation of the box is with the ratio of lengths BB / AB.
If the time interval that has elapsed between when the dye is at AB and when it is
at AB is t , then the distance from B to B will be Ut . Then we might write
shape change
=
Ut/H,
where H is the layer thickness. Then we would be using the tangent of the angle
BAB as our measure of deformation. For our viscous fluid we need the rate of
change of the shape or the rate of deformation. This can be found by dividing by
t ,so
rate of deformation
=
U/H.
You can see that this quantity is a spatial gradient of velocity - in other words, it
is the rate of change of (horizontal) velocity with (vertical) position. It may be a
little confusing at first that the direction in which the velocity changes (vertical)
is not the same as the direction of the velocity (horizontal), but if you look at
Figure 4.1(a) it should make sense.
Velocity gradient turns out to be a generally useful quantity to measure rates of
deformation. The technical terminology for a quantity that measures deformation
is called strain . It follows that a quantity that measures rate of deformation is called
a strain rate , so this is an alternative name for our velocity gradient. Here I will
use the symbol s for strain rate. For consistency with the more general technical
treatment that applies in two and three dimensions, which I will spare you here, I
will include a factor of one-half in the definition of strain rate for Figure 4.1(a), so
we get to the following definition of rate of deformation, or strain rate:
U
2 H .
=
s
(4.1)
Now let's turn to the force causing the deformation. A force must be applied to
the top plate in order to keep it moving. The moving plate then imparts a force into
the adjacent fluid. The force imparted into the top of the deformed box is depicted
in Figure 4.1(b) as F . The magnitude of this force depends on the length, L ,ofthe
box. For example, a second, adjacent box would also have a force F imparted into
it, and the total force imparted into both boxes would have to be 2 F in order to
cause the same rate of strain. However, the deformation of each box is the same.
Therefore what counts is the force per unit area that is applied to the fluid. We are
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