Laboratory Teaching Exercises
23
Such a system of flow lines and equipotential lines is what is called the flow
net. In each flow net, the flow lines and the equipotential lines intersect at
right angles.
This important feature of a flow net can be explained as follows. Just as water
flowing downhill naturally follows the steepest path, so does water flowing
between equipotential lines follow the path of maximum gradient. By
definition, gradient is the difference in potential between two equipotential
lines divided by the distance between the lines. If the lines are parallel the
maximum gradient will occur where the distance between the lines is least, ie.
along any line which is perpendicular to the equipotential lines. This is true
even if the equipotential lines are infinitely close to one another.
Now imagine two adjacent equipotential lines which are infinitely close to one
another but not parallel. The shortest distance between the two lines at any
point along one of them can be established as follows. Using the intersection
of the two equipotential lines as a centre, describe an imaginary arc that
passes through the point in question and also intersects the second
equipotential line. The chord of that arc which is contained between the two
equipotential lines can be shown to be the shortest distance between the two
lines at the point in question. Hence, the gradient will be maximum along that
chord. However, by definition, the two lines are infinitely close together at that
point.
So the difference in length between the arc and the chord is infinitely small.
Hence, at that point, the gradient along the arc is equal to the gradient along
the chord. By construction, the arc is perpendicular to each equipotential line
at its intersection with that line. It follows, then, that all lines representing the
paths of maximum gradient intersect all lines of equipotential at right angles.
Since flow follows the path of maximum gradient, it, too is perpendicular to all
equipotential lines.
c. Rate of Seepage
Now consider that same portion of the water that seeps out of basin I and into
basin II (see diagram above) in a given time will pass through an area dF
bounded by the two flow lines (a
s
a
t
and b
s
b
t
) shown. This seepage, Q, will be
measured in gallons per unit of time per running foot (ie. per foot normal to
the plane of the section). If we designate that portion of the seepage passing
through dF as dq we get from Darcy's Law:
…. (4)
where dh is the potential drop between the two equipotential lines, the only
unknown in equation (4).
If we choose equipotential lines such that the area dF resembles a square,
then the distance dm is approximately equal to ds, and equation (4) reduces
to:
dq = kdh …. (5)
