Video Transcript
The diagram shows the flow of the
fluid past a circular obstacle. The gray lines represent the
direction of the fluid flow. Black regions represent solid
obstacles to the flow. In which of the two regions within
the dashed lines is the fluid flow faster?
So we’re imagining here that the
fluid in this diagram is flowing left to right. And as it does so, a solid obstacle
to that flow, right here, comes up. We could imagine this as being
something like a rock in a riverbed. And we know that such a rock causes
the fluid around it to flow differently. And indeed, we see that the gray
lines indicating the direction of fluid flow do change as the fluid avoids this
solid obstacle. Downstream of that obstacle, we see
these two regions marked out one and two. And we want to know in which of
them is the fluid flow faster on average.
Before we answer this question
though, of in which region the fluid flow is faster, let’s consider in which of the
two regions is the fluid flow more turbulent. We can recall that fluid flow is
more turbulent when a fluid’s speed and direction change more. So if we wanna figure out in which
region, one or two, the flow is more turbulent, we can look to see in which one do
the fluid speed and direction change more.
We can figure this out by
considering the gray lines, what we could call the streamlines of the fluid in this
flow. Notice how, in region one, these
two flow lines approach one another across this region, while in region two, the
flow lines are nearly parallel with one another. This might seem like a small
difference. But the changes in fluid direction
we see in region one greater than the direction changes we see in region two
indicate a more turbulent flow in region one. And indeed, this agrees with our
intuition.
We would expect a region in our
fluid right behind a large solid obstacle to be more turbulent, whereas farther
downstream, farther away from this obstacle, we expect the flow to smooth out. Okay, so if the flow in region one
is more turbulent than that in region two, how does that help us answer this
question of in which region does the fluid flow faster?
Consider again the streamlines in
region one. If we were to draw exaggerated
velocity vectors for parts of these streamlines in this region, they might look like
this. These vectors that we’ve drawn in,
even though they may overstate the motion of the fluid in this dimension, show us
that, nonetheless, because of these streamlines getting closer and closer together,
fluid in one part of this region will be pushing against fluid in the other
part. That is, fluid on this side of our
dashed line will be pushing on fluid on this side of the line, and vice versa.
Because the fluid in region one
isn’t only moving left to right but is instead also moving up and down we could say
from this perspective, some of that fluid velocity will be negated as the fluid
flows against itself. This will lead on average across
the whole region to a slowing down of this fluid. This is in contrast to the fluid in
region two. Instead of some fluid in the region
pushing against fluid in other parts of it, it’s all essentially moving in lockstep
and therefore fairly rapidly from left to right. Because the fluid in region one
resists its own motion more than the fluid in region two, we can say that it’s the
fluid in region two which flows faster.