Video Transcript
The diagrams shown represent the
boundaries of the layers of a viscous fluid, which move at different speeds when the
liquid flows through a container. Which of the diagrams correctly
represents the boundaries of the layers for the liquid flowing through a container
in which the top of the container is moved relative to the base of the
container?
In our two diagrams, (A) and (B),
we see cross sections of layers of fluid. We want to pick which diagram
correctly represents the boundaries of the layers for a liquid that flows through a
container in which the top of the container is moved relative to the container’s
base. So say that this surface is the top
of the container and this is its base. We can imagine a scenario where the
top of the container moves while the base stays still. In that case, we want to know how
the layers of fluid in between the top and bottom would respond. If we think about the very top and
the very bottom layers of liquid, those layers will effectively stick with the parts
of the container they’re in contact with.
Therefore, the top-most layer of
fluid will move to the right with a speed roughly equal to the speed of the
container’s top, while the bottom-most layer of liquid will be stationary. In between these layers of liquid,
the other layers will move in such a way as to create a smooth speed gradient; that
is, the speeds of layers of liquid will vary something like this. Therefore, the boundaries, these
lines we’ve drawn that separate the layers of liquid, will remain intact. These boundaries will all be
oriented in a horizontal plane to agree with the relative motion of the container
top and bottom. All of this suggests that diagram
(A) correctly represents the boundaries of these fluid layers.
Let’s think for a moment about what
kind of movement of the container would be necessary to create diagram (B). If the container was, say, a box
like this, then if the entire box was stationary while fluid moved through it, say
like water through a pipe, or if the entire container altogether was in motion, then
the boundaries between the layers of liquid in the container would look like as is
shown in diagram (B). In diagram (B) then, we see the
boundaries of layers of liquid if the container all moved as a unit or stayed still
as a unit. In our actual scenario though, the
top of the container is moving relative to the container’s base. It’s indeed true, then, that
diagram (A) accurately represents the boundaries of the layers of liquid in this
container.
