Video Transcript
A volume of a nonviscous fluid
is contained between two parallel horizontal plates, as shown in the
diagram. The plate above the volume of
the fluid moves horizontally at a speed 𝑣 one. Horizontal layers of the fluid
move at speeds 𝑣 two to 𝑣 six. Which of the following
correctly describes the relationship between the speeds of the layers?
Before we get to our answer
options, let’s consider our diagram, which shows us a stationary plate separated
from a moving plate by five layers of a fluid. We’re also shown that the
moving plate has a speed 𝑣 one, while layer one of the fluid has a speed 𝑣
two, layer two has a speed 𝑣 three, and so on, all the way down to a layer five
with a speed 𝑣 six. We want to consider the
relationship between the speeds of these different layers. And a critical fact to keep in
mind is that the fluid we’re working with is nonviscous. A nonviscous fluid is an
idealized case where there is no friction between the layers of the fluid.
This means that layer one
exerts no frictional force on layer two which exerts no frictional force on
layer three and so on through all the fluid layers. Knowing this, we want to
identify the correct relationship for the speeds of these layers, in other
words, 𝑣 two, 𝑣 three, 𝑣 four, 𝑣 five, and 𝑣 six. So let’s now look at our answer
options.
Option (a) says that 𝑣 two is
greater than 𝑣 three is greater than 𝑣 four is greater than 𝑣 five is greater
than 𝑣 six. Option (b) says that 𝑣 six is
greater than 𝑣 five is greater than 𝑣 four is greater than 𝑣 three is greater
than 𝑣 two. Option (c) says 𝑣 four is
greater than 𝑣 five, 𝑣 five is equal to 𝑣 two, 𝑣 six is equal to 𝑣 one, and
𝑣 three is greater than 𝑣 two, while (d) says 𝑣 four is less than 𝑣 five, 𝑣
five is equal to 𝑣 two, 𝑣 six is equal to 𝑣 one, and 𝑣 three is less than 𝑣
two. And lastly, option (e) says
that all the speeds of the layers are equal.
Now, the key fact in all of
this, as we saw earlier, is that we’re working with a nonviscous fluid. This means that the fluid
layers don’t influence one another through friction. And that means it’s impossible
for any one of these layers to move in a way that’s different from any of the
others. To see why that’s so, let’s
pick a layer, let’s pick layer two, and let’s imagine that this layer is moving
along left to right faster than layers one and three. If that was the case, if these
layer speeds were unequal, then layer two would exert a frictional force on
layers one and three. It would have to because the
molecules in this layer of the fluid are moving faster. But because our fluid is
nonviscous, that can’t be.
None of the layers exerts a
frictional force on any of the others, which means that rather than thinking of
this fluid as five separate layers, we can really think of it as one single
layer. It all moves together and all
at the same exact speed. Therefore, whatever the speed
of layer one, for example, that’s 𝑣 two, this must equal the speed of layer two
and the speed of layer three and four and five. And we see that out of our
answer options, it’s option (e) which claims that all the layers move with the
same speed. This must be the case for a
nonviscous fluid.
