Question Video: Understanding the Movement of Layers of a Nonviscous Fluid Physics

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.