Question Video: Comparing Speeds of Layers in a Fluid Physics

Video Transcript

A volume of a viscous fluid is contained between two parallel horizontal plates, as shown in the diagram. The plate above the volume of fluid moves horizontally at a speed 𝑣 one. Horizontal layers of the fluid move in laminar flow at speeds 𝑣 two to 𝑣 six. Which of the following correctly describes the relationship between the speeds of the layers? (A) 𝑣 two equals 𝑣 three equals 𝑣 four equals 𝑣 five equals 𝑣 six. (B) 𝑣 six is greater than 𝑣 five is greater than 𝑣 four is greater than 𝑣 three is greater than 𝑣 two. (C) 𝑣 two is greater than 𝑣 three is greater than 𝑣 four is greater than 𝑣 five is greater than 𝑣 six. (D) 𝑣 four is greater than 𝑣 five; 𝑣 five equals 𝑣 two; 𝑣 six equals 𝑣 one; 𝑣 three is greater than 𝑣 two. And (E) 𝑣 four is less than 𝑣 five; 𝑣 five equals 𝑣 two; 𝑣 six equals 𝑣 one; and 𝑣 three is less than 𝑣 two.

On the right side of our diagram, we see all six of these speeds 𝑣 one through 𝑣 six. 𝑣 one is the speed of the plate that’s moving to the right on the top of these layers of fluid. The bottom plate has no speed listed because we’re told it’s stationary. The layers of fluid between the plates, layers one through five, move with speeds 𝑣 two through 𝑣 six. We want to choose which of our five answer options correctly describes the relationship between these speeds. Let’s clear some space to work.

And we can, first of all, note that the top and the bottom layer of the fluid — those are layers one and five — will match the speeds of the plates that they’re in contact with. That is, we expect that the speed of the moving plate 𝑣 one be equal to the speed of the topmost layer 𝑣 two. Likewise, we expect that 𝑣 six, the speed of layer five, which is in contact with the stationary plate, be zero. Writing down what we know, we can say that 𝑣 one equals 𝑣 two and 𝑣 six is zero. Notice that in answer options (D) and (E), the speed 𝑣 six is said to be equal to the speed 𝑣 one. But because the moving plate is moving — that is, it has a speed that’s not equal to zero — it can’t be true that 𝑣 six, which is zero, equals 𝑣 one. For this reason, we can eliminate options (D) and (E) from consideration.

Because the top plate in our scenario has a nonzero speed and the bottom plate does have a speed of zero and that the flow between these layers is laminar, we can expect that the relative speeds of the layers of fluid between the plates might look like this. At the top is the speed of layer one, then the speed of layer two, then layer three, then layer four, and then layer five. Recall that layer five has a speed 𝑣 six, which we’ve said equals zero because it matches the speed of the stationary plate.

So, going from layer one all the way down to layer five of our fluid, we see a smooth decrease in speed. Since these layers do not all have the same speed, we can cross off answer option (A). And since these speeds decrease in magnitude from 𝑣 two to 𝑣 three to 𝑣 four to 𝑣 five to 𝑣 six, we know that option (B), which has the inequalities acting in the opposite direction, also can’t be correct. Answer choice (C) accurately describes the relative speeds of our fluid layers. 𝑣 two is greater than 𝑣 three which is greater than 𝑣 four which is greater than 𝑣 five which is greater than 𝑣 six.