Video Transcript
A viscous fluid flows in a pipe of constant cross-sectional area perpendicular to the flow direction. The diagram shows a cross section of the pipe parallel to the flow direction. The fluid moves in horizontal layers in laminar flow at speeds 𝑣 one to 𝑣 five. Which of the following correctly describes the relationship between the speeds of the layers? (A) 𝑣 one equals 𝑣 two equals 𝑣 three equals 𝑣 four equals 𝑣 five. (B) 𝑣 three is greater than 𝑣 four; 𝑣 four equals 𝑣 two; 𝑣 four is greater than 𝑣 five; 𝑣 five equals 𝑣 one. (C) 𝑣 one is greater than 𝑣 two is greater than 𝑣 three is greater than 𝑣 four is greater than 𝑣 five. (D) 𝑣 one equals 𝑣 five; 𝑣 one is greater than 𝑣 two; 𝑣 two equals 𝑣 four; 𝑣 two is greater than 𝑣 three. (E) 𝑣 one is less than 𝑣 two is less than 𝑣 three is less than 𝑣 four is less than 𝑣 five.
In our diagram, we see these five layers of fluid that are moving between two stationary sections of pipe. From our perspective, these layers are all flowing to the right with speeds from 𝑣 one to 𝑣 five. We’re told that overall the fluid moves in horizontal layers — those are layers one, two, three, four, and five — in laminar flow. In this type of fluid flow, the layers will remain separate from one another. They won’t mix. We want to figure out which of our five answer options correctly ranks the speeds of these five layers.
To begin figuring that out, let’s clear some space on screen and remind ourselves that these two sections of the pipe, shown in gray, are not moving. That detail is important because it turns out that the layers in direct contact with these sections of pipe, layers one and five, will tend to match the speed of the stationary pipe. That is, so long as layers one and five of this fluid flow are fairly thin, their speed will be zero. In contrast to this, the middle layer, layer three, has the least resistance to flow. We expect the speed of layer three, that’s 𝑣 three, to be the highest speed of any of the layers. The remaining layers, layers two and four, will have speeds somewhere in between the speeds of the layers on either side of them.
We could sketch out the relative speeds of these five layers as follows. First, we show the speeds of layers one and five. As we’ve said, these speeds attempt to match the speed of the stationary pipe on either side. Midway between these layers, layer three demonstrates the greatest speed. Then the speeds of layers two and four are somewhere in between these maximum and minimum speed values. The speeds of layers two and four, that is, 𝑣 two and 𝑣 four, are equal, as are the speeds of layers one and five. We can now go through our answer options, starting with option (A) to see which one is correct.
Answer option (A) says that the speeds of all five layers are the same. We know, though, that that is not the case. Though the speeds of some of the layers are the same as the speeds of others, there is still a variation in speeds among our five layers. Moving on to answer choice (B), this option says, first, that 𝑣 three is greater than the four. Looking at our sketch of the speeds of these layers, we agree with that statement. The arrow representing the speed of layer three is longer than that representing the speed of layer four. Answer option (B) also says that 𝑣 four equals 𝑣 two. Since the arrows representing the speeds of these two layers have the same length, both ending along this same vertical line, we agree with this statement as well.
Next, this option says that 𝑣 four is greater than 𝑣 five. Our sketch confirms this also. 𝑣 five, we believe, is zero, and 𝑣 four is greater than that. Lastly, option (B) says that 𝑣 five equals 𝑣 one. Since both of those speeds end up along the same vertical line, we can agree with this statement too. All four of the claims of answer option (B) are correct. To confirm that this will be our answer, let’s review the remaining choices. Answer option (C) says that 𝑣 one is greater than 𝑣 two, which is greater than 𝑣 three, which is greater than 𝑣 four, which is greater than 𝑣 five. However, we’ve seen that it’s 𝑣 three that is the greatest of all the speeds, so it’s not accurate to say that 𝑣 one or 𝑣 two is greater than 𝑣 three. We won’t choose answer choice (C).
Answer choice (D) first says that 𝑣 one is equal to 𝑣 five, and that’s true. Both those speeds have the same value. Next, answer choice (D) says that 𝑣 one is greater than 𝑣 two. This goes against the result that we found. We found that 𝑣 one is zero, while 𝑣 two is nonzero. 𝑣 one then is not greater than 𝑣 two, and so we won’t choose answer choice (D). Lastly, answer option (E) says that 𝑣 one is less than 𝑣 two, which is less than 𝑣 three, which is less than 𝑣 four, which is less than 𝑣 five. Once again, we find an answer choice which claims that 𝑣 three is not the greatest speed of all. We know, though, that it actually is that that layer moves faster than any other. So we won’t choose answer option (E) either.
This confirms that answer option (B) is the best choice. The relationship between the speeds of these layers is that 𝑣 three is greater than 𝑣 four; 𝑣 four equals 𝑣 two; 𝑣 four is greater than 𝑣 five; and 𝑣 five equals 𝑣 one.
