# Oscillation of Two Fluid Interface in a U-tube: The Story before Statics

Attached above is an example problem from College Physics by Nicholas Giordano. Of course, the problem is extremely simple - and I solved it too; however, I've a conceptual doubt.

Here's a part of the explanation given in the book:

Consider the pressure at point B. This point is inside liquid 1 and is just below the boundary that separates the two liquids. Because we consider point B to be at the same depth as point A (also located in liquid 1, but on the other side of the U-tube), the pressures at points A and B are the same. We next consider point C; this point is in liquid 2, and just above the boundary between the liquids. Because points B and C are extremely close together, the pressures at these two locations must be the same. You can see how that is true by imagining a thin (massless) plastic sheet stretched along the boundary between liquids 1 and 2. Because the system is at rest, this sheet is in translational equilibrium; hence, the total force on the sheet must be zero. That can only be true if the pressure is the same on the top and bottom surfaces of the sheet. We have thus concluded that the pressure is the same at points A, B, and C.

Now, for a very similiar problem in Physics by Resnick, Halliday & Krane (abbreviated RHK hereafter), something else is added as well-

For concluding that the pressure is same at B and C, RHK uses the same argument, that of translational equilibrium of the interface. Just after, it also mentions the following:

When we first pour oil in the tube, however, there may be a difference in pressure and an unbalanced force that would cause the system to move, until it reached the static situation as shown in the figure.

I'm not able to picture this phenomenon clearly. What kind of motion is RHK talking about? My instructor said there would be oscillations of the interface, which would eventually die down due to energy loss to the vessel and surroundings.

1. Will these oscillations be simple harmonic? (Damped, I guess?)
2. Under what assumptions, if any, would these oscillations be simple harmonic?
3. What effect does the viscosity of the two fluids have on these oscillations?
4. Is it necessary to include viscosity in our analysis? I ask this because such problems usually start with the assumption of ideal fluids, which are inviscid by definition.
5. Can we talk about these oscillations in terms of an equation of motion, for the interface? A detailed analysis would help me understand the concept better, and it will also clear up doubts of many (my classmates, at least) in such grey areas of fluid mechanics, which no text I've come across, addresses.

Thanks a lot.

• -1 Not clear what difficulty you are having with these questions. What is your conceptual doubt? Surely you can guess what effect viscosity (or the lack of it) will have? Your instructor explained what the 2nd quote means. Why don't you ask him also about the effect of viscosity? ... You ask for a detailed analysis, which is off topic. For such (thinly disguised) exercises you must show your attempt and ask about a conceptual difficulty. – sammy gerbil Mar 12 '18 at 22:55
• – sammy gerbil Mar 12 '18 at 23:39
• @sammy gerbil, I'm sorry but it doesn't look like it is off topic. Also, if instructors could answer everything then we wouldn't have Physics SE. My question is simple, I wish to learn more about the nature of these oscillations and possibly derive an equation describing the oscillation of the interface. I don't care about the original problem's answer, which I've already gotten as the problem is really easy - so didn't post the working. Moreover, I attached that problem only so that I'm able to explain the motivation behind my question, nothing else. – arya_stark Mar 13 '18 at 2:24
• What's funny is that the two links you've attached in your comment, do not answer my question + they're not even related to what I'm asking. Please reconsider your downvote and better think it through next time. – arya_stark Mar 13 '18 at 2:26
• You have not identified your conceptual doubt. You are asking for a solution to the problem, or help solving it, but you have shown no effort. If you can solve the "simple" problem what is your difficulty with this one? What don't you understand about the nature of the oscillations? – sammy gerbil Mar 13 '18 at 8:59

First of all, as mentioned by Sammy, the answers of this post should be somewhat helpful: Oscillations in U-tube

I will take the approach of using energy conservation to answer your question. First, I denote point $B$ of the equilibrium position to be of height $H$, and I will define $h_1=y_L$ and $h_2=y_R$ as given by the problem.

Furthermore, I will denote the deviation from the equilibrium position as $h$, oil density as $\rho_o$, water density as $\rho_w$, and the area of the cross section as $A$. Then the potential energy can be written as:

$$V=\frac{\rho_wg(H+h_1-h)^2S}{2}+\frac{\rho_wg(H+h)^2S}{2}+\rho_ogh_2(H+\frac{h_2}{2}+h)S$$

The first term is due the potential energy from the water in the "left side" of the tube. The second term is due to the water in the "right side" of the tube. The third term is due to the oil.

And the kinetic energy can be written as:

$$T=\frac{1}{2}\rho_wLS(\dot{h})^2+\frac{1}{2}\rho_oh_2S(\dot{h})^2$$

The sum of the two has to equal to some constant energy:

$$T+V=S(\frac{\rho_wg(H+h_1-h)^2}{2}+\frac{\rho_wg(H+h)^2}{2}+\rho_ogh_2(H+\frac{h_2}{2}+h)+\frac{1}{2}\rho_wL(\dot{h})^2+\frac{1}{2}\rho_oh_2(\dot{h})^2)=E$$

If you take the time derivative of both sides, you find:

$$\dot{h}[2\rho_wgh+(\rho_wL+\rho_0h_2)\ddot{h}]=0$$

where I've used the fact that $\rho_0h_2=\rho_wh_1$. So in fact, this is a simple harmonic oscillator assuming ideal liquids.

As for the remainder of your questions. I think the best way to intuitively grasp them is to think of viscosity as a source of energy dissipation. And whenever you have energy dissipation, the oscillation will be damped in some form.

• There's a little problem here, why did you assume T + V = constant? When you did that, you were bound to obtain the equation of a simple harmonic oscillator, so it was no surprise. – arya_stark Mar 13 '18 at 14:32
• This is based on the assumption that we have ideal liquids, so there will be no viscosity thus energy loss. This is not true in the presence of viscosity, as mentioned in my last paragraph. – PeaBrane Mar 13 '18 at 14:35
• Also, T + V = constant doesn't necessarily mean that we will obtain a simple harmonic oscillation. – PeaBrane Mar 13 '18 at 14:37
• Is it right to imagine this as the oscillation of an interface, such as when wine is slowly poured into a glass? Also, the oscillations - if simple harmonic, would go on forever! How then, can we talk about the in text problem, for which it is necessary to attain a steady state? – arya_stark Mar 13 '18 at 14:38
• Can you elaborate what you mean by oscillation of an interface? – PeaBrane Mar 13 '18 at 14:40