[PhD Qualifier] Liquid in a rotating arm

• confuted
In summary, the problem involves a bent glass tube with one arm immersed in a liquid and the other in air. The tube is rotated at a constant speed and the height of the liquid in the vertical arm is determined. Using Bernoulli's principle, the change in pressure from the centrifugal force can be calculated and integrated to find the pressure in the air above the water. This pressure is then compared to the atmospheric pressure to determine the height of the liquid.
confuted

Homework Statement

An open glass tube of uniform bore (uniform inner diameter) is bent into the shape of an "L". One arm is immersed into a liquid of density $$\rho$$ and the other arm of length l remains in the air in a horizontal orientation. The tube is rotated with constant angular speed $$\omega$$ about the axis of the vertical arm. Show that the height y to which the liquid rises in the vertical arm is
$$y=\frac{P_0}{g\rho}\left(1-exp\left[-\frac{\omega^2 l^2 M}{2RT}\right]\right)$$​
where $$P_0$$ is the atmospheric pressure and M is the molecular weight of air and R is the universal gas constant. Assume that the air pressure inside open end of the tube is atmospheric pressure, as shown (see attached).

Homework Equations

pV=nRT
$$\rho_{air}=\frac{m}{V}=\frac{MP_0}{RT}$$ (from ideal gas law)

The Attempt at a Solution

I thought about trying to kludge something together with the centripetal acceleration causing a difference in pressure, but that didn't work out for me. Then I looked up Bernoulli's principle, and I don't see how I'm going to get an exponential out of that. What's the appropriate law here?

Attachments

• liquid.jpg
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I think that I've figured it out. You do need Bernoulli's equation. The pressure will be decreasing as you come in from the farthest out point of the arm. The rise in the water level will counteract this decrease in pressure. Now we just need to figure out the change in pressure going from the outside of the arm and working our way in.

An infinitesimal change in pressure from the dynamical pressure term (velocity pressure) will be:
$$dP= \rho_{air} v dv$$.

However, $$v=r\omega$$ and $$\rho_{air}=\frac{MP}{RT}$$
So we get:
$$dP=\frac{MPr\omega^2}{RT} dr$$
Rearrange and integrate:
$$\int_{P_0}^P \frac{dP'}{P'}=\int_l^0 \frac{M\omega^2 r}{RT} dr$$
which yields
$$P=P_0 exp(-\frac{M\omega^2 l^2}{2R T})$$
What I have found is the pressure in the air above the water inside the tube. Take the difference between this and $$P_0$$ and set it equal to $$\rho g h$$ (which is making up for the decrease in pressure) and you have the answer. Let me know if you see things wrong with my reasoning.

1. What is the purpose of studying liquid in a rotating arm for a PhD Qualifier?

The purpose of studying liquid in a rotating arm for a PhD Qualifier is to understand the dynamics and behaviors of liquids under rotational motion. This knowledge can have practical applications in various fields such as fluid mechanics, material science, and industrial processes.

2. How does the rotation of the arm affect the behavior of the liquid?

The rotation of the arm creates centrifugal forces that can cause the liquid to move and form unique patterns and shapes. It can also affect the surface tension, viscosity, and density of the liquid, leading to different behaviors compared to when the liquid is at rest.

3. What are some common methods used to study liquid in a rotating arm?

Some common methods used to study liquid in a rotating arm include numerical simulations, experimental setups using rotating devices, and theoretical models. Each method has its advantages and limitations, and a combination of these approaches is often used to gain a better understanding of the system.

4. What are some potential applications of studying liquid in a rotating arm?

Studying liquid in a rotating arm can have various applications, such as in the design of centrifuges for separating substances, predicting the behavior of liquids in space, and understanding the dynamics of hurricanes and other rotating weather phenomena. It can also help improve the efficiency of industrial processes that involve the use of rotating equipment.

5. How does the study of liquid in a rotating arm contribute to the overall understanding of fluid dynamics?

The study of liquid in a rotating arm provides valuable insights into the fundamental principles of fluid dynamics, such as the effects of rotation, shear, and turbulence. It also helps researchers develop and test new theories and models that can be applied to a wide range of fluid systems, from microfluidics to large-scale industrial processes.

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