Why Does Detergent Cause a Cotton Loop to Expand on Water?

[sunquick]

Homework Statement

A loop of cotton floats on the surface of some water. A little detergent is dropped onto the water surface inside the loop, and the loop opens out and becomes circular. Explain why this happens. Draw a diagram showing the forces acting on a short length of the circumference of the loop has been added. Explain how the short length is in mechanical equilibrium

Homework Equations

To calculate the force on the string
$\textbf{F} = \gamma L$
$\gamma$ is the surface tension in the water-soap boundary
$L$ is the length of string we're considering.

$dW = \gamma dA$

The Attempt at a Solution

From a mechanics point of view, because the detergent lowers the surface tension inside the loop, the water outside the loop pulls the string with a larger force. From an energy point of view, I think that the water surface is minimized by maximizing the area of detergent, and that is achieved by turning the loop into a circle.

I don't know how to put this description into a mathematical statement, since I don't understand fluid mechanics very well. That's why I'm also struggling with the part where I'm asked to explain the mechanical equilibrium. Considering that the water outside has a larger pull than the detergent inside, and they are acting on the same length of string, I don't see how mechanical equilibrium can be achieved.

As a side question,are there any good books dealing with surface tension effects and how they can be explained mathematically? The ntroductory fluid mechanics books I looked into mention surface tension only in the first chapters and don't go into great detail about it.

 

[Chestermiller]

You have it doped out correctly. But another feature of the system is that the loop is going to be under tension. This is what it will take to make good on the mechanical equilibrium. Take a small arc of the loop, and let T be the tension at the ends of the section. Because the section of the loop is curved, the net force of the tension on the section will be radially inward. This will balance the difference in surface tension forces between the outside and the inside of the loop.

Chet

 

[sunquick]

So it will be an analogue of the pressure difference between the inside and the outside of a bubble of liquid. The tension of the string plays a similar role to the pressure difference in this case.

 

[Chestermiller]

So it will be an analogue of the pressure difference between the inside and the outside of a bubble of liquid. The tension of the string plays a similar role to the pressure difference in this case.

Yes.

 

[HalfLostAndSearching]

Thank you for your question. I can provide a response to your content and offer some guidance.

Firstly, the phenomenon you described is known as the Marangoni effect, where a difference in surface tension causes a flow of liquid. In this case, the detergent lowers the surface tension inside the loop, causing a flow of water towards the center of the loop, pulling the string and opening it into a circular shape.

To explain this mathematically, we can use the Laplace-Young equation, which relates the surface tension to the curvature of the interface between two fluids. In this case, the detergent creates a lower surface tension in the water-soap boundary, causing a higher curvature and thus a larger force pulling the string. This can be represented by the equation F = γΔL, where γ is the surface tension and ΔL is the change in length of the string due to the detergent.

As for the mechanical equilibrium, we can consider the forces acting on a short length of the string in the loop. The forces acting on this length are the tension from the string itself, the force from the detergent pulling the string, and the force from the water outside the loop pulling the string. These forces must be balanced for mechanical equilibrium to be achieved.

Regarding your side question, there are many books that deal with surface tension effects and their mathematical explanations. Some recommended titles include "Surface Tension in Microsystems: Engineering Below the Capillary Length" by Adrian Neild, "Surface Tension: Fundamentals and Applications" by Marek Kosmulski, and "Intermolecular and Surface Forces" by Jacob Israelachvili. I would also suggest looking into fluid mechanics textbooks that specifically focus on surface tension, such as "Fluid Mechanics: Fundamentals and Applications" by Yunus Cengel and John Cimbala.

I hope this response has been helpful in understanding the surface tension problem and providing some resources for further study. As a scientist, it is important to continuously seek knowledge and understanding in our respective fields, and I encourage you to keep exploring and learning about surface tension and its effects.