A soap bubble of radius 38.2 mm is blown on the end of a narrow tube of length 11.2 cm and internal diameter 1.08 mm. The other end of the tube is exposed to the atmosphere. Find the time taken for the bubble radius to fall to 21.6 mm. Assume Poiseuille flow in the tube. (For the surface tension of the soap solution use 2.50 × 10^–2 N/m; the viscosity of air is 1.80 × 10^–5 N∙ s/m².)

To find the time taken for the bubble radius to fall to 21.6 mm, we first need to understand the concept of Poiseuille flow and how it applies in this scenario.

Understanding Poiseuille Flow

Poiseuille flow refers to the steady flow of a viscous fluid through a cylindrical tube. It is characterized by a parabolic velocity profile, with the highest velocity at the center of the tube and decreasing towards the walls. The flow rate is directly proportional to the pressure difference and the fourth power of the radius of the tube.

Calculating the Time Taken

Given that the soap bubble is blown at one end of the tube, we can assume that the pressure inside the bubble is greater than atmospheric pressure. As the bubble radius decreases, the pressure inside the bubble decreases, causing the bubble to move towards the other end of the tube.

Using the principles of Poiseuille flow, we can calculate the time taken for the bubble radius to fall to 21.6 mm. We can use the formula:

t = (8 * viscosity * length * radius^2) / (3 * pressure difference)

Calculation Steps:

• Calculate the initial pressure difference using the surface tension of the soap solution and the initial radius of the bubble.
• Calculate the final pressure difference using the surface tension of the soap solution and the final radius of the bubble.
• Substitute the values into the formula to find the time taken for the bubble radius to fall to 21.6 mm.

By following these steps and calculations, you can determine the time it will take for the soap bubble radius to decrease to 21.6 mm in the narrow tube.