Consider a vertical cylindrical container partially filled...

Name: Problem 41, Chapter 11: Fundamentals of Thermal-Fluid Sciences
Uploaded: 2021-12-22T15:31:32-08:00
Duration: 1 min 9 s
Channel: Hast Aggarwal
Description: Consider a vertical cylindrical container partially filled with water. Now the cylinder is rotated about its axis at a specified angular velocity, and rigid-body motion is established. Discuss how the pressure will be affected at the midpoint and at the edges of the bottom surface due to rotation.

Consider a vertical cylindrical container partially filled with water. Now the cylinder is rotated about its axis at a specified angular velocity, and rigid-body motion is established. Discuss how the pressure will be affected at the midpoint and at the edges of the bottom surface due to rotation.

Key Concepts

Hydrostatic Pressure

In fluids at rest, hydrostatic pressure increases linearly with depth due to the weight of the fluid above. This concept remains important even as additional forces come into play in rotating systems, as it establishes the baseline pressure distribution in the vertical direction.

Rigid Body Rotation in Fluids

When a fluid is rotated such that every particle moves with the same angular velocity, it is said to be in rigid-body rotation. This assumption simplifies the analysis by ensuring that there are no velocity gradients within the fluid, meaning the entire fluid mass rotates uniformly with a linear velocity profile proportional to the distance from the axis.

Centrifugal Force

In a rotating reference frame, an apparent outward force, known as the centrifugal force, acts on the fluid elements. This force is proportional to the square of the angular velocity and increases with radial distance from the axis, leading to a radial pressure gradient that is not present in non-rotating fluids.

Pressure Distribution in a Rotating Fluid

The combination of gravitational and centrifugal effects leads to a pressure distribution that varies both vertically and radially. Specifically, in a rotating cylindrical container, the pressure at the bottom of the container is higher near the edges compared to the midpoint because the centrifugal force increases with distance from the center, augmenting the pressure beyond that predicted by gravitational hydrostatic pressure alone.

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