The Critical Angular Velocity ωc

The critical angular velocity, ωc, of a cylinder inside another cylinder containing liquid is the angular velocity at which turbulence occurs. Turbulence is a state of fluid flow characterized by chaotic and irregular motion. It is associated with high fluid velocities and is often observed in situations where there is a significant difference in the velocity of adjacent fluid layers.

Factors Affecting ωc

The critical angular velocity ωc depends on three main factors: viscosity η, density ρ, and the distance d between the walls of the cylinder. Let's explore each factor in detail:

Viscosity (η)

Viscosity is a measure of a fluid's resistance to flow. It determines how easily a fluid can be deformed or how it responds to shear forces. In the context of the critical angular velocity, viscosity plays a crucial role in determining the flow behavior of the liquid between the cylinders.

Density (ρ)

Density is the measure of mass per unit volume of a substance. It describes how compact or spread out the molecules of a substance are. In the case of the critical angular velocity, density influences the overall mass and inertia of the fluid between the cylinders. The density of the liquid affects how the fluid responds to the rotational motion of the inner cylinder.

Distance between the walls (d)

The distance between the walls of the cylinder, denoted as d, is another important factor influencing the critical angular velocity. The smaller the distance between the walls, the more confined the space for the fluid to flow. This confinement affects the flow patterns and can lead to increased turbulence at lower angular velocities.

Expression for ωc

The expression for the critical angular velocity ωc can be derived by considering the balance between the centrifugal force acting on the liquid and the viscous drag force. The centrifugal force is proportional to the angular velocity ω and the radius of the inner cylinder, while the viscous drag force is proportional to the viscosity η, the velocity gradient, and the area of contact between the cylinders.

By equating these two forces, we can obtain an expression for ωc:

ωc ∝ √(η / (ρd²))

This equation shows that the critical angular velocity is inversely proportional to the square root of the product of viscosity and the square of the distance between the walls. It implies that as viscosity or the distance between the walls increases, the critical angular velocity decreases.

In conclusion, the critical angular velocity ωc of a cylinder inside another cylinder containing liquid depends on the viscosity η, density ρ, and the distance d between the walls of the cylinder. By considering the balance between centrifugal and viscous drag forces, an expression for ωc can be derived.