A body of liquid rotates with constant and uniform angular...

A body of liquid rotates with constant and uniform angular velocity A about a vertical axis, under the action of gravity only. By integrating the equation of motion show that the free surface is a paraboloid of revolution, with axis vertical, and that the vorticity at any point of the fluid is equal to $\mathbf{A}$.

Key Concepts

Equations of Motion in Fluids

The equations of motion describe how fluid parcels accelerate under the influence of forces. In fluid dynamics, these equations are derived from Newton’s second law and incorporate pressure gradients, gravitational forces, and inertial effects. When a fluid is in a rotating frame, additional fictitious forces such as the centrifugal force must be taken into account, leading to modified forms of the momentum equation that are essential for analyzing rotating flows.

Force Balance in Rotating Fluids

In fluids undergoing rotation, the balance of forces is paramount. The gravitational force acts downward, while the centrifugal force, which arises due to the circular motion, acts radially outward. The requirement for mechanical equilibrium along the fluid’s free surface leads to a balance between these forces and the pressure gradient. This balance, when integrated over the fluid domain, determines the pressure distribution and the shape of the free surface.

Hydrostatic Pressure Distribution

In a steady or slowly varying flow, the pressure at any point within the fluid adjusts to balance the applied body forces such as gravity and centrifugal effects. This balance is described by a hydrostatic pressure distribution that is modified in a rotating system. The pressure becomes a function of both the vertical and radial positions, and its gradient is crucial for determining the free surface profile of the fluid.

Free Surface Formation and Paraboloid Geometry

The free surface of a fluid is defined as the interface where the pressure equals the ambient pressure, typically atmospheric. In a rotating fluid, the interplay between gravitational and centrifugal forces causes this free surface to deform. Integration of the pressure distribution reveals that the surface adopts the shape of a paraboloid of revolution. This geometric form reflects the fact that the centrifugal force increases with radial distance, thereby lifting the fluid more at the edges than at the center.

Vorticity in Fluid Dynamics

Vorticity is a measure of the local rotation of the fluid, defined as the curl of the velocity field. In the context of a rotating fluid, particularly in the case of rigid-body rotation, the vorticity is uniform throughout the flow. This uniform vorticity encapsulates the rotation imposed on the fluid and plays an important role in understanding the dynamics and stability of rotating flows. In such scenarios, the vorticity is directly related to the angular velocity of the fluid.

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