Rotating liquid in a cylindrical vessel – identify the flow type When a vertical-axis rotation depresses the free surface at the center and raises it near the wall, what is this flow called?

Introduction:
Rotating fluids in a cylindrical container develop a characteristic free-surface shape: depressed at the axis and elevated near the wall, forming a paraboloid. This question assesses whether you can identify the underlying flow category—namely, a vortex flow—and recognize the physics of forced vortices.

Given Data / Assumptions:

• Rigid-body rotation about the vertical axis (forced vortex).
• Incompressible fluid with a visible free surface.
• Steady rotation after spin-up transients have decayed.

Concept / Approach:
In a forced vortex, every fluid particle rotates with the same angular velocity. The radial pressure gradient balances the centrifugal effect: dp/dr = rho * r * omega^2. Integrating this hydrostatically in the vertical direction yields a paraboloid free surface z = (omega^2 * r^2) / (2g) + constant, depressed at r = 0 and raised at larger r. This is the textbook signature of vortex flow under forced rotation.

Step-by-Step Solution:

Verification / Alternative check:
Plotting z versus r^2 gives a straight line in a forced vortex, confirming the paraboloid form and the vortex nature of the motion.

Why Other Options Are Wrong:

• steady flow: can be steady or unsteady; does not name the rotational nature.
• turbulent flow: turbulence may or may not occur; this label misses the rotational structure.
• uniform flow: implies uniform velocity over a section, not valid here.
• compressible flow: not a defining feature of this phenomenon.

Common Pitfalls:
Confusing free vortex (omega varies with r) with forced vortex (rigid-body rotation). The described paraboloid under container rotation is the forced vortex case.

Final Answer:
vortex flow