Forced Vortex in a Rotating Cylindrical Vessel
Begin with the liquid at rest, then increase the rotational speed. The free surface becomes a paraboloid, reaches the rim, spills, and eventually develops a dry core when the theoretical vortex vertex falls below the tank bottom.
At rest · horizontal free surface
Rotation Controls
0.0
RPM
Use the increase button repeatedly. The simulation automatically removes spilled liquid once the free surface reaches the vessel rim.
Vessel and Liquid
Initial water depth is constrained below the tank height. Gravity is taken as 9.81 m/s².
Progression
1
At restω = 0. The free surface is horizontal at h₀.
2
Contained paraboloidThe center drops and the liquid rises at the wall without spilling.
3
Overflow beginsThe rim elevation is fixed at H; additional RPM spills more liquid.
4
Vortex below the vesselThe theoretical vertex is below the bottom and a dry central core forms.
Forced-Vortex Relations
z(r) = zc + ω²r²/(2g)
Δz = ω²R²/(2g)
ω = 2πN/60
Before overflow: average depth = h₀
After overflow: z(R) = H
Δz = ω²R²/(2g)
ω = 2πN/60
Before overflow: average depth = h₀
After overflow: z(R) = H
Angular speed, ω0 rad/s
Center / vertex elevation3.000 m
Wall elevation3.000 m
Water remaining100.0%
Spill begins at29.9 RPM
Vertex reaches bottom at42.3 RPM
Dry-core radius0.000 m
Water spilled0.000 m³
At rest
The vessel is stationary, so the liquid free surface remains horizontal at the initial depth.