The Levitating Oil Drop Chamber · AI Generator #GEN-0A0D128D

Physics · fluid-mechanics

Problem

A vertical cylindrical chamber of radius R=5.0  cmR = 5.0\;\text{cm} is filled with water and rotated about its axis at angular velocity ω\omega. Small oil drops of density ρoil=800  kg/m3\rho_{oil} = 800\;\text{kg/m}^3 and radius r=0.50  mmr = 0.50\;\text{mm} are injected at the bottom center of the chamber. Due to the centrifugal effect, the water surface forms a paraboloid, and the oil drops can become suspended at a fixed height above the bottom. When the chamber rotates at ω=12  rad/s\omega = 12\;\text{rad/s}, an oil drop injected at the center reaches a stable equilibrium position at height h=8.0  cmh = 8.0\;\text{cm} above the bottom, where it neither rises nor sinks. At this equilibrium position, determine the local slope angle θ\theta of the water surface (measured from horizontal) where the oil drop is located. Take ρwater=1000  kg/m3\rho_{water} = 1000\;\text{kg/m}^3, g=9.8  m/s2g = 9.8\;\text{m/s}^2, and assume the oil drop reaches its terminal velocity instantly in the rotating reference frame.

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