The Levitating Water Drop · AI Generator #GEN-2CEBDE78
Physics · fluid-mechanics
Problem
A fascinating demonstration involves suspending a small water droplet in mid-air using a carefully controlled upward air flow. The setup consists of a vertical circular tube of radius R=2.0cm through which air flows upward at speed v0=8.0m/s. A spherical water droplet of radius r=1.5mm and density ρw=1000kg/m3 is introduced into the airstream.
The droplet experiences three forces: gravitational force downward, viscous drag from the air (given by Stokes' law Fd=6πηrvrel where η=1.8×10−5Pa⋅s is the dynamic viscosity of air and vrel is the relative speed between the droplet and air), and a buoyant force due to the air's reduced pressure above the droplet.
The key insight is that as the droplet moves through the flowing air, it creates a local disturbance in the airflow. Using Bernoulli's principle, the air moving around the droplet at speed v0 creates a pressure difference. The pressure below the droplet (where air is nearly stagnant) is approximately atmospheric pressure p0, while the pressure above the droplet (where air flows at speed v0) is reduced according to Bernoulli's equation.
Find the equilibrium height above the tube entrance where the droplet will hover motionless. At this height, the air speed has decreased to v0A0/A(h) due to the tube's expansion, where A0=πR2 is the tube's cross-sectional area and A(h)=π(R+0.1h)2 accounts for the tube's slight flaring with height h (in meters). Use ρair=1.2kg/m3 and g=9.8m/s2.