The Levitating Water Drop · AI Generator #GEN-7DD8FE52

Physics · fluid-mechanics

Problem

A small water droplet is held suspended in air by a vertical air jet from a nozzle below. The air jet has a circular cross-section with diameter d=4.0  mmd = 4.0\;\text{mm} at the nozzle exit, where the air speed is v0=25  m/sv_0 = 25\;\text{m/s}. The droplet hovers at a height h=15  cmh = 15\;\text{cm} above the nozzle, where it sits at the center of the now-expanded air jet. At the hovering position, the air jet has expanded to a diameter D=12  mmD = 12\;\text{mm}, and the dynamic pressure of the air creates an upward force that exactly balances the droplet's weight. The droplet can be modeled as a sphere that experiences a drag force Fd=12ρairv2ACdF_d = \frac{1}{2}\rho_{\text{air}} v^2 A C_d, where AA is the droplet's cross-sectional area, vv is the local air speed, and Cd=0.47C_d = 0.47 is the drag coefficient for a sphere. Assuming the air jet expands uniformly as it rises and that air density remains constant at ρair=1.2  kg/m3\rho_{\text{air}} = 1.2\;\text{kg/m}^3, find the mass of the water droplet. Take g=9.8  m/s2g = 9.8\;\text{m/s}^2.

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