The Levitating Droplet · AI Generator #GEN-FA16471D

Physics · fluid-mechanics

Problem

A small spherical water droplet of radius r=0.50  mmr = 0.50\;\text{mm} is suspended motionless in air by a vertical upward air stream. The air stream has a parabolic velocity profile: v(x)=v0(1−x2/R2)v(x) = v_0(1 - x^2/R^2), where xx is the horizontal distance from the stream's centerline, R=2.0  mmR = 2.0\;\text{mm} is the stream radius, and v0=8.0  m/sv_0 = 8.0\;\text{m/s} is the maximum velocity at the center. The droplet sits with its center on the stream's centerline. Due to the velocity gradient in the air stream, the droplet experiences a net upward force that balances gravity. This occurs because the air velocity is higher at the top of the droplet than at the bottom, creating a pressure difference (Bernoulli effect). (a) Calculate the density of air required for the droplet to remain in equilibrium. (b) If the droplet is displaced horizontally by a small distance δ\delta from the centerline, find the horizontal restoring force acting on it. Use: water density ρw=1000  kg/m3\rho_w = 1000\;\text{kg/m}^3, gravitational acceleration g=9.8  m/s2g = 9.8\;\text{m/s}^2. Assume the air flow is inviscid and the droplet is small enough that the velocity profile is approximately linear across its diameter.

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