A cylindrical laboratory tank of water

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Convection, rotation and geophysical flows · Convection, rotation and geophysical flows

Fluid Dynamics, Ch Rotating fluids: geostrophy, Taylor columns and Ekman layers

Problem

A cylindrical laboratory tank of water, depth H=15 cmH = 15\,\mathrm{cm}, spins steadily at Ω=2.0 rad s−1\Omega = 2.0\,\mathrm{rad\,s^{-1}}. Its rotation rate is then reduced by a small increment ΔΩ≪Ω\Delta\Omega \ll \Omega, so that the fluid is left rotating slightly faster than the container. Take the kinematic viscosity of water as ν=1.0×10−6 m2 s−1\nu = 1.0\times10^{-6}\,\mathrm{m^2\,s^{-1}}. ((a)) Predict the pattern of secondary flow this sets up inside the tank, and say why it appears near the floor rather than in the interior. ((b)) Compute the spin-down time τE∼H/νΩ\tau_E \sim H/\sqrt{\nu\Omega} and the purely diffusive time H2/νH^2/\nu, and say by what factor the tank equilibrates faster than diffusion alone would allow.

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