Water ( , ) fills the narrow gap between two vertical cylinders of inner radius and gap

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Instability, transition and turbulence · Instability, transition and turbulence

Fluid Dynamics, Ch Hydrodynamic instability: the normal-mode method

Problem

Water (μ=1.00×10−3 Pa s\mu = 1.00\times10^{-3}\,\mathrm{Pa\,s}, ρ=998 kg m−3\rho = 998\,\mathrm{kg\,m^{-3}}) fills the narrow gap between two vertical cylinders of inner radius R1=3.0 cmR_1 = 3.0\,\mathrm{cm} and gap d=3.0 mmd = 3.0\,\mathrm{mm}, with the outer cylinder at rest. Taylor vortices appear once the Taylor number Ta=Ω2R1d3/ν2Ta = \Omega^2 R_1 d^3/\nu^2 reaches the critical value Tac≈1.7×103Ta_c \approx 1.7\times10^{3}. ((a)) Find the angular velocity Ωc\Omega_c of the inner cylinder at onset. ((b)) State the corresponding rotation period.

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