Taylor-Couette Flow

Fluid between rotating coaxial cylinders — laminar to vortex to turbulent transitions

Inner Ω₁: 1.0 (Ta=0)
Outer Ω₂: 0.0
Gap η=r₁/r₂: 0.50

Taylor-Couette flow transitions from laminar Couette flow (azimuthal velocity u_θ = Ar + B/r) to Taylor vortex flow at Ta_c≈1708 (Taylor, 1923). The Taylor number Ta = Ω₁²r₁(r₂-r₁)³/ν². Centrifugal instability (Rayleigh criterion) develops when ∂(ru_θ)²/∂r < 0. Further increasing rotation yields wavy vortices, modulated waves, and ultimately turbulence.