Taylor-Couette Flow

What is modelled

Viscous fluid fills the gap between two coaxial cylinders. For low speeds the flow is purely azimuthal (circular Couette flow). When the inner cylinder is fast enough that centrifugal force beats viscous damping, axisymmetric vortex cells (Taylor cells) develop, one pair per two gap widths of height.

Equations

• Exact base flow ω(r) = A + B / r², A = (ωo ro² - ωi ri²)/(ro² - ri²), B = (ωi - ωo) ri² ro²/(ro² - ri²).
• Narrow-gap Taylor number Ta = 4 (ωi - ωo)² ri d³ / ν², reference onset Ta ≈ 1708 (outer cylinder at rest).
• Wall torque τ = 4 π μ h B.
• Amplitude (Landau form, heuristic) dA/dt = g A - 0.92 A³ + forcing with g = 0.76 (Ta/1708 - 1).

Checks run in the page

• No-slip at both walls and the azimuthal momentum equation d/dr [r³ dω/dr] = 0.
• The model turns unstable at 1.02 and stable at 0.98 of the speed where Ta = 1708.
• Integrating the wall shear stress reproduces the closed-form torque.
• The amplitude equation settles at its fixed point.

Honest simplifications

The base flow is exact, but the vortex cells are a prescribed divergence-free pattern whose strength follows a reduced-order amplitude equation; this is not a Navier-Stokes solution and shows no wavy-vortex or turbulent regimes. The 1708 threshold is only exact for a narrow gap with the outer cylinder at rest.

• Choose Laminar, then Onset, then Vortices: watch the amplitude graph and the cells in the axial section appear.
• Try Counter-rotating and Co-rotating: Rayleigh's criterion on the second graph tells you when angular momentum decreases outward.
• Increase the viscosity until the white dot on the Taylor-number graph falls below 1708.