Math 2.0

Fluid mechanics Definition

Taylor–Couette instability

Builds on Navier–Stokes equationsHydrodynamic stabilityReynolds number

Definition

Consider a viscous incompressible fluid with kinematic viscosity ν>0\nu > 0 in the gap R1<r<R2R_1 < r < R_2 between two coaxial cylinders, in cylindrical coordinates (r,θ,z)(r, \theta, z). The inner cylinder rotates at angular velocity Ω1\Omega_1 and the outer one at Ω2\Omega_2. The Navier–Stokes equations with no-slip conditions have the steady solution called circular Couette flow, with azimuthal velocity v(r)=Ar+Br,A=Ω2R22−Ω1R12R22−R12,B=(Ω1−Ω2)R12R22R22−R12.v(r) = A r + \frac{B}{r}, \qquad A = \frac{\Omega_2 R_2^2 - \Omega_1 R_1^2}{R_2^2 - R_1^2}, \qquad B = \frac{(\Omega_1 - \Omega_2) R_1^2 R_2^2}{R_2^2 - R_1^2}. The Taylor–Couette instability is the loss of stability of circular Couette flow as Ω1\Omega_1 increases past a critical value. This value depends on ν\nu, Ω2\Omega_2, R1R_1 and R2R_2. Beyond this value the flow is unstable to axisymmetric disturbances. A new steady flow appears, the Taylor vortex flow, made of toroidal vortices periodic in zz.

Remarks

For an inviscid fluid, Rayleigh's circulation criterion says that a rotating flow v(r)v(r) is stable to axisymmetric disturbances exactly when (rv)2(r v)^2 does not decrease with rr. For circular Couette flow with Ω2=0\Omega_2 = 0 and Ω1>0\Omega_1 > 0, the quantity rvr v decreases from R12Ω1R_1^2 \Omega_1 to 00 across the gap. So the inviscid flow is unstable, and viscosity delays the instability to a finite critical rotation rate. The threshold is usually expressed by a Taylor number or a Reynolds number Ω1R1(R2−R1)/ν\Omega_1 R_1 (R_2 - R_1) / \nu based on the gap width.

Example: with the outer cylinder at rest, there is a critical inner rotation rate beyond which Taylor vortices appear, and Taylor's linear theory predicted it in agreement with his experiments. Non-example: with the inner cylinder at rest and the outer cylinder rotating, Ω1=0<Ω2\Omega_1 = 0 < \Omega_2, the quantity (rv)2(r v)^2 increases outward. So the flow satisfies Rayleigh's criterion, and there is no centrifugal instability of this kind.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Hydrodynamic stability, concept, no evidence levelHydrodynamic stabilityconceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptReynolds number, concept, no evidence levelReynolds numberconceptTaylor–Couette instability, definition, no evidence levelTaylor–Couette instab…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hydrodynamic stability concept
  2. Navier–Stokes equations concept
  3. Reynolds number concept
  4. Taylor–Couette instability definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor–Couette_flow
About this page
TagHFHE56NYH1
URIm2://HFHE56NYH1
Kinddefinition
WritingW0 stub
MSC76E07
DefinesTaylor–Couette instability
UsesNavier–Stokes equations, Hydrodynamic stability, Reynolds number
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.