In fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal. This basic state is known as circular Couette flow, after Maurice Marie Alfred Couette, who used this experimental device as a means to measure viscosity. Sir Geoffrey Ingram Taylor investigated the stability of Couette flow in a ground-breaking paper. Taylor's paper became a cornerstone in the development of hydrodynamic stability theory and demonstrated that the no-slip condition, which was in dispute by the scientific community at the time, was the correct boundary condition for viscous flows at a solid boundary. Taylor showed that when the angular velocity of the inner cylinder is increased above a certain threshold, Couette flow becomes unstable and a secondary steady state characterized by axisymmetric toroidal vortices, known as Taylor vortex flow, emerges. Subsequently, upon increasing the angular speed of the cylinder the system undergoes a progression of instabilities which lead to states with greater spatio-temporal complexity, with the next state being called wavy vortex flow. If the two cylinders rotate in opposite sense then spiral vortex flow arises. Beyond a certain Reynolds number there is the onset of turbulence. Circular Couette flow has wide applications ranging from desalination to magnetohydrodynamics and also in viscosimetric analysis. Different flow regimes have been categorized over the years including twisted Taylor vortices and wavy outflow boundaries. It has been a well researched and documented flow in fluid dynamics.
Flow description A simple Taylor–Couette flow is a steady flow created between two rotating infinitely long coaxial cylinders. Since the cylinder lengths are infinitely long, the flow is essentially unidirectional in steady state. If the inner cylinder with radius R 1 {\displaystyle R_{1}} is rotating at constant angular velocity Ω 1 {\displaystyle \Omega _{1}} and the outer cylinder with radius R 2 {\displaystyle R_{2}} is rotating at constant angular velocity Ω 2 {\displaystyle \Omega _{2}} as shown in figure, then the azimuthal velocity component is given by
v θ = A r + B r , A = Ω 1 μ − η 2 1 − η 2 , B = Ω 1 R 1 2 1 − μ 1 − η 2 {\displaystyle v_{\theta }=Ar+{\frac {B}{r}},\quad A=\Omega _{1}{\frac {\mu -\eta ^{2}}{1-\eta ^{2}}},\quad B=\Omega _{1}R_{1}^{2}{\frac {1-\mu }{1-\eta ^{2}}}}
where
μ = Ω 2 Ω 1 , η = R 1 R 2 . {\displaystyle \mu ={\frac {\Omega _{2}}{\Omega _{1}}},\quad \eta ={\frac {R_{1}}{R_{2}}}.}
Rayleigh's criterion Lord Rayleigh studied the stability of the problem with inviscid assumption i.e., perturbing Euler equations. The criterion states that in the absence of viscosity the necessary and sufficient condition for distribution of azimuthal velocity v θ ( r ) {\displaystyle v_{\theta }(r)} to be stable is
Φ ≡ 1 r 3 d d r ( r v θ ) 2 ≥ 0 {\displaystyle \Phi \equiv {\frac {1}{r^{3}}}{\frac {d}{dr}}(rv_{\theta })^{2}\geq 0}
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