On the instability of viscous flow in a rapidly rotating pipe

Abstract

The stability of almost fully developed viscous flow in a rotating pipe is considered. In cylindrical polar co-ordinates (r, ø, z) this flow has the velocity components\[ \{W_0o(1),\quad\Omega r[1+o(\epsilon)],\quad W_0[1-r^2/r^2_0+o(1)]\},_{+}^{+} \]where ε =W_o/2Ωr₀and is bounded externally by the rigid cylinderr = r₀, which rotates about its axis with angular velocity Ω. In the limit of small ε, the disturbance equations can be solved in terms of Bessel functions and it is shown that, in that limit, the flow is unstable for Reynolds numbersR=W_or₀/vgreater thanR_c[asymp ] 82[sdot ]9. The unstable disturbances take the form of growing spiral waves, which are stationary relative to the rotating cylinder and the critical disturbance atR = R_chas azimuthal wave-number 1 and axial wavelength 2πr₀/ε. Furthermore, it is shown that the most rapidly growing disturbance forR > R_chas an azimuthal wave-number which increases withR. Some of the problems involved in testing the results by experiment are discussed and a possible application to the theory of vortex breakdown is mentioned. In an appendix this instability is shown to be an example of inertial instability.