(normalized) $z$ with a period $1$.

The value of $\rho$ changes between $\rho=1$ (at the inner cylinder) and $\rho=1/\eta$ (at the outer cylinder). For $\om(\rho,z)$ we have: (,z) = - +1 + (z) (- 1 ). \[vpc2\] A perturbation consists of two parts. First, the inner cylinder is propagating upward (in the axial direction) with a constant (slow) speed $\dot{z} = \eps$. Second, the shape of the outer cylinder deviates slightly from being exactly circular. The full 3-D flow is given by &=& ( - 1 ) ,\
&=& ( 1+ / ), \[vpc3\]\
&=& ( ,z) - 1 ( 2- 1 ) . In (\[vpc3\]), $0<\eps\ \ll 1$ is a small parameter, while $\kp \sim 1$ defines a characteristic ratio of the two perturbations. By changing the value of $\kp$ we can vary the properties of the resonance phenomena (see Subsect. 5.3 below). One can see that the axial velocity, $\dot{z}$, equals $\eps$ at $\rho=1$ and vanishes at $\rho=1/\eta$. A limiting case $\kp = 0$ corresponds to the outer cylinder being circular and the axial symmetry of the flow. Note, that the reason for choosing both perturbations to be of the same order is that it is the interaction 