Unstable viscous wall modes in rotating pipe flow

Linear stability of flow in a rotating pipe is studied. These flows depend on two parameters, which can be taken as the axial Reynolds number Re and the rotating rate, q. In the region of Re much greater than 1 and q = O(1), the most unstable modes are concentrated near the pipe wall. These wall modes are found to satisfy a simpler set of equations containing two parameters rather than four as in the full linear stability problem. The set of equations is solved numerically and asymptotically over a wide range of the parameters. In the limit of Re approaching infinity, the eigenvalue goes to the inviscid limit. The eigenfunction shows a two-layer structure. It reaches the inviscid limit over the main part of the domain, while near the wall of the pipe, the eigenfunction is represented by a viscous solution of the boundary layer type.