The stability of viscous flow between rotating concentric cylinders with an axial flow

Abstract The stability of viscous flow between concentric cylinders with the inner cylinder rotating and with an axial flow due to an axial pressure gradient is considered. The analysis is motivated, in part, by recent papers by Hasoon & Martin (1977) and Chung & Astill (1977). Results are given for radius ratios ɳ = R1/R2 = 1 (the small-gap limit), 0.95, and 0.9, and for values of the axial Reynolds number R up to 100. Here R1 and R2 are the radii of the inner and outer cylinders, respectively. For the small-gap problem, the results are compared with those obtained by approximating the angular velocity and/or the axial velocity by average values. Two of the conclusions of the paper are that the small-gap problem is a valid approximation to the ɳ ≠ 1 problem for ɳ near 1, and that the approximation of the axial velocity by its average value leads to qualitatively and quantitatively incorrect results. In addition, we find that for axial Reynolds numbers of about ninety a second mode of instability arises and there is a discontinuity in the axial wavenumber of the critical disturbance as a function of the Reynolds number.