The rotating flows of von Ka´rma´n and Bo¨dewadt are re‐examined. Complete sets of boundary data are defined for both problems in accordance with practical flow models. First‐order approximations are defined as solutions of ordinary boundary‐value problems and equivalent Volterra integral equations. The new solutions depend on three‐parameter Reynolds numbers and eliminate discrepancies of the old solutions. Numerical results are displayed for various Reynolds numbers. The critical Reynolds number below which Bo¨dewadt flows are laminar, attached, and nonoscillating is computed.