The onset of hydrodynamical instability induced by impulsive spin-down to rest in a cylinder containing a Newtonian fluid is analyzed by using the propagation theory. The primary transient swirl flow is laminar but for an initially high rotating speed secondary motion sets in at a certain time. It is found here that the critical Reynolds number Re-C = 320, below which the flow is unconditionally stable. For Re > Re-C the dimensionless critical time T, to mark the onset of a fastest growing instability is presented as a function of the Reynolds number Re. Available experimental data and also predictions show that deviation of the velocity profiles from their primary ones occurs starting from a certain time tau approximate to 4 tau(C). This means that secondary motion is detected experimentally at this characteristic time. It seems that during tau(C) <= tau <= 4 tau(C) secondary motion is relatively very weak. (c) 2006 Elsevier Ltd. All rights reserved.