Is the bathtub whirlpool an instability ?

The spontaneous genesis of a free-surface whirlpool represents a classical fluid dynamic phenomenon, but the reason for the onset is not completely clarified yet. It has long been argued that the rotation is due to a gradual concentration of vorticity that comes from the far-field region and is convected along the axis of the rising whirlpool [Andrade, 1963]. A second hypothesis is that the vortex onset is caused by a flow instability. The latter possibility was investigated in the experimental work by Kawakubo et al.(1978) and in the numerical simulations by Sanmiguel-Royas & Fernandez-Feria (2006) and Sanmiguel-Royas & Fernandez-Feria (2002) where an axisymmetric flow was assumed. In the present study it is shown that, introducing non-uniform inlet boundary conditions, as the Reynolds number increases an instability appears and leads to a swirling motion. A linear stability analysis of the steady base flow is performed in order to calculate the least stable eigenvalue (and the related eigenfunction) of the linearized equations and to determine the critical Reynolds number Rec at which the instability first arises.