AZIMUTHAL INSTABILITY OF DIVERGENT FLOWS

We investigate a new mechanism for instability (named divergent instab ility), characterized by the formation of azimuthal cells, and find it to be a generic feature of three-dimensional steady axisymmetric flow s of viscous incompressible fluid with radially diverging streamlines near a planar or conical surface. Four such flows are considered here: (i) Squire-Wang flow in a half-space driven by surface stresses; (ii) recirculation of fluid inside a conical meniscus; (iii) two-cell regi me of free convection above a rigid cone; and (iv) Marangoni convectio n in a half-space induced by a point source of heat (or surfactant) pl aced at the liquid surface. For all these cases, bifurcation of the se condary steady solutions occurs: for each azimuthal wavenumber m = 2,3 ,..., a critical Reynolds number (Re) exists. The intent to compare w ith experiments led us to investigate case (iv) in more detail. The re sults show a non-trivial influence of the Prandtl number (Pr): instabi lity does not occur in the range 0.05 < Pr < 1; however, outside this range, Re(m) exists and has bounded limits as Pr tends to either zero or infinity. A nonlinear analysis shows that the primary bifurcations are supercritical and produce new stable regimes. We find that the ne utral curves intersect and subcritical secondary bifurcation takes pla ce; these suggest the presence of complex unsteady dynamics in some ra nges of Re and Pr. These features agree with the experimental data of Pshenichnikov & Yatsenko (Pr = 10(3)).