INSTABILITIES OF CONICAL FLOWS CAUSING STEADY BIFURCATIONS

A new stability approach is developed for a wide class of strongly non -parallel axisymmetric flows of a viscous incompressible fluid. This a pproach encompasses all conical flows, and all steady and weakly unste ady disturbances, while prior studies were limited to specific flows a nd particular disturbances. A specially derived form of the Navier-Sto kes equations allows the exact reduction of the linear stability probl em to a system of ordinary differential equations. We found that distu rbances originating at the boundaries of a similarity region cause a v ariety of steady bifurcations. Consideration of the still fluid allows disturbances to be classified into inner, outer and global modes, dep ending on the boundary conditions perturbed. Then we identify and stud y modes which cause bifurcation as the Reynolds number increases. The study provides improved understanding of (a) azimuthal symmetry breaki ng, (b) genesis of swirl, (c) onset of heat convection, (d) hydromagne tic dynamo, (e) hysteretic transitions, and (f) jump flow separation. We also discover and analyse two new bifurcations: (g) fold catastroph es and (h) appearance of radial oscillations in swirl-free jets. The s tability analysis reveals that bifurcations (a), ((I) and df) are caus ed by inner perturbations, bifurcations (b), (d), (e) and (g) by outer perturbations, and bifurcation (h) by global perturbations. We deduce amplitude equations to describe the nonlinear spatiotemporal developm ent of disturbances near the critical Reynolds numbers for (b) and (g) . Disturbances switching between the basic and secondary steady states are found to grow monotonically with time.