On the structure of steady axisymmetric Navier-Stokes flows with a stream function having multiple local extrema in its definite-sign domains

B. V. Pal'tsev ; M. B. Solov'ev ; I. I. Chechel'

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 11, pp. 1869-1893

Résumé

A new high-order accurate method and a corresponding computer program developed previously by the first and third authors for the numerical solution of the axisymmetric stationary Dirichlet boundary value problem for the Navier–Stokes equations in spherical layers at low Reynolds numbers were used to reliably study the structure of certain flows with a stream function in a meridional plane having multiple local extrema in its positive-sign domains. Regimes of rotation of the boundary spheres were detected that ensure this flow pattern: the inner sphere rotates at a constant angular velocity, while the outer sphere rotates at zenith-angle-dependent angular velocities. To describe the structure of these flows, the domain where the stream function is positive was partitioned into subdomains (circulation zones) by the separatrices of the saddle points of the stream function, which generate manifolds of unstable initial points of trajectories. Unexpected phenomena in the circulation of such flows were discovered. Examples were presented that illustrate the behavior of fluid particle trajectories. The computed trajectories were shown to be of high accuracy even on long time intervals.