This paper considers the flow of an incompressible, viscous fluid forced by the independent rotation of two (bounding) infinite, parallel planes. The flow field is assumed to have a radial self-similarity of Von Kármán form and the relevant governing equations are derived with no assumptions of rotational symmetry. An exact class of solutions to the Navier–Stokes equations is shown to exist, corresponding to nonlinear, non-axisymmetric states. These steady, non-axisymmetric solutions appear through symmetry breaking of the classical axisymmetric steady states. The locus of bifurcation points is determined numerically and a number of limiting cases are described asymptotically. The initial-value problem is considered in the context of the self-similar equations. It is shown that unsteady calculations can break down at a finite time with the development of a singularity in the (exact) system of equations. An asymptotic description is given in the neighbourhood of the breakdown event. The structure of the singularity consists of an inviscid core flow to which an infinity of solutions are possible within the framework of the same asymptotic description. Whether a singularity is approached, or a steady/periodic axisymmetric state is achieved (and even the qualitative details of the singularity) is dependent on the initial conditions for some parameter regimes.
Finite-time singularity Rotating disk Symmetry breaking
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