Scheichl, B., Klettner, C. A., & Smith, F. T. (2025, June 26). Stewartson’s collision problem on a sphere and on spheroids [Conference Presentation]. British Mathematical Colloquium and British Applied Mathematics Colloquium 2025 (BMC-BAMC 2025), Exeter, United Kingdom of Great Britain and Northern Ireland (the). https://doi.org/10.34726/9999
We consider the flow induced by a rigid sphere and, in an extension of our study, oblate/prolate spheroids spinning about their axes of rotational symmetry in an otherwise quiescent Newtonian fluid of uniform properties that fills an unbounded domain. Here we distinguish between the unsteady flow due to (impulsive) start-up from rest and the (asymptotically attained) steady one. The associated Reynolds number (Re), as the only parameter at play, shall take on arbitrarily large values. We tackle this classical problem by solving the full Navier-Stokes equations numerically using a finite-volume technique as well as by rigorous asymptotic analysis. The most intriguing and still controversially debated questions concern the existence of a steady state for all values of Re and here specifically of two symmetric toroidal vortices astride the equatorial plane, engendered by the colliding longitudinal wall jets that emanate from the poles. Stewartson (1958) proposed a structure of stationary collision, but its self-consistent completion faces severe challenges. In contrast to claims made very recently, we demonstrate theoretically why Smith & Duck’s (1977) conflicting alternative structure, resorting to free viscous-inviscid interaction, is to be favoured. However, we furthermore show the inexistence of a perfectly stationary flow. This is also substantiated by our detection of an upper bound of Re for stationarity, the equatorial finite-time break-up that terminates the solution of the wall layer problem and the first steps of its regularisation, revealing a permanently unsteady radially ejected jet.
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Forschungsschwerpunkte:
Computational Fluid Dynamics: 50% Modeling and Simulation: 50%
