Streamline tori in a precessing sphere at small Reynolds numbers

Abstract

The flow structure in a precessing sphere is investigated analytically and numerically at small Reynolds numbers. The spin angular velocity Ωs and the precession angular velocity Ωp are constant in magnitude and perpendicular to each other. The Navier–Stokes equation for an incompressible viscous fluid written in the precession frame is solved analytically in a power series of the Reynolds number Re(=a2Ωs/ν), where a is the sphere radius and ν is the kinematic viscosity of fluid. Steady solutions are obtained up to O(Re3) for arbitrary fixed values of the Poincaré number Γ(=Ωp/Ωs). By calculating the orbit of passive particles advected in these steady flows, we find that the individual particles move along circles centred on and perpendicular to the spin axis with small deviation of O(Re3Γ2) over one revolution and that in a long time span of O(1/(ΩsRe3Γ2)) (or O(a6Ωs2Ωp2/ν3) all the particles densely cover the whole tori, of thickness of O(Re), expressed as (x(y2+z2) in the Cartesian coordinate (x, y, z) system with x for the spin axis and z for the precession axis. There is a separatrix sphere of radius and of thickness O(Re), the fluid particles just above (or below) which can come close to the surface (or the centre) of the sphere.)