The flow past a sphere is of great interest in many engineering applications, from sedimentation processes and particle transport to biological systems . In particular, we focus our attention on the dynamics of freely falling (rising) bodies under the effect of the gravitational field . D. Fabre et al.  investigated the occurrence of steady oblique trajectory characterized by a weak deviation from the vertical and by a rotation rate. We further investigate this problem by considering the flow past a transversely rotating sphere embedded in a uniform flow using DNS, global linear stability and weakly nonlinear analysis. As the Reynolds number Re and the rotation rate are increased, the flow experiences different Hopf bifurcations. We investigate in detail the range Re ~ [150; 300] and Omega ~ [0:0; 1:2]. Two neutral stability curves are obtained and the resulting destabilizing global eigenmodes are discussed. Direct, adjoint and sensitivity problems are solved for both branches. Furthermore, we also perform a weakly nonlinear analysis and show that for small rotation rates, the problem can be modeled with an amplitude equation describing an imperfect pitchfork bifurcation.
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