The flow past a freely rotating sphere

We consider the flow past a sphere held at a fixed position in a uniform incoming flow but free to rotate around a transverse axis. A steady pitchfork bifurcation is reported to take place at a threshold (Formula presented.) leading to a state with zero torque but nonzero lift. Numerical simulations allow to characterize this state up to (Formula presented.) and confirm that it substantially differs from the steady-state solution which exists in the wake of a fixed, non-rotating sphere beyond the threshold (Formula presented.). A weakly nonlinear analysis is carried out and is shown to successfully reproduce the results and to give substantial improvement over a previous analysis (Fabre et al. in J Fluid Mech 707:24–36, 2012). The connection between the present problem and that of a sphere in free fall following an oblique, steady (OS) path is also discussed.