  We study numerically the relativistic Bondi-Hoyle accretion of an ideal gas
onto a Kerr fixed background space-time on the equatorial plane with s-lab
symmetry. We use both Kerr-Schild (KS) and Boyer-Lindquist (BL) coordinates. We
particularly focus on the study of the flip-flop motion of the shock cone
formed when the gas is injected at supersonic speed. The development of the
flip-flop instability of the shock cone in the relativistic regime was reported
recently for the first time. We reproduce the flip-flop behaviour found in the
past when BL coordinates are used, and perform similar numerical experiments
using horizon penetrating KS coordinates. We find that when using KS
coordinates the shock cone oscillates, however such oscillations are not of the
flip-flop type and their amplitude decrease with resolution.
