Investigating the instability of the torus around the blackhole in the presence of the radial and angular velocity perturbations

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We have performed the numerical simulation of the perturbed relativistic torus around the black hole by solving the general relativistic hydrodynamics equations on the equatorial plane. We have monitored the different simple models by applying the radial and angular velocity perturbations to the steady state torus around the black hole. The instabilities are triggered by adding the subsonic velocity fields over the radial and angular velocities of the stable torus. We have found the disk instability so-called the Papaloizou-Pringle instability and excited modes of the Quasi-Periodic Oscillations (QPOs) in the thin disk approximation case. The significant oscillations and oscillation modes in the mass accretion rates are observed around the rotating black hole with a radial velocity perturbation. The angular velocity perturbation can not create the significant variations and QPOs in the nonselfgravitating relativistic torus around the nonrotating and rotating black holes. The results observed from the radial velocity perturbation might be used to explain the Papaloizou-Pringle instability observed in $NGC$ $1068$