  The accretion of matter onto celestial bodies like black holes and neutron
stars is a natural phenomenon that releases up to $40\%$ of the matter's
rest-mass energy, which is considered a source of radiation. In active galactic
nuclei and X-ray binaries, huge luminosities are observed as a result of
accretion. Using isothermal fluid, we examine the accretion and geodesic motion
of particles in the vicinity of a spherically symmetric black hole spacetime in
the Einstein-$SU(N)$ non-linear sigma model. In the accretion process, the
disk-like structure is produced by the geodesic motion of particles near the
black hole. We determine the innermost stable circular orbit, energy flux,
radiation temperature, and radioactive efficiency numerically. In the
equatorial plane, we investigate the mobility of particles with stabilities
that form circular orbits. We examine perturbations of a test particle by using
restoring forces and particle oscillations in the vicinity of the black hole.
We analyze the maximum accretion rate and critical flow of the fluid. Our
findings demonstrate how parameter $N$ influences the circular motion of a test
particle as well as the maximum accretion rate of the black hole in the
Einstein-$SU(N)$ non-linear sigma model.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology