The Meissner effect states that when we cool the superconductor below the critical temperature and we apply a magnetic field at its direction, then there is field expulsion inside the superconductor. For example, this happens when we have a superconducting sphere. I have been reading about flux quantization and there are two examples, one with a superconducting cylinder and one with a superconducting ring. But when we apply a magnetic field, it is not expelled but rather there is magnetic flux inside the cylinder and the ring. Does the Meissner effect not have effect when we are dealing with superconducting materials that have holes?
Let's take a superconducting (type-I) hollow cylinder below its transition temperature. Now if you apply a magnetic field there will be a persistent current on the surface of the superconductor (SC) to expel the magnetic field from the interior of the body of the SC. This same surface current produces flux that cancels the magnetic field inside of the hole. This the Meissner-ochsenfeld effect.
However, if you apply the magnetic field before you cool down the sC below its critical temperature the body of the SC becomes perfectly diamagnetic (Meissner state) but there will be a flux through the hole and this flux is maintained in the hole by circulating supercurrents on the interior walls moving in the opposite direction. This was all for a type-I superconductor.
For a type-II SC, there are there different states of the matter when it comes to the application of the magnetic field to a SC. As shown below there is a mixed state where you have normal regions in the SC where magnetic fields can penetrate and the amount of flux in these regions is quantized.
The Meissner effect does play a role in a ring (or cylinder) geometry. Imagine a solenoid that passes through the ring. For some finite (but small) magnetic field, the superconductor will screen out the magnetic field in the solenoid, so that the flux through the center remains zero. This happens because there are surface currents on the ring, but the magnetic field deep inside the superconductor will remain zero. This is the Meissner effect in the ring geometry.
You may also find this link helpful: https://thiscondensedlife.wordpress.com/2020/06/04/meissner-effect-as-amplified-atomic-diamagnetism/