Confusion about relativistic fluids

According to Schutz book, in special relativity general fluids in the MCRF (Momentarily Commoving Reference Frame) are characterized by no bulk flow of the fluid element and no spatial momentum in the particles. The energy-momentum tensor elements T0i (flux of energy across i-surface) are not zero because, although there is no spatial momentum in the particles (no motion since we are in the MCRF), there is heat flow. But how can the other elements Ti0 (momentum density) be not null if the there isn’t particle motion inside the fluid element? Also, T_ij is not null either, so there is momentum flux across the boundaries even with no particle motion?
In the case of perfect fluids in the MCRF there isn’t heat flow and also there isn’t particle motion, so T0i is now zero. However the pressure, due to the random motion of particles is not zero.
Is this random motion not taken into account when it’s said that there isn’t particle motion in the MCRF? In order words, when it is said that motion doesn’t exist in the MCRF do they mean that CM motion or bulk motion only do not exist? So that the energy transfer (for non-perfect fluids) and pressure (for all fluids) are not due to particles transfer (since we are in the MCRF) but only collisions between particles in the fluid element with particles in its neighboring elements (in the case of non-perfect fluids)?
 

atyy

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Also, T_ij is not null either, so there is momentum flux across the boundaries even with no particle motion?
Maybe they balance each other out, something like the "vector sum" being zero?
 

atyy

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Maybe they balance each other out, something like the "vector sum" being zero?
I think that's wrong. It seems more like the fluid being at momentarily rest, but it could have an acceleration.
 

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