Moving fluids are generally in a state of non-equilibrium. However, in fluid dynamics, people generally assume a state of local thermodynamic equilibrium and argue that in such a condition, equilibrium thermodynamic concepts such as pressure, temperature, entropy, internal energy etc. can be defined for a local fluid element. As such, the moving fluid can be considered as a continuum of local thermodynamic states and relations from equilibrium thermodynamics can be applied locally.
I am having trouble understanding this concept. For example, for a system in global equilibrium, pressure is defined as normal stress acting at a point, and it is equal in all directions.
Now consider a moving fluid. If we assume local equilibrium for this fluid, we can define a local property called pressure in the same way it was defined for a system in global equilibrium. This means that in a moving fluid, pressure at a point should be the normal stress at that point, and it should be equal in all directions (which is clearly not the case).
Perhaps my idea of local equilibrium or my conception of pressure is flawed. Can someone clarify my doubts? Thanks!