Assumption of local thermodynamic equilibrium in fluid dynamics 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!
 A: The local stress tensor for a Newtonian fluid is usually represented as the isotropic "thermodynamic pressure" tensor, plus the viscous stress tensor.  The viscous stress tensor is not necessarily isotropic, and its components are proportional to the Newtonian viscosity, with an isotropic bulk viscosity stress tensor contribution.  The local "thermodynamic pressure" is determined by the equation of state, applied locally in terms of the local specific volume and local temperature.  For an incompressible fluid, the local "pressure" p at all locations is determined up to an arbitrary constant value that is obtained by applying the boundary conditions on the flow.  The local thermodynamic equilibrium assumption is an approximation, but one that has been found to be exceedingly accurate, based on centuries of experimental and practical evidence.
A: Your question seems to be: normal stress inside a fluid in motion is not an isotropic quantity, but pressure as it is used in thermodynamic equations is an isotropic quantity, so how do we define pressure at a point inside a fluid in motion? In a flow, pressure at a point is defined as the average of the normal stresses (with an additional minus sign). In fact we just take the isotropic part of the stress tensor and define it to be pressure (see Fluid Mechanics by Kundu & Cohen). We use this pressure so defined in the thermodynamic equations. The justification for this definition, as far as I know, comes from its success in practical applications, but I am not sure if there is theoretical justification for it.
