We know from the molecular theory of the ideal gas that in an enclosed container pressure is interpreted as the momentum transfer from the gas molecules in due of its collision with the walls. The molecules will move faster if the kinetic energy of those is increased by increasing temperature. In a way pressure is also an energy density that is the average kinetic energy contained of the gas molecules divided by the total volume enclosed in the container. This is just another way to state the equation of state for an ideal gas.
However when we study the Navier Stokes Equations in the fluid mechanics and the incompressibility assumption is made, the pressure P there is defined as the static pressure. We associate the static pressure at a point in the fluid flow field as the mass of the fluid lying above that (commonly known as hydrostatic approximation).
Now I think we run into two different definitions of the pressure one from molecular gas theory and other from continuum theory of the fluid flow. If the fluid is taken as the continuum then ofcourse pressure is not simply the momentum transfer due to the molecular collisions with the wall as in the enclosure example. But again it is difficult to imagine, a variable can have two different definitions at the same time.
My question is "How exactly do we reconcile these two views of the pressure, one from molecular gas theory and another from continuum fluid mechanics?"