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hydrostatic pressure is the scalar and isotropic pressure exerted by a fluid at rest at a certain point inside the fluid itself on the neighboring fluid parcels or on on an elementary differential surface area dA. This pressure is also the thermodynamic pressure and linearly depends on the temperature and depth inside the fluid (for liquids).

The term "static" pressure refers to the pressure of a fluid that is moving at speed V and measured in a direction that is not along the fluid's direction of motion: if we measured the pressure along the direction of motion we would have to take the "dynamic pressure" contribution into account. For example, if the fluid was brought to rest (at a stagnation point), the total pressure would still be isotropic and increased by the dynamic pressure factor. So the term "static" does not really mean the pressure of a fluid that is not moving, correct?

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