Definition
Pressure in incompressible flow
The scalar field whose negative spatial gradient supplies the pressure force per unit mass.
In the constant-density incompressible equations, the normalized pressure is the physical pressure divided by the constant density. Its contribution to acceleration is
Pressure is a scalar field; its spatial gradient is a vector field. In the evolution problem, pressure is an unknown enforcing compatibility with the incompressibility constraint.
Additive freedom
Replacing by leaves its spatial gradient unchanged. A normalization, such as zero spatial mean on a periodic domain, fixes this freedom. On a disconnected domain there can be one additive function of time on each component.
Regularity and reconstruction
To speak of a classical pressure gradient, the necessary spatial derivatives must exist. Pressure can instead be interpreted distributionally in weaker formulations. Taking the divergence of the momentum equation leads to a pressure Poisson equation, developed with the harmonic-analysis prerequisites.