Pressure-gradient force

Formalism

Consider a cubic parcel of fluid with a density ${\displaystyle \rho }$, a height ${\displaystyle dz}$, and a surface area ${\displaystyle dA}$. The mass of the parcel can be expressed as, ${\displaystyle m=\rho dAdz}$. Using Newton's second law, ${\displaystyle F=ma}$, we can then examine a pressure difference ${\displaystyle dP}$ (assumed to be only in the ${\displaystyle z}$-direction) to find the resulting force, ${\displaystyle F=-dPdA=\rho adAdz}$.

The acceleration resulting from the pressure gradient is then, $${\displaystyle a=-\frac{1}{\rho }\frac{dP}{dz}.}$$

The effects of the pressure gradient are usually expressed in this way, in terms of an acceleration, instead of in terms of a force. We can express the acceleration more precisely, for a general pressure ${\displaystyle P}$ as, $${\displaystyle \overrightarrow{a}=-\frac{1}{\rho }\overrightarrow{\mathrm{\nabla }}P.}$$

The direction of the resulting force (acceleration) is thus in the opposite direction of the most rapid increase of pressure.