Pressure gradient ($\nabla p$)

The pressure gradient is the vector $\nabla p$, pointing in the direction in which pressure increases fastest. Its magnitude is pressure change per unit distance, with units $\mathrm{Pa}{\mathrm{m}}^{-1}$. Across a small fluid element, larger pressure on one face than its opposite produces a net pressure gradient force per unit volume $-\nabla p$.

For $p(x)=p_0-Gx$ with $G>0$, the pressure force points along positive $x$. In a long pipe, it can balance viscous resistance and sustain steady flow. In a stationary vertical column, it instead balances gravity, so a nonzero pressure gradient does not by itself prove that fluid must move. The Cauchy momentum equation determines acceleration from all the forces together. Comparing pressure measurements at separated positions estimates the mean gradient over their spacing.

Along a boundary-layer streamline directed by increasing $s$, an adverse pressure gradient is a rise in pressure in the direction of flow. The associated pressure gradient force opposes the velocity. In a steady inviscid outer stream the Euler equation gives $UdU/ds=-{\rho }^{-1}dp/ds$, so a positive gradient decelerates a positive outer velocity. Within a viscous boundary layer the slower fluid near the wall has less streamwise momentum and can reverse while the outer stream still moves forward. The onset of flow separation is therefore associated with vanishing and then reversed wall shear stress. A pressure rise does not by itself locate separation: the full boundary-layer solution and upstream history also matter.

A spatially varying pressure exerts net force per unit volume $-\nabla p$ on a small fluid element. Its contribution to acceleration is $-\nabla p/\rho$. Pressure increases in the direction opposite to this force.