Newton's Laws for Fluids and the Pressure Gradient Force

Newton's second law also applies to fluids. When pressure differs across a fluid element, the unbalanced forces cause acceleration.

Setup

Consider a small fluid element with cross-sectional area $A$ and length $\Delta x$. The pressure on the left face is $P_1$, and on the right face is $P_2$.

Since pressure is force per unit area, the forces on each face are:

The net force in the positive $x$-direction is:

Applying Newton's Second Law

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