Bernoulli's Equation

Bernoulli's equation expresses conservation of energy per unit volume for an ideal fluid flowing along a streamline.

Consider a fluid element of volume $V$ and mass $m = \rho V$. This element has kinetic energy $\frac{1}{2}mv^2$ and gravitational potential energy $mgh$. Dividing by volume gives energy densities:

$$\text{Kinetic energy density} = \frac{1}{2}\rho v^2$$ $$\text{Gravitational potential energy density} = \rho g h$$

Pressure $P$ has units of $\text{Pa} = \text{N/m}^2 = \text{J/m}^3$, which is energy per unit volume. It represents work done by pressure forces per unit volume.

Bernoulli's equation states that the total energy density is constant along a streamline:

$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$

For two points along the same streamline:

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$

This unifies ideas you already know. The continuity equation relates speed to cross-sectional area. Hydrostatic pressure describes $\rho g h$ for fluids at rest. Bernoulli's equation extends these to moving fluids: when speed increases, pressure or height must decrease to keep the sum constant.