Fluid Dynamics: Continuity and Bernoulli's Equation

Fluids obey a traffic law: what flows in must flow out. Squeeze the pipe and the fluid must speed up ($A_1v_1 = A_2v_2$) — and, counterintuitively, its pressure drops where it moves fastest. That inverse pairing lifts airplane wings, curves soccer balls, and powers perfume atomizers.

For an ideal fluid (incompressible, non-viscous, steady flow):

Continuity equation (conservation of mass):

$$A_1 v_1 = A_2 v_2$$

Where the pipe narrows, the fluid speeds up.

Bernoulli's equation (conservation of energy along a streamline):

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

Applications

Real fluids — Viscosity introduces friction. For laminar flow in a pipe:

Poiseuille's law:

$$Q = \frac{\pi r^4 \Delta P}{8\eta L}$$

Reynolds number predicts flow regime:

$$\text{Re} = \frac{\rho v D}{\eta}$$

• $\text{Re} < 2000$: laminar flow.
• $\text{Re} > 4000$: turbulent flow.

Physics link: Bernoulli's equation is just the work-energy theorem applied to a fluid element. Each term represents a form of energy per unit volume.

Common pitfall: Bernoulli’s equation holds along a streamline of smooth, steady flow with negligible friction. Applying it across turbulent, viscous or unconnected regions — like directly comparing two different pipes — produces confident nonsense.