Fluid Dynamics: Continuity and Bernoulli's Equation

Continuity and Ideal Fluids

Fluids obey a traffic law: what flows in must flow out. Squeeze the pipe and the fluid must speed up, and counterintuitively, its pressure drops where it moves fastest.

For an ideal fluid (incompressible, non-viscous, steady flow), the continuity equation enforces conservation of mass:

$$A_1 v_1 = A_2 v_2$$

Where pipe area $A$ shrinks, flow speed $v$ increases.

Energy conservation along a streamline is given by Bernoulli's equation:

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

Here $P$ is pressure, $\rho$ is density, $v$ is speed, $g$ is gravity, and $h$ is height. Each term is energy per unit volume.

Real Fluids and Viscosity

Real fluids introduce viscosity (friction). For laminar flow in a round pipe, Poiseuille's law governs volumetric flow rate $Q$:

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

Where $r$ is radius, $\Delta P$ is pressure drop, $\eta$ is dynamic viscosity, and $L$ is pipe length. Flow rate scales heavily with radius ($r^4$).

The Reynolds number ($\\text{Re}$) predicts whether flow remains smooth or breaks down:

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

Where $D$ is pipe diameter. Regimes divide as follows:

Common pitfall: Bernoulli's equation applies only along a single streamline of smooth, steady, frictionless flow. Applying it across turbulent or unconnected regions yields nonsense.