Fluids in Motion

Ideal Flow

Continuity, $A_1v_1 = A_2v_2$, makes a fluid speed up where a pipe narrows. Bernoulli's equation, $P + \tfrac{1}{2}\rho v^2 + \rho gh = \text{constant}$ is energy conservation for ideal flow: faster means lower pressure.

Real Flow and the Circulation

Viscous flow obeys Poiseuille's law: flow scales with $r^4$, so small changes in vessel radius control blood flow. Blood is slowest in the capillaries, where the total cross-sectional area is largest, and the biggest pressure drop is across the arterioles.

Key Takeaways

• Narrower means faster means lower pressure (ideal flow).
• Halving a vessel's radius cuts flow 16-fold (viscous flow).