Bernoulli's Principle & Fluid Flow

Bernoulli's principle describes the trade-off between pressure, kinetic energy, and gravitational potential energy in a steady, incompressible, non-viscous fluid flow along a streamline. When a fluid speeds up (e.g., through a constriction), its pressure must drop to conserve energy.

Two companion equations work together:

• Continuity equation (mass conservation): flow rate is constant, so a narrower pipe means faster flow.
• Bernoulli's equation (energy conservation): the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant.

Applications include the Venturi meter (measuring flow speed from pressure drop), the Pitot tube (aircraft airspeed), carburetors, aspirators, and aerofoil lift.

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

Continuity equation:

$A_1 v_1 = A_2 v_2$

Venturi meter — for a horizontal pipe ($h =$ const):

$v_1 = A_2\sqrt{\frac{2(P_1 - P_2)}{\rho(A_1^2 - A_2^2)}}$

Water flows through a pipe that narrows from cross-section $A_1 = 8\,\text{cm}^2$ to $A_2 = 2\,\text{cm}^2$. Inlet velocity $v_1 = 1\,\text{m/s}$, inlet pressure $P_1 = 1.5 \times 10^5\,\text{Pa}$, $\rho = 1000\,\text{kg/m}^3$, horizontal pipe.

Continuity: $v_2 = \frac{A_1}{A_2}v_1 = \frac{8}{2} \times 1 = 4\,\text{m/s}$

Bernoulli:

$P_2 = P_1 + \frac{1}{2}\rho(v_1^2 - v_2^2) = 1.5\times10^5 + \frac{1}{2}(1000)(1 - 16)$

$P_2 = 1.5\times10^5 - 7500 = 1.425\times10^5\,\text{Pa}$

The pressure drops by 7,500 Pa as the flow accelerates.

Aircraft wings are shaped so air flows faster over the top surface, creating lower pressure there and net upward lift. Carburetors use the Venturi effect to draw fuel into the airstream. The human cardiovascular system relies on Bernoulli effects: a partially blocked artery speeds up blood flow at the constriction, dropping pressure and potentially causing the artery walls to collapse further — a dangerous positive feedback called the Venturi effect in stenosis.

1. A pipe of cross-section $20\,\text{cm}^2$ narrows to $5\,\text{cm}^2$. If the inlet velocity is $2\,\text{m/s}$, what is the outlet velocity?

2. Using Bernoulli's equation, explain why a fast-moving river can erode its banks even without the water physically hitting them.