Fluid Dynamics and Bernoulli Applications

Driving question

How do area, speed, pressure, and height trade off as fluid moves?

Before: prediction

Water flows steadily from a wide pipe into a section with half the cross-sectional area. Predict the new speed and whether continuity alone determines the pressure change.

Hook: squeeze the path, speed up the flow

For steady incompressible flow, the volume flow rate is constant:

$Q=Av,$

$A_1v_1=A_2v_2.$

If area halves, speed doubles. This follows from conservation of mass, not from Bernoulli's equation.

Worked example: pipe narrowing

Water travels at $2.0\text{ m/s}$ through area $6.0\text{ cm}^2$ and enters area $2.0\text{ cm}^2$:

$v_2=\frac{A_1}{A_2}v_1=\frac{6.0}{2.0}(2.0)=6.0\text{ m/s}.$

The flow rate is $Q=A_1v_1=(6.0\times10^{-4})(2.0)=1.2\times10^{-3}\text{ m}^3/\text{s}$.

Bernoulli as energy per volume

For steady, incompressible, negligible-viscosity flow along one streamline with no pump or turbine work,

$P+\frac12\rho v^2+\rho gy=\text{constant}.$

The terms represent pressure energy, kinetic energy, and gravitational potential energy per unit volume. At equal height, greater speed corresponds to lower static pressure under these assumptions.

Worked example: equal-height pressure change

For water, $\rho=1000\text{ kg/m}^3$, with $v_1=2.0\text{ m/s}$ and $v_2=6.0\text{ m/s}$ at equal height:

$P_1-P_2=\frac12\rho(v_2^2-v_1^2)=\frac12(1000)(36-4)=1.60\times10^4\text{ Pa}.$

The faster narrow section has pressure lower by $16.0\text{ kPa}$ in the ideal model.

Efflux from a tank

Apply Bernoulli between a large tank's free surface and a small outlet a vertical distance $h$ below. Both points are at atmospheric pressure, and the surface speed is negligible:

$v=\sqrt{2gh}.$

This is Torricelli's law. At $h=0.45\text{ m}$, $v\approx2.97\text{ m/s}$.

During: retrieval pause

In a horizontal ideal pipe, area becomes one-third as large. By what factor does speed change? Which Bernoulli term must decrease?

Reading applications carefully

A Venturi meter relates pressure difference to speed change through continuity plus Bernoulli. A Pitot tube compares stagnation pressure and static pressure. Real atomizers and aerodynamic systems involve streamline shape, viscosity, entrainment, circulation, and pressure fields; “faster fluid always means lower pressure” is not a universal rule.

When Bernoulli fails or needs extension

• viscosity dissipates mechanical energy;
• pumps add energy and turbines remove it;
• compressible flow changes density;
• unsteady or turbulent flow may violate the simple streamline model;
• Bernoulli constants can differ across streamlines in rotational flow.

Misconception checks

• Continuity relates area and speed; it does not by itself give pressure.
• Pressure is not “used up” merely because a pipe narrows.
• Bernoulli applies between chosen points under stated assumptions, not everywhere automatically.
• Larger flow speed does not mean fluid particles have larger pressure in every real situation.
• Gauge and absolute pressure must not be mixed silently.

After: evaluate a claim

A student says, “Air moving faster always has lower pressure.” Rewrite this as a scientifically defensible conditional statement and name two assumptions that matter.

AP-style synthesis

Choose a system and streamline, mark areas and heights, apply continuity first, write Bernoulli with every retained term, use consistent pressure references, and check whether predicted losses or Reynolds effects make the ideal result implausible.

Further learning

See sources.yaml. All examples, questions, and diagrams are original and do not imply College Board endorsement.