Fluid mechanics
The idea
Fluids transmit force as pressure — force per unit area — and three working tools cover most first-year problems: hydrostatics (p = p₀ + ρgh for depth h in a fluid of density ρ), continuity (Av is constant along a narrowing pipe because mass is conserved), and Bernoulli's equation, which is conservation of energy per unit volume for steady, incompressible, nonviscous flow: p + ½ρv² + ρgy stays constant along a streamline. Buoyancy follows from hydrostatics: the upward force equals the weight of displaced fluid.
Read Bernoulli's equation as an energy budget. The ½ρv² term is kinetic energy density and ρgy is gravitational potential energy density, so pressure plays the role of the budget's flexible account: where the fluid speeds up, pressure must drop to pay for the kinetic energy. Continuity tells you where the speeding up happens — narrow sections force faster flow.
Intuition often insists that squeezing flow into a narrow pipe raises the pressure there, like pinching a hose. It is the reverse: the constriction is the low-pressure region, because fluid must accelerate into it and only a pressure drop can provide that forward push. The pinched-hose spray feels forceful because of speed, not pressure.
Worked example
Water (density 1000 kg/m³) flows steadily through a horizontal pipe that narrows from 4.0 cm diameter to 2.0 cm diameter. In the wide section the speed is 2.0 m/s and the gauge pressure is 150 kPa. Find the speed and gauge pressure in the narrow section.
- Apply continuity with areas proportional to diameter squared: A₁/A₂ = (4.0/2.0)² = 4, so the speed in the narrow section is v₂ = 4 × 2.0 = 8.0 m/s.
- Since the pipe is horizontal, the ρgy terms cancel and Bernoulli reduces to p₁ + ½ρv₁² = p₂ + ½ρv₂².
- Solve for p₂: p₂ = p₁ + ½ρ(v₁² − v₂²) = 150000 + 0.5 × 1000 × (2.0² − 8.0²) = 150000 + 500 × (4 − 64).
- Evaluate: 500 × (−60) = −30000 Pa, so p₂ = 150000 − 30000 = 120000 Pa = 120 kPa.
- Interpret the sign: the pressure dropped by 30 kPa exactly where the water sped up — the pressure difference is the force per area that accelerated the water into the constriction, so a drop is required, not a rise.
Answer. The water moves at 8.0 m/s in the narrow section, where the gauge pressure falls to 120 kPa.
Check your understanding
- Why must the pressure be lower in the narrow section — what force would otherwise accelerate the water?
- Which of Bernoulli's assumptions (steady, incompressible, nonviscous) fails first in a real garden hose, and what does viscosity do to the pressure along the pipe?
- How does the continuity equation express conservation of mass, and what changes for a compressible gas?
- How would you use these tools to explain why a partially blocked artery is dangerous beyond the reduced flow itself?
Build the foundations first
Fluid mechanics builds on these concepts. If any feel shaky, start there.