Pressure is force per unit area (P = F/A). In a fluid at rest, pressure at a depth h is P = P₀ + ρgh.
Pascal's principle states that pressure applied to a confined fluid transmits equally in all directions
- the basis of hydraulic systems.
P = P₀ + ρgh
P₀ = surface pressure (101,325 Pa at sea level)ρ = fluid
density (water: 1000 kg/m³)Every 10 m of water ≈ 1 atm extra pressure
Worked Example 1
Pressure at Depth - Submarine
Problem: What is the water pressure at the Titanic wreck site
(3,800 m depth)? Seawater density = 1,025 kg/m³.
Calculate pressure
P = P₀ + ρgh = 101,325 + (1025)(9.81)(3800)
= 101,325 + 38,249,850
P = 38.35 MPa ≈ 378 atmospheres
Answer: 378 atmospheres. That's 38.35 million Pascals - enough to
crush a standard submarine. The Titanic's hull sections were flattened by this immense pressure on the
seafloor.
2 Buoyancy - Archimedes' Principle
An object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it
displaces. If the buoyant force exceeds the object's weight, it floats.
Problem: Ice has density 917 kg/m³, seawater 1025 kg/m³. What
fraction of an iceberg is below the surface?
Fraction submerged
Fraction = ρ_ice / ρ_seawater = 917 / 1025
= 0.895 = 89.5% underwater
Answer: 89.5% of an iceberg is below the waterline - only the
"tip" (10.5%) is visible. This is a direct application of Archimedes' principle and is why icebergs are
so dangerous to ships.
3 Bernoulli's Equation - Energy Conservation in Fluids
For incompressible, inviscid flow along a streamline, the total energy (pressure + kinetic + potential)
is constant. As fluid speeds up, its pressure drops - this is how wings generate lift and carburetors
mix fuel.
Viscosity is a fluid's resistance to flow - "internal friction." High viscosity: honey, oil. Low
viscosity: water, air. The Reynolds number Re = ρvL/μ determines whether flow is laminar (smooth, Re
< 2300 in pipes) or turbulent (chaotic, Re > 4000).
Water (20°C)
μ = 1.0 × 10⁻³ Pa·s
Air (20°C)
μ = 1.81 × 10⁻⁵ Pa·s
Honey
μ ≈ 2–10 Pa·s
Blood
μ ≈ 3–4 × 10⁻³ Pa·s
Motor oil
μ ≈ 0.1–0.3 Pa·s
Worked Example 3
Pipe Flow - Hagen-Poiseuille
Problem: Water (μ = 0.001 Pa·s) flows through a 2 cm diameter, 5
m long pipe with 10 kPa pressure drop. What is the flow rate?
Hagen-Poiseuille equation
Q = πr⁴ΔP/(8μL) = π(0.01)⁴(10000)/(8×0.001×5)
= π × 10⁻⁸ × 10000 / 0.04
Q = 7.85 × 10⁻⁴ m³/s = 0.785 L/s
Answer: 0.785 L/s. Notice the r⁴ dependence - doubling the pipe
radius increases flow by 16×. This is why arteries narrowed by plaque (atherosclerosis) cause such
dramatic blood flow reduction.
5 Continuity Equation - Mass Conservation
For incompressible flow, what flows in must flow out. When a pipe narrows, fluid speeds up to maintain
the same volume flow rate.
A₁v₁ = A₂v₂
A = cross-sectional area, v = fluid velocityNarrow pipe
→ faster flowGarden hose nozzle: half the area → double the speed
6 Navier-Stokes Equations & Turbulence
The Navier-Stokes equations are the complete description of viscous fluid flow - Newton's second law
applied to fluids. Solving them for turbulent flow remains one of the unsolved Millennium Prize Problems
in mathematics (worth $1 million).
ρ(∂v/∂t + v·∇v) = −∇P + μ∇²v + ρg
Left: inertial forces (acceleration)Right: pressure +
viscous + gravity forcesExact solutions exist only for simple geometries
Why turbulence is hard:
Turbulent flow is chaotic, sensitive to initial conditions, and spans many length scales
simultaneously. We can simulate it numerically (CFD) but cannot solve it analytically in general.
Richard Feynman called it "the most important unsolved problem of classical physics."