§ 00 · Visual overview

Water keeps
its own books.

Every river, wing, and heartbeat runs on the same short ledger: mass is conserved, energy is traded, momentum is paid for. Each idea here — continuity, Bernoulli's bargain, the Reynolds number, drag — lives as a simulation you can push, throttle, and replay. The equations are bookkeeping. The motion is anything but.

Continuity → Bernoulli → Vortex street → Terminal velocity →
tracer particles tracers born in the eddies An ideal 2-D flow: a steady stream threading a lattice of wandering vortices. No particle is ever lost — watch how they crowd together where the flow squeezes.
§ 01 · Conservation of mass

Squeeze the pipe, and the water must hurry

Water doesn't compress and doesn't vanish. So whatever volume enters a pipe each second must leave it each second — no exceptions, no delays. Narrow the exit and the fluid has one option: speed up, in exact inverse proportion to the area. Put your thumb over a garden hose and you are enforcing a conservation law.

A₁v₁ = A₂v₂ = Q   —  same liters per second through every cross-section
slow fluid fast fluid Each dash is a fluid parcel; its length and color track its speed. Count parcels crossing any vertical line per second — the count never changes.
Inlet speed v₁ (d₁ = 8 cm)
–m/s
Outlet speed v₂
–m/s
Speed-up factor
–×
Area ratio A₁/A₂
–×

Notice the two readouts agree: speed scales with area, and area scales with diameter squared. Halving the diameter quadruples the speed. This single rule explains nozzle jets, why rivers race through canyon narrows, and why blood creeps through capillaries: your aorta branches into billions of tiny tubes whose total area is ~500× larger, so the flow slows 500× — exactly slow enough for oxygen to step off.

§ 02 · Conservation of energy

Fast fluid is cheap fluid

Continuity says the throat of a venturi flows faster. But the pump pushes no harder — so where does the kinetic energy come from? It's bought from the pressure. Along a streamline, pressure + kinetic energy per volume is a fixed budget: speed up, and the pressure must drop. The manometer columns below are the fluid's bank statement.

P + ½ρv² + ρgh = constant   —  the budget along a streamline
pressure columns fast fluid Static pressure at three stations, shown as standpipe columns. Push the flow speed up and watch the throat column dive — if it hits the vapor line, the water itself boils.
Throat speed v₂
–m/s
Pressure drop ΔP
–kPa
Throat pressure (abs)
–kPa
Status
–

Now push v₁ up with a narrow throat. The throat pressure crashes toward zero, and below 2.3 kPa water boils at room temperature — bubbles of vapor flash into existence and collapse violently downstream. This is cavitation, the flow's way of declaring bankruptcy. The same trade — speed for pressure — is measured by venturi flow meters, feeds carburetors and perfume atomizers, and holds a spinning ball's curveball path together.

§ 03 · Viscosity vs inertia

One number decides: honey or hurricane

Take a flow's speed and size, and divide by its viscosity. That single ratio — the Reynolds number — predicts the flow's personality: whether it creeps politely around obstacles or tears itself into vortices. Below is a real fluid solver: water streaming past a cylinder. Drag Re across four orders of magnitude and watch the same geometry change species.

Re = ρvL / μ   —  inertia ÷ viscosity, the flow's personality score
dye released above the line dye released below A live Navier–Stokes simulation. At high Re the wake sheds vortices alternately — the Kármán vortex street — at a rhythm the fluid chooses itself.
Regime
–
Shedding frequency
–Hz (sim)
Strouhal number fD/v
–
Viscosity (sim)
–

At Re ≈ 100+ the wake locks into the Kármán vortex street — vortices peeling off alternately at a clock-steady frequency near St ≈ 0.2. That rhythm sings in power lines on windy days, and in 1940 it shook the Tacoma Narrows Bridge apart. The deeper point: Re is the similarity number. A model wing in a wind tunnel at the real wing's Re flies the same flow — that's why wind tunnels work, and why a bacterium swimming (Re ≈ 10⁻⁵) lives in a world where water feels like tar.

§ 04 · Drag & terminal velocity

Everything that falls signs a treaty

Drop anything into a fluid and a negotiation begins: gravity pulls, drag pushes back — gently and linearly if the flow around the object is viscous (Stokes), brutally and quadratically if it's inertial (Newton). The treaty is signed at terminal velocity, where the forces balance and acceleration ends. Drop spheres below and read the fine print.

Fdrag = 6πμrv  (Stokes, Re≲1)   ·   Fdrag = ½ρCdAv²  (Newton, Re≳1000)
velocity v(t) terminal velocity vt Left: the sphere in its fluid (streaks show relative flow — the replay is slowed to fit). Right: its velocity approaching the treaty line.
Terminal velocity vt
–
Reynolds number at vt
–
Drag regime
–
Time to 95% of vt
–

Try a 0.1 mm sphere in glycerin: it reaches terminal velocity in microseconds and Stokes' clean linear law rules — this is how Millikan weighed the electron with falling oil drops. Then drop a 2 cm steel ball in air: Re hits the tens of thousands, drag goes quadratic, and vt soars. In between the readout says “mixed” — nature ignores our tidy limits. And note what density does: a denser sphere falls faster, but only as the square root — the fluid always takes its cut. Set the sphere lighter than the fluid and the whole treaty runs in reverse: it becomes a bubble.

§ 05 · Practice

Now you run the numbers

Each problem needs one idea from above and arithmetic you can do on paper. Hints reveal one step at a time — try before you peek, and use the simulators to check yourself.

1 · The thumb on the hose

A garden hose of inner diameter 2.4 cm delivers water at 1.5 m/s. You press your thumb over the end, leaving an opening equivalent to a 1.2 cm diameter. How fast does the jet leave — and what stays exactly the same?

Hint 1 — Continuity: A₁v₁ = A₂v₂, and area goes as diameter squared: A ∝ d².
Hint 2 — The diameter halves, so the area shrinks by (2.4/1.2)² = 4×. The speed must grow by the same factor.
Answer — v₂ = 4 × 1.5 = 6 m/s. What's unchanged is the flow rate Q ≈ 0.68 L/s — your thumb rearranges the flow, it doesn't create water. Check it in § 01: set d₂ to half of d₁ and watch the speed-up factor pin to 4.0.

2 · The wing's invoice

Air (ρ = 1.2 kg/m³) streams over a wing at 70 m/s on top and 60 m/s underneath. The wing's area is 20 m². Use Bernoulli to estimate the lift force — and roughly what mass can it hold up?

Hint 1 — Along a streamline, P + ½ρv² is constant, so the pressure difference between the two faces is ΔP = ½ρ(v_top² − v_bot²).
Hint 2 — ΔP = ½ · 1.2 · (70² − 60²) = 0.6 · 1300 = 780 Pa. Force = pressure difference × area.
Answer — F = 780 × 20 ≈ 15.6 kN — about 1.6 tonnes held up by a 780 Pa whisper (less than 1% of atmospheric pressure). Lift isn't one big push; it's a small pressure imbalance over a large area. (Full aerodynamics also needs circulation and Newton's third law — Bernoulli is the accountant, not the whole story.)

3 · Is your blood turbulent?

Blood (ρ ≈ 1050 kg/m³, μ ≈ 3.5 mPa·s) flows through the aorta (d = 2.5 cm) at a mean speed of 0.2 m/s. Compute Re. Laminar flow typically holds below Re ≈ 2300 in a pipe — so which side is your aorta on?

Hint 1 — Re = ρvd/μ. Watch the units: μ = 3.5 mPa·s = 3.5 × 10⁻³ Pa·s, d = 0.025 m.
Hint 2 — Re = 1050 × 0.2 × 0.025 / 0.0035. Numerator: 5.25.
Answer — Re ≈ 1500 — laminar, but not by a wide margin. During peak systole v triples and Re briefly crosses the line; a narrowed (stenotic) valve pushes it far past, and the resulting turbulence is audible as a heart murmur. A stethoscope is a Reynolds-number detector. Compare regimes in § 03.

4 · The gentle raindrop

A raindrop of radius 1 mm falls through air (ρ = 1.2 kg/m³, Cd ≈ 0.47). Estimate its terminal velocity — and explain why rain doesn't hurt.

Hint 1 — At terminal velocity, weight = quadratic drag: mg = ½ρC_dAv², with m = (4/3)πr³ρ_w and A = πr².
Hint 2 — Solve: v_t = √(8 r ρ_w g / (3 ρ C_d)). Plug in r = 10⁻³ m, ρ_w = 1000.
Answer — v_t ≈ √(8·10⁻³·1000·9.8 / (3·1.2·0.47)) ≈ 6.8 m/s — about 25 km/h, reached after falling only a few meters. Without air, rain from a 2 km cloud would arrive at ~200 m/s, like gravel from a shotgun. Drag is why weather is survivable. Verify in § 04: water-density sphere, r = 1 mm, in air.