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Fluids in Motion: Continuity & Bernoulli

Source lecture(s): SC133 Lec 19

Intuition

Put your thumb over a garden hose: the water speeds up. Why? The same volume must squeeze through a smaller opening each second — continuity. And where a fluid speeds up, its pressure drops — Bernoulli, which is just energy conservation written per unit volume of flowing fluid. Two principles, one breath: they explain airplane wings, chimney draft, perfume atomizers, and why shower curtains attack you.

Continuity: mass conservation

For steady incompressible flow through a tube of varying cross-section:

\[\boxed{\,A_1 v_1 = A_2 v_2\,}\]

Volume flow rate \(Q = Av\) (\(\text{m}^3/\text{s}\)) is the same everywhere along the tube — narrow means fast. This is why rivers race through gorges and dawdle across plains.

Bernoulli's equation

Along a streamline of steady, incompressible, frictionless flow:

\[\boxed{\,P + \tfrac12\rho v^2 + \rho g y = \text{constant}\,}\]

Three terms, three energies per unit volume: pressure (work done by squeezing), kinetic, gravitational. Full derivation and applications: equation page and the graduate treatment in PC316.

The core trade-off: at constant height, fast ⇒ low pressure. Blow across a strip of paper and it rises into the fast, low-pressure stream.

Worked example: Torricelli's tank

Water drains through a small hole a depth \(h\) below the surface of an open tank. Exit speed?

Apply Bernoulli from surface (large area ⇒ \(v \approx 0\), \(P = P_\text{atm}\)) to hole (\(P = P_\text{atm}\)):

\[\rho g h = \tfrac12 \rho v^2 \Rightarrow \boxed{v = \sqrt{2gh}}\]

— identical to free fall from height \(h\): the water "falls" through the pressure field. (Where should the hole be to spray farthest? Halfway down — worked out in PC316.)

Worked example: Venturi meter

Pipe narrows from \(A_1\) to \(A_2\); a manometer reads the pressure drop \(\Delta P\). Continuity + Bernoulli give

\[v_1 = A_2\sqrt{\frac{2\Delta P}{\rho\,(A_1^2 - A_2^2)}}\]

— flow measured with no moving parts. The same throat-suction runs carburetors and perfume atomizers.

Where Bernoulli earns its reputation

  • Wings: air over the curved top travels faster ⇒ lower pressure above ⇒ lift. (The full story needs circulation — PC316 potential flow — but the pressure-speed link is genuine.)
  • Chimneys & burrows: wind across the top lowers pressure and draws air up — prairie dogs engineer their mounds for exactly this.
  • Roofs in storms: fast wind above, still air inside ⇒ net upward push; roofs peel, not crush.

Common mistakes

  • Applying Bernoulli across a pump, fan, or heater — it's frictionless streamline energy bookkeeping; devices that add energy break the constant.
  • Using it in strongly viscous or turbulent regions (long thin pipes obey Poiseuille instead).
  • Forgetting continuity first. Almost every Bernoulli problem is a two-equation system: continuity fixes the speeds, Bernoulli converts to pressures.
  • "Fast air sucks." Low pressure doesn't pull; the surrounding higher pressure pushes. Same physics, honest language.

Knowledge graph position

Prerequisites: Fluid statics, Conservation of energy. Leads to: the whole of PC316 Fluid Mechanics.

Quiz

Q1 (computational). Water flows at 2 m/s in a 4 cm-diameter pipe that narrows to 2 cm. Speed in the narrow section?

Answer

\(v_2 = v_1 (d_1/d_2)^2 = 2\times4 = 8\,\text{m/s}\) — area goes as diameter squared.

Q2 (conceptual). Two boats moving parallel and close are pushed together. Why?

Answer

Water funneled between the hulls speeds up (continuity), dropping the pressure between them below the outside pressure — the outer water pushes the hulls together. The same effect once sank ships during refueling alongside.

Q3 (multiple choice). In the narrow throat of a horizontal Venturi tube, the pressure is: (a) highest (b) lowest (c) unchanged

Answer

(b). Fastest flow ⇒ lowest pressure (at fixed height) — the defining Bernoulli trade.