Equation of ContinuityFoundation

Concept

An incompressible fluid cannot pile up: whatever volume enters a pipe section each second must leave it. So where the pipe narrows, the fluid speeds up — $A_1v_1 = A_2v_2$. Speed scales with the square of the diameter ratio. Faster flow then means lower pressure (Bernoulli), the heart of the Venturi effect.

Key formula

Derivation

Mass conservation for steady flow: $\rho A_1 v_1\,dt = \rho A_2 v_2\,dt$. Cancel ρ (incompressible) and dt.

With $A = \pi d^2/4$, halving the diameter quarters the area and quadruples the speed. The pressure drop follows from Bernoulli on a level pipe: $\Delta P = \tfrac{\rho}{2}(v_2^2 - v_1^2)$.

Scenarios to explore

• Equation of Continuity — A₁v₁ = A₂v₂ — why fluids race through the narrow part.

Real-world applications

• Garden-hose thumb trick and nozzle design.
• Venturi meters measure flow from the pressure drop.
• Blood flow speeds up at arterial stenoses — a diagnostic signal.

JEE exam tips

• Q = Av is the conserved quantity — compute it once, reuse everywhere.
• Falling water column narrows: v increases as it falls, so A must shrink (that's why streams taper).
• Combine with Bernoulli for Venturi/atomizer problems — continuity gives v₂, Bernoulli gives ΔP.

Common mistakes

• Scaling speed with d instead of d² (area, not diameter).
• Thinking the narrow section has HIGHER pressure — it's lower (fast = low P).
• Applying continuity across a junction without adding the branch flows.

Exam traps to avoid

• Continuity is volume-per-second conservation, valid even in non-viscous ideal flow only if incompressible.
• The manometer in a Venturi reads P₁ − P₂ = ½ρ(v₂² − v₁²), not ½ρv₂².