Continuity equation — derivation (conservation of mass), ρAv = const: Visual walkthrough

Step 1 — What is a "flowing tube" and what are we measuring?

WHAT. Picture fluid (water, air, honey) sliding along inside a pipe. We slice the pipe with two flat imaginary windows and call them section 1 (inlet) and section 2 (outlet). At each window we care about three numbers.

WHY these three? Because the thing we ultimately conserve is mass, and mass at a window is built from exactly these: how much room ( $A$ ), how fast it refills ( $v$ ), how heavy per room ( $ρ$ ). Nothing else is needed.

PICTURE. The pipe narrows from a wide mouth to a slim neck. Two red windows mark the sections; each is labelled with its own $A$ , $v$ , $ρ$ .

Figure — Continuity equation — derivation (conservation of mass), ρAv = const

Step 2 — How far does a slab of fluid travel in a tiny time?

WHAT. Freeze a thin slab of fluid sitting right at section 1. Let a tiny stretch of time pass — we name it $Δ t$ (the symbol $Δ$ , Greek "delta", just means "a small change in"; here "a small chunk of time"). During $Δ t$ the slab drifts forward by a distance we call $Δ x_{1}$ .

WHY this step? This is the plainest fact in motion: distance = speed × time. It is the bridge that turns a speed (which we can measure) into an actual chunk of fluid (which has volume and mass). Without it we could never count "how much crossed".

WHY these units multiply cleanly. $m/s \times s = m$ — the seconds cancel, leaving metres. The symbol arithmetic itself confirms we get a distance.

PICTURE. The slab at section 1, drawn twice: pale at its start, bright after time $Δ t$ . A magenta arrow of length $Δ x_{1} = v_{1} Δ t$ links them.

Step 3 — Turn that distance into a volume

WHAT. The slab that swept past the window is a little cylinder: its flat face is the window (area $A_{1}$ ) and its length is the distance it moved ( $Δ x_{1}$ ). Its volume is:

WHY this step? Volume of any straight cylinder = (area of its base) × (its length). We already found the length in Step 2, so we just multiply. This is the amount of room of fluid that crossed the window.

WHY not something more complicated? Because over a tiny $Δ t$ the slab is short and straight — no curving, no fancy shape. The cylinder formula is exact in the limit of small $Δ t$ .

PICTURE. The same slab now shaded as a solid cylinder: face $A_{1}$ , length $Δ x_{1}$ , volume filled in violet.

Step 4 — Turn that volume into a mass

WHAT. Density tells us how many kilograms live in each cubic metre. Multiply density by the volume of the slab to get the slab's mass, which we call $Δ m_{1}$ .

WHY this step? Mass = density × volume, always. This finally gives us the mass that flowed in during $Δ t$ — the quantity conservation of mass actually cares about. Notice the chain: speed → distance → volume → mass. Each step converted one measurable thing into the next.

PICTURE. The violet cylinder from Step 3, now stamped with its mass label $Δ m_{1} = ρ_{1} A_{1} v_{1} Δ t$ , with $ρ_{1}$ , $A_{1}$ , $v_{1}$ , $Δ t$ each colour-tagged where they appear.

Step 5 — Do the identical thing at the exit

WHAT. Everything in Steps 2–4 repeats at section 2, using its own numbers $ρ_{2}, A_{2}, v_{2}$ . In the same $Δ t$ , the mass leaving through section 2 is:

WHY the same $Δ t$ ? Crucial: we watch both windows during the identical stretch of time. Otherwise we'd be comparing apples with a different-sized batch of apples. Same clock, both ends.

PICTURE. Both windows shown together. Left cylinder (violet) = mass in; right cylinder (orange), thinner but longer, = mass out. Both labelled with their mass formulas and the shared $Δ t$ .

Step 6 — The box in between can't hoard or lose fluid

WHAT. Look at the region of pipe trapped between the two windows. It is already full of fluid and it isn't stretching or leaking. So whatever mass enters on the left in time $Δ t$ must leave on the right in that same $Δ t$ .

$Δ m_{1} = Δ m_{2} ⟹ ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t$

WHY this is the heart of it. This single equality is conservation of mass — see Conservation of mass. Every other line was bookkeeping to get us here.

PICTURE. The trapped "box" of pipe outlined in navy. A magenta arrow labelled mass in points into it; an equal orange arrow labelled mass out points out. A balance-scale icon between them reads $Δ m_{1} = Δ m_{2}$ .

Step 7 — Cancel the common time and read the law

WHAT. Both sides carry the same factor $Δ t$ . Divide it out — it was only ever a scaffolding to let us count a slab, and it appears identically on both sides.

WHY $Δ t$ vanishes. It was the same tiny time on both windows, so it never affected the balance — only the size of the two slabs equally. Dropping it leaves a statement about rates, true at every instant, not just for one chosen slab.

The incompressible special case. If the fluid barely changes density (water, most liquids — Incompressible vs compressible flow), then $ρ_{1} = ρ_{2}$ and they cancel too: $A_{1} v_{1} = A_{2} v_{2} = Q \frac{v _{2}}{v _{1}} = \frac{A _{1}}{A _{2}}$ Small area ⇒ big speed. Squeeze to speed.

PICTURE. The equation $ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t$ with both $Δ t$ struck through in magenta, arrow to the boxed final law.

Step 8 — The degenerate cases: check the law never breaks

WHAT. A good law must survive its extreme inputs. Four to check.

WHY show these? So you never meet a pipe geometry the derivation didn't cover. Wide, narrow, zero, still, compressing — the single boxed law handles them all.

PICTURE. Four mini-panels: (A) straight pipe equal arrows, (B) same width but a denser packed exit with a shorter speed arrow, (C) pipe pinching to a point with a huge speed arrow, (D) a still pipe with crossed-out arrows.

The one-picture summary

Everything in one frame: the chain speed → distance → volume → mass → balance → law, drawn along a narrowing pipe.

Recall Feynman retelling — the whole walkthrough in plain words

Water is running through a pipe that gets skinnier. I stand at the fat end and watch a thin slice of water for one tick of my watch. In that tick it slides forward by (its speed × the tick) — that's a little cylinder of water. I know how big the cylinder is (its face is the pipe's area, its length is how far it slid), and I know how heavy each bucketful of water is (that's density), so I know exactly how many kilograms slipped past me. Now I run to the skinny end and do the very same measurement during the very same tick. Here's the trick: the chunk of pipe between my two spots is already full of water and it isn't swelling or springing a leak — so the kilograms that walked in on the fat side had to be the exact same kilograms that walked out the skinny side. Writing "mass in = mass out" and crossing off the common tick of time gives $ρ A v =$ the same number everywhere. For water, density barely budges, so it simplifies to area × speed = constant — which is why pinching the pipe makes the water shoot out faster. Squeeze to speed.

Active recall

Why can we cancel

Δ t

from both sides in Step 7?

It is the same tiny time interval at both windows, so it multiplies both slabs equally and drops out, leaving a statement about rates.

In Step 4, what chain converted speed into mass?

speed → distance (

v Δ t

) → volume (

A v Δ t

) → mass (

ρ A v Δ t

In Case B (fixed area, density doubles), what happens to speed?

It halves;

ρ_{1} v_{1} = ρ_{2} v_{2}

with

ρ_{2} = 2 ρ_{1}

gives

v_{2} = v_{1} /2

As area

A_{2} \to 0

(Case C), what does the exit speed do?

It blows up toward infinity —

v_{2} = v_{1} A_{1} / A_{2} \to \infty

(a nozzle jet).

What single physical principle does Step 6 encode?

Conservation of mass — the full, leak-free box lets in exactly what leaves.

Connections

Parent topic
Conservation of mass — the principle behind Step 6.
Volume flow rate and discharge — the incompressible form $Q = A v$ .
Incompressible vs compressible flow — why Step 7's simplification needs constant $ρ$ .
Bernoulli's equation — uses the speeds continuity provides.
Venturi meter — Step 8 Case C in a real instrument.
Streamlines and stream tubes — the "imaginary tube" our derivation quietly assumed.