Intuition The one core ideaA flowing fluid is just a crowd of tiny particles that never appear from nowhere and never vanish. So the amount of "stuff" passing any slice of the pipe each second is the same at every slice — and once you accept that, the whole continuity equation writes itself.
This page assumes you have seen nothing . Before you can read the parent derivation we will build every letter it uses, one at a time, each with a picture and a reason it exists.
A fluid is anything that flows and takes the shape of its container — liquids (water) and gases (air). Picture a river of tiny marbles all sliding past one another.
Definition Flow / flow speedFlow means the fluid is moving. Flow speed is how fast the fluid at one spot travels — like the speed of the water surface you'd measure by dropping a leaf and timing it.
Why we need this: the whole topic is about how fast fluid moves at different places in a pipe. "Speed at a place" is the object we track.
Δ Δ "a small change in" or "a little bit of". So Δ t Δ x not a multiplication — Δ t
Intuition Why bother with tiny slices?To catch fluid in the act of crossing a line, we freeze a very short moment Δ t
In the figure, the fluid slides forward by the short red distance Δ x Δ t
v Speed v v = Δ t Δ x . Δ x = v Δ t v = 20 Δ t = 3 Δ x = 60
Why the topic needs it: Step 1 of the derivation asks how far a slab of fluid travels in time Δ t Δ x = v Δ t
Recall Why is it "v Δ t v /Δ t multiply , not divide. Double the time, double the distance. Longer time means further travel ::: distance grows with time, so multiply
Definition Cross-sectional area, symbolA area A m 2
For a circle of radius r A = π r 2 .
r 2 r two-dimensional amount — it grows in both directions as the circle gets bigger. Double the radius and you double the width and the height, so the area goes up by 2 × 2 = 4 squared .
Look at the two circles: the right one has double the radius but four times the shaded area — that is the r 2
Common mistake "Half the radius ⇒ half the area."Why it feels right: radius is the number we're given. Reality: area follows r 2 quarter the area. Fix: always square the radius ratio: A 1 / A 2 = ( r 1 / r 2 ) 2
Why the topic needs it: A
Definition Volume, symbolV Volume is the amount of 3-D space something fills, in cubic metres (m 3 A Δ x Δ V = A Δ x . A Δ x
Combine with Step 2 of the derivation — since Δ x = v Δ t Δ V = A v Δ t .
The slab in the figure is the window A Δ x
Why the topic needs it: this slab volume is the fluid that actually crossed the boundary in one blink Δ t
Definition Density, symbolρ ρ mass per unit volume — how much stuff is crammed into each cubic metre:
ρ = volume mass = V m , units kg/m 3 . m = ρ V ρ
So the mass of our fluid slab is:
Δ m = ρ Δ V = ρ A v Δ t .
Intuition Why density is the star of the compressible caseWater barely squashes — its ρ ρ mass , and mass = ρ V ρ Incompressible vs compressible flow ).
Why the topic needs it: the law conserves mass , not volume. Density is what converts the volume slab into a mass slab.
m Mass m never created or destroyed (see Conservation of mass ). Fluid entering a sealed pipe section must leave it — it has nowhere else to go.
Definition Mass flow rate, symbolm ˙ m "per second" (rate). So m ˙ mass passing a section each second :
m ˙ = Δ t Δ m = Δ t ρ A v Δ t = ρ A v ( kg/s ) . Δ t
Why the topic needs it: m ˙ = ρ A v is the continuity equation. Conservation of mass says this number is the same at every section: ρ 1 A 1 v 1 = ρ 2 A 2 v 2
Definition Volume flow rate, symbolQ Q volume delivered each second:
Q = Δ t Δ V = A v ( m 3 / s ) . Q Volume flow rate and discharge .
Definition Streamline & stream tubeA streamline is the path a fluid particle follows — like a single strand of dye in the flow. A bundle of streamlines forms a stream tube , an imaginary pipe with no fluid crossing its walls. See Streamlines and stream tubes .
Why the topic needs it: continuity doesn't need a real metal pipe. Any stream tube counts as the sealed "box", so the same ρ A v = const
1 2 A 1 , v 1 , ρ 1 section 1 "; the subscript 2 two different places along the same tube — a wide spot and a narrow spot.
Saying ρ A v = constant any cross-section you like, compute ρ A v same number. That single number travelling unchanged down the pipe is the mass flow rate.
Speed v = distance over time
Slab distance = v times dt
Volume = A times slab distance
Density rho = mass over volume
Continuity rho A v = const
Read it bottom-up: every arrow is a symbol you now own, feeding into the continuity equation at the base.
Test yourself — cover the right side. If any fails, re-read that section before the parent note.
What does the symbol Δ t "A tiny slice of time"; it is one single quantity, not Δ × t
Distance moved by fluid at speed v Δ t Δ x = v Δ t
Area of a circular pipe face of radius r A = π r 2
If radius halves, what happens to area? Area becomes one quarter (area ∝ r 2
What is density ρ Mass per unit volume, ρ = m / V
Volume of a fluid slab of face A Δ x Δ V = A Δ x = A v Δ t
Mass of that slab? Δ m = ρ Δ V = ρ A v Δ t
What does the dot in m ˙ m ˙ "Per second"; mass flow rate m ˙ = ρ A v
What is Q Volume flow rate Q = A v ρ
Which quantity is conserved for ALL fluids? Mass flow rate m ˙ = ρ A v
What do subscripts 1 2 Two different cross-sections (places) along the same tube.
Parent topic (Hinglish) — the derivation these foundations feed.Conservation of mass — the one law behind everything here.Volume flow rate and discharge — deep dive on Q = A v Incompressible vs compressible flow — when you may drop ρ Streamlines and stream tubes — the imaginary pipe.Bernoulli's equation — needs the speeds continuity gives.Venturi meter — continuity in action. 
