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

WHAT is the continuity equation?

The quantity $\rho A v$ is the mass flow rate (kg/s). The quantity $A v$ is the volume flow rate $Q$ (m³/s).

WHY must it be true? (Conservation of mass)

HOW to derive it from first principles

We track the mass that crosses a cross-section in a tiny time $\Delta t$ .

Step 1 — How much fluid crosses a section in time $\Delta t$ ? At cross-section 1, fluid moves at speed $v_1$ . In time $\Delta t$ it advances a distance $\Delta x_1 = v_1\,\Delta t.$ Why this step? Speed × time = distance — that's the slab of fluid that sweeps past the boundary.

Step 2 — Volume of that slab. The slab is a cylinder of area $A_1$ and length $\Delta x_1$ : $\Delta V_1 = A_1\,\Delta x_1 = A_1 v_1 \,\Delta t.$ Why this step? Volume = area × length; this is the volume that entered.

Step 3 — Mass of that slab. $\Delta m_1 = \rho_1 \,\Delta V_1 = \rho_1 A_1 v_1\,\Delta t.$ Why this step? Mass = density × volume. This is the mass entering through section 1.

Step 4 — Same logic at section 2 (mass leaving). $\Delta m_2 = \rho_2 A_2 v_2\,\Delta t.$

Step 5 — Apply conservation of mass (steady state). No mass accumulates between the sections, so mass in = mass out: $\Delta m_1 = \Delta m_2 \;\Rightarrow\; \rho_1 A_1 v_1\,\Delta t = \rho_2 A_2 v_2\,\Delta t.$ Why this step? The "box" between the sections is full and unchanging, so whatever flows in must flow out.

Step 6 — Cancel $\Delta t$ . $\boxed{\rho_1 A_1 v_1 = \rho_2 A_2 v_2}$ For incompressible flow $\rho_1=\rho_2$ , divide it out: $\boxed{A_1 v_1 = A_2 v_2 = Q}$

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

Worked examples

Worked example Example 3 — Compressible gas (don't drop

$\rho$ !) Air enters a duct ( $A_1 = 0.5\text{ m}^2$ ) at $\rho_1 = 1.2\text{ kg/m}^3$ , $v_1 = 10\text{ m/s}$ . Downstream it's compressed to $\rho_2 = 2.4\text{ kg/m}^3$ in area $A_2 = 0.5\text{ m}^2$ . Find $v_2$ .

Use the full form $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ : $v_2 = \frac{\rho_1 A_1 v_1}{\rho_2 A_2} = \frac{1.2\times0.5\times10}{2.4\times0.5} = \frac{6}{1.2} = 5\text{ m/s}.$ Why this step? Here $\rho$ changes, so $Av$ alone is not conserved — mass flow $\rho A v$ is. Doubling density at fixed area halves the speed.

Common mistakes (steel-manned)

Recall Feynman: explain it to a 12-year-old

Imagine a busy hallway full of kids walking shoulder to shoulder. Where the hallway is wide, kids stroll slowly. Where it squeezes into a narrow doorway, the same number of kids per second still has to pass — so they have to hurry. Water in a pipe is exactly like this: nobody is created or vanishes, so a narrow spot forces the fluid to speed up. Squeeze it, it speeds; widen it, it slows.

Active recall

What physical law is the continuity equation a statement of?

Conservation of mass (no mass created/destroyed).

Write the general continuity equation.

\rho_1 A_1 v_1 = \rho_2 A_2 v_2 = \text{const}

(mass flow rate constant).

What form holds for incompressible fluids and why?

A_1 v_1 = A_2 v_2

, because

\rho

is constant and cancels.

In a narrowing pipe, does the fluid speed up or slow down?

Speeds up;

v \propto 1/A

If radius halves (incompressible), what happens to speed?

Speed becomes 4× (

v\propto 1/r^2

What is the volume flow rate

Q

and its units?

Q = Av

, units m³/s.

Why can't you use

A_1v_1=A_2v_2

for a compressing gas?

Because

\rho

changes; only

\rho A v

(mass flow) is conserved.

Derive distance a fluid slab moves in time

\Delta t

at speed

v

\Delta x = v\,\Delta t

, so volume

=A v\,\Delta t

, mass

=\rho A v\,\Delta t

What quantity is conserved for ALL fluids (compressible too)?

Mass flow rate

\dot m=\rho A v

Connections

Bernoulli's equation — continuity gives the speeds Bernoulli needs.
Volume flow rate and discharge
Incompressible vs compressible flow
Conservation of mass — parent principle.
Venturi meter — direct application of $A_1v_1=A_2v_2$ .
Streamlines and stream tubes

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, continuity equation ka core idea bilkul simple hai: fluid na to create hota hai na destroy. Jitna mass har second pipe ke ek section se andar jaa raha hai, utna hi mass dusre section se bahar nikalna chahiye — warna fluid kahin jam ho jata ya gayab ho jata. Isi conservation of mass ko maths me likhte hain: $\rho A v = \text{constant}$ . Yahan $\rho$ density, $A$ cross-section area, aur $v$ speed hai.

Derivation samajhna easy hai. Time $\Delta t$ me fluid $v\,\Delta t$ distance chalta hai, to volume $A v\,\Delta t$ banta hai, aur mass $\rho A v\,\Delta t$ . Section 1 ka mass = section 2 ka mass, $\Delta t$ cancel kar do, ho gaya $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ . Agar fluid incompressible hai (jaise paani), to $\rho$ same hai, cancel ho jaata hai, aur bachta hai $A_1 v_1 = A_2 v_2$ .

Iska practical matlab: jahan pipe patli hoti hai (chhota $A$ ), wahan fluid tez bhaagta hai (bada $v$ ). Yahi reason hai ki hose ke aage angootha lagao to paani zor se chhoot ta hai. Bas yaad rakhna — area $r^2$ ke proportional hota hai, to radius aadha karne par speed 4 guna ho jaati hai.

Ek important warning: gases ke liye, jahan density change hoti hai (compression), $A v$ wala short form mat use karna — wahan poora $\rho A v$ (mass flow rate) hi conserve hota hai. Exam me yeh trap aksar aata hai, isliye hamesha pehle decide karo fluid compressible hai ya nahi.

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

WHAT is the continuity equation?

WHY must it be true? (Conservation of mass)

HOW to derive it from first principles

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Fluid Mechanics

Connections