Continuity Equation
Mass Conservation in Fluid Flow — Derivation, Formula & Solved Problems
Last Updated: March 2026
📌 Key Takeaways
- The continuity equation expresses conservation of mass for flowing fluids.
- General form: ρ₁A₁V₁ = ρ₂A₂V₂ (mass flow rate is constant along a streamtube).
- Incompressible form: A₁V₁ = A₂V₂ (volume flow rate is constant when density is constant).
- Volume flow rate: Q = AV (m³/s). Mass flow rate: ṁ = ρAV (kg/s).
- When a pipe narrows, velocity increases. When it widens, velocity decreases.
- The continuity equation is always used together with Bernoulli’s equation to solve most fluid flow problems.
1. The Concept — Mass Cannot Disappear
The continuity equation is simply the principle of mass conservation applied to fluid flow. In steady flow, the mass entering any section of a pipe or channel per unit time must equal the mass leaving. No fluid accumulates inside, and no fluid is created or destroyed.
Imagine water flowing through a garden hose. If 2 litres per second enters the hose at one end, 2 litres per second must exit at the other end (assuming no leaks and steady flow). If the hose nozzle is narrower than the hose, the water must speed up to maintain the same flow rate through the smaller opening. This everyday observation is exactly what the continuity equation describes mathematically.
2. Derivation
Consider a streamtube (an imaginary tube formed by streamlines) with cross-section A₁ at the inlet and A₂ at the outlet. For steady flow:
Mass entering per second = Mass leaving per second
ρ₁ × A₁ × V₁ × Δt = ρ₂ × A₂ × V₂ × Δt
Continuity Equation — General Form
ρ₁A₁V₁ = ρ₂A₂V₂
Or: ṁ = ρAV = constant along a streamtube
Where: ρ = fluid density (kg/m³), A = cross-sectional area (m²), V = average flow velocity (m/s), ṁ = mass flow rate (kg/s)
For incompressible fluids (liquids, and gases at low Mach numbers), density is constant (ρ₁ = ρ₂), so the equation simplifies to:
Continuity Equation — Incompressible Flow
A₁V₁ = A₂V₂
Or: Q = AV = constant
Where Q = volume flow rate (m³/s)
This is the form used in the vast majority of engineering problems and exam questions. It says that the product of area and velocity is constant — if the area decreases by half, the velocity doubles.
3. Incompressible vs Compressible Forms
| Aspect | Incompressible Flow | Compressible Flow |
|---|---|---|
| Density | Constant (ρ₁ = ρ₂) | Variable (ρ changes with P and T) |
| Continuity equation | A₁V₁ = A₂V₂ | ρ₁A₁V₁ = ρ₂A₂V₂ |
| Conserved quantity | Volume flow rate Q | Mass flow rate ṁ |
| Applies to | All liquids, gases at Mach < 0.3 | High-speed gas flows (Mach > 0.3) |
| Examples | Water in pipes, low-speed air ducts | Jet engines, rocket nozzles, high-speed aerodynamics |
GATE tip: Unless the problem explicitly involves high-speed gas flow or compressibility effects, use the incompressible form A₁V₁ = A₂V₂. This applies to virtually all pipe flow and hydraulics problems.
4. Volume Flow Rate vs Mass Flow Rate
Volume Flow Rate
Q = A × V
SI unit: m³/s. Also common: litres per second (L/s), litres per minute (LPM).
1 m³/s = 1000 L/s = 60,000 LPM
Mass Flow Rate
ṁ = ρ × A × V = ρ × Q
SI unit: kg/s
For a circular pipe of diameter D:
A = πD²/4
Q = (πD²/4) × V
5. Applications
| Application | How Continuity Applies |
|---|---|
| Pipe diameter change | Narrower pipe → higher velocity, wider pipe → lower velocity. A₁V₁ = A₂V₂. |
| Venturi meter | Throat has smaller area → velocity increases → pressure drops (by Bernoulli). Continuity + Bernoulli together give flow rate. |
| Nozzle | Area decreases along nozzle → velocity increases. Used in jets, garden hoses, fire hoses. |
| Branching pipe | Flow into junction = sum of flows out: Q₁ = Q₂ + Q₃ (for incompressible flow). |
| Rivers and channels | When a river narrows, water speeds up. When it widens into a lake, water slows down. |
6. Worked Numerical Examples
Example 1: Pipe Diameter Change
Problem: Water flows through a pipe that narrows from 200 mm diameter to 100 mm diameter. If the velocity in the larger section is 2 m/s, find the velocity in the smaller section and the volume flow rate.
Solution
A₁ = π(0.2)²/4 = 0.03142 m², A₂ = π(0.1)²/4 = 0.007854 m²
A₁V₁ = A₂V₂ → V₂ = V₁ × A₁/A₂ = 2 × 0.03142/0.007854 = 2 × 4
V₂ = 8 m/s
Q = A₁V₁ = 0.03142 × 2 = 0.0628 m³/s = 62.8 L/s
Note: halving the diameter quadruples the velocity (since area ∝ D²).
Example 2: Branching Pipe
Problem: A 300 mm pipe splits into two branches — one 200 mm and one 150 mm diameter. Velocity in the main pipe is 3 m/s. If velocity in the 200 mm branch is 2.5 m/s, find the velocity in the 150 mm branch.
Solution
Qmain = Q1 + Q2
Amain = π(0.3)²/4 = 0.07069 m²
A₁ = π(0.2)²/4 = 0.03142 m², A₂ = π(0.15)²/4 = 0.01767 m²
Qmain = 0.07069 × 3 = 0.2121 m³/s
Q₁ = 0.03142 × 2.5 = 0.07854 m³/s
Q₂ = 0.2121 − 0.07854 = 0.1335 m³/s
V₂ = Q₂/A₂ = 0.1335/0.01767 = 7.56 m/s
Example 3: Mass Flow Rate — Compressible
Problem: Air enters a duct at ρ₁ = 1.2 kg/m³, A₁ = 0.5 m², V₁ = 10 m/s. It exits at ρ₂ = 0.9 kg/m³ through an area of 0.4 m². Find the exit velocity.
Solution
ρ₁A₁V₁ = ρ₂A₂V₂
1.2 × 0.5 × 10 = 0.9 × 0.4 × V₂
6 = 0.36 × V₂
V₂ = 16.67 m/s
Mass flow rate = ρ₁A₁V₁ = 6 kg/s (same at both sections).
7. Common Mistakes Students Make
- Using diameter instead of area: A₁V₁ = A₂V₂ uses area (A = πD²/4), not diameter. If you halve the diameter, the area reduces by a factor of 4 (not 2), so velocity quadruples.
- Forgetting unit conversions: Diameter is often given in mm, but area must be in m². Convert before calculating: 200 mm = 0.2 m.
- Using volume flow rate for compressible flow: For gases with changing density, Q₁ ≠ Q₂. Only mass flow rate ṁ = ρAV is conserved. Use the general form ρ₁A₁V₁ = ρ₂A₂V₂.
- Not accounting for all branches at junctions: At a branching point, the sum of outflows must equal the inflow: Qin = Qout1 + Qout2 + … Students sometimes forget a branch.
- Assuming velocity is uniform across the cross-section: The continuity equation uses the average velocity. In real pipe flow, velocity varies from zero at the wall (no-slip) to maximum at the centre. Q = A × Vavg, not A × Vmax.
8. Frequently Asked Questions
What is the continuity equation in fluid mechanics?
The continuity equation states that mass is conserved in a flowing fluid. For steady, incompressible flow: A₁V₁ = A₂V₂ — the product of cross-sectional area and velocity is constant along a flow path. When a pipe narrows, velocity increases; when it widens, velocity decreases. This equation is derived directly from the principle that no mass can be created or destroyed during flow.
What is the difference between volume flow rate and mass flow rate?
Volume flow rate Q = AV (m³/s) measures volume passing per second. Mass flow rate ṁ = ρAV (kg/s) measures mass passing per second. For incompressible fluids where density is constant, both are conserved and either can be used. For compressible fluids, only mass flow rate is conserved — volume flow rate changes when density changes.
Why does fluid speed up in a narrower pipe?
Because the same mass of fluid must pass through every cross-section per second (mass conservation). A smaller cross-sectional area means the fluid must move faster to maintain the same flow rate. From A₁V₁ = A₂V₂: if A₂ is smaller, V₂ must be larger. This is why nozzles accelerate fluid and diffusers decelerate it.