Fluid Flow Formulas
Reference for fluid flow formulas: continuity equation, Bernoulli's equation, and Poiseuille's law for viscous pipe flow with units and worked examples.
The Formulas
Continuity Equation: A₁ × v₁ = A₂ × v₂
Poiseuille's Law: Q = (π × r⁴ × ΔP) / (8 × μ × L)
Fluid flow formulas describe how liquids and gases move through pipes and channels. The continuity equation ensures mass is conserved. Poiseuille's law gives flow rate in long, straight pipes.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Q | Volumetric flow rate | m³/s |
| A | Cross-sectional area of pipe | m² |
| v | Flow velocity | m/s |
| r | Pipe radius | m |
| ΔP | Pressure difference | Pa (Pascals) |
| μ | Dynamic viscosity of the fluid | Pa·s |
| L | Length of the pipe | m |
Example 1
Water flows at 2 m/s through a pipe with radius 0.05 m. Find the flow rate.
A = π × r² = π × 0.05² = 0.00785 m²
Q = A × v = 0.00785 × 2
= 0.0157 m³/s ≈ 15.7 liters/s
Example 2
A pipe narrows from 10 cm to 5 cm diameter. If v₁ = 1 m/s, find v₂.
A₁ × v₁ = A₂ × v₂
A₁ = π(0.05)² = 0.00785 m², A₂ = π(0.025)² = 0.00196 m²
v₂ = (A₁ × v₁) / A₂ = (0.00785 × 1) / 0.00196
= 4 m/s (velocity quadruples when diameter halves)
When to Use Them
Use fluid flow formulas when:
- Designing plumbing, irrigation, or hydraulic systems
- Sizing pumps and calculating required flow rates
- Analyzing blood flow in medical applications
- Calculating pressure drops in piping networks
Key Notes
- Continuity equation: A₁v₁ = A₂v₂: For incompressible flow, the volumetric flow rate Q = Av is constant along a pipe. A narrower section must have higher velocity to conserve mass. This is why blood moves faster through narrowed arteries.
- Bernoulli's equation: P + ½ρv² + ρgh = constant: Along a streamline in steady, incompressible, inviscid flow. A speed increase must be accompanied by a pressure decrease — the principle behind aircraft lift, venturi meters, and carburetor operation.
- Flow rate Q = Av: Volume per unit time (m³/s or L/min). Mass flow rate ṁ = ρQ. These are the fundamental quantities in pipe sizing, pump selection, and process engineering.
- Laminar vs turbulent flow: Reynolds number Re = ρvD/η determines the regime. Re < 2,300: laminar (smooth, predictable). Re > 4,000: turbulent (chaotic, higher friction losses). Between 2,300–4,000: transitional. Turbulent flow requires significantly more pumping energy.
- Applications: Fluid flow principles govern pipe network design, aircraft aerodynamics, HVAC duct sizing, heart valve design, chemical reactor throughput, and irrigation system layout.