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Poiseuille's Law Calculator

Calculate flow rate, velocity, and pressure drop for laminar flow in circular pipes using Poiseuille's law

Parameters

Paⓘ
mⓘ
mⓘ
Pa·sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Flow Rate:
0.00;m3/s0.00;m³/s
Average Velocity:
0.31;m/s0.31;m/s
Maximum Velocity:
0.63;m/s0.63;m/s
Flow Resistance:
4074366.54;Pa⋅s/m34074366.54;Pa·s/m³
Reynolds Number:
3125.00;3125.00;

Examples

Example 1: Water in Small Pipe

Water flowing through a 1cm diameter pipe with 100 Pa pressure difference.

  • Flow Rate: 0.000.00
  • Average Velocity: 0.310.31
  • Maximum Velocity: 0.630.63
  • Flow Resistance: 4074374.004074374.00

Example 2: Oil in Large Pipe

Oil flowing through a 10cm diameter pipe with 1000 Pa pressure difference.

  • Flow Rate: 0.000.00
  • Average Velocity: 0.130.13
  • Maximum Velocity: 0.250.25
  • Flow Resistance: 1018594.001018594.00

Example 3: Blood in Capillary

Blood flowing through a capillary with 50 Pa pressure difference.

  • Flow Rate: 0.000.00
  • Average Velocity: 0.000.00
  • Maximum Velocity: 0.000.00
  • Flow Resistance: 636619772.00636619772.00

Visualization

Poiseuille's Law

Poiseuille's law describes the volumetric flow rate of a fluid through a circular pipe under laminar flow conditions. It was derived by Jean Léonard Marie Poiseuille and is fundamental to understanding fluid flow in pipes, blood vessels, and other cylindrical conduits.

The law states that the volumetric flow rate Q is proportional to the pressure difference ΔP, the fourth power of the pipe radius r, and inversely proportional to the pipe length L and fluid viscosity μ: Q = (πr⁴ΔP)/(8μL). This relationship shows that flow rate is extremely sensitive to pipe diameter.

Key assumptions of Poiseuille's law include: (1) Laminar flow (Reynolds number < 2300); (2) Incompressible fluid; (3) Steady flow; (4) No-slip boundary conditions at pipe walls; (5) Circular cross-section; (6) Constant viscosity. The law is most accurate for long, straight pipes.

The velocity profile in laminar pipe flow is parabolic, with maximum velocity at the center and zero velocity at the walls. The average velocity is half the maximum velocity, and the flow rate is related to average velocity by Q = A·v_avg, where A is the cross-sectional area.

Poiseuille's law has applications in engineering design, including water distribution systems, oil pipelines, blood flow analysis, and microfluidics. It's essential for predicting pressure drops, pump requirements, and flow rates in pipe networks.

Key Concepts

  • Poiseuille's Law: Q = (πr⁴ΔP)/(8μL)
  • Flow Rate: Q = A·v_avg = πr²·v_avg
  • Average Velocity: v_avg = Q/A = (r²ΔP)/(8μL)
  • Maximum Velocity: v_max = 2v_avg
  • Pressure Drop: ΔP = (8μLQ)/(πr⁴)
  • Resistance: R = (8μL)/(πr⁴)

Real-World Applications

  • Water Distribution: Designing municipal water systems
  • Oil Pipelines: Calculating flow rates and pressure drops
  • Blood Flow: Understanding cardiovascular dynamics
  • Microfluidics: Designing lab-on-chip devices
  • Heat Exchangers: Optimizing fluid flow for heat transfer

Explore Further

More fluid mechanics tools

Physics Equations

Poiseuille's Law:
Q=πr4ΔP8μLQ = \frac{\pi r^4 \Delta P}{8\mu L}
Flow Rate:
Q=Avavg=πr2vavgQ = A v_{avg} = \pi r^2 v_{avg}
Average Velocity:
vavg=r2ΔP8μLv_{avg} = \frac{r^2 \Delta P}{8\mu L}
Maximum Velocity:
vmax=2vavgv_{max} = 2v_{avg}
Pressure Drop:
ΔP=8μLQπr4\Delta P = \frac{8\mu L Q}{\pi r^4}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Pipe Radius

First, we calculate the radius from the diameter:

Equation:

r=d2r = \frac{d}{2}

Calculation:

r=0.012=0.005 mr = \frac{0.01}{2} = 0.005 \text{ m}

Explanation:

The radius is half the diameter of the pipe.

2

Step 2: Apply Poiseuille's Law

Use Poiseuille's law to calculate flow rate:

Equation:

Q=πr4ΔP8μLQ = \frac{\pi r^4 \Delta P}{8\mu L}

Calculation:

Q=π(0.005)4(100)8(0.001)(1)=2.45e−5 m3/sQ = \frac{\pi (0.005)^4 (100)}{8(0.001)(1)} = 2.45e-5 \text{ m}^3/\text{s}

Explanation:

This gives the volumetric flow rate through the pipe.

3

Step 3: Calculate Average Velocity

Find the average velocity using the flow rate and cross-sectional area:

Equation:

vavg=QA=Qπr2v_{avg} = \frac{Q}{A} = \frac{Q}{\pi r^2}

Calculation:

vavg=2.45e−5π(0.005)2=0.312 m/sv_{avg} = \frac{2.45e-5}{\pi (0.005)^2} = 0.312 \text{ m/s}

Explanation:

The average velocity is flow rate divided by cross-sectional area.

4

Step 4: Calculate Maximum Velocity

In laminar flow, maximum velocity is twice the average velocity:

Equation:

vmax=2vavgv_{max} = 2v_{avg}

Calculation:

vmax=2(0.312)=0.625 m/sv_{max} = 2(0.312) = 0.625 \text{ m/s}

Explanation:

This follows from the parabolic velocity profile in laminar flow.

Frequently Asked Questions (FAQ)

What is Poiseuille's law?

Poiseuille's law describes the volumetric flow rate of a fluid through a circular pipe under laminar flow conditions. It shows that flow rate is proportional to pressure difference and the fourth power of pipe radius.

When does Poiseuille's law apply?

Poiseuille's law applies to laminar flow (Re < 2300) in long, straight, circular pipes with incompressible fluids and constant viscosity. It assumes no-slip boundary conditions at pipe walls.

Why is flow rate so sensitive to pipe diameter?

Flow rate is proportional to the fourth power of pipe radius (r⁴), so doubling the diameter increases flow rate by 16 times. This is because larger pipes have both more area and less wall friction per unit volume.

What is the velocity profile in laminar pipe flow?

The velocity profile is parabolic, with maximum velocity at the center and zero at the walls. The average velocity is half the maximum velocity, and flow is smooth with no mixing between layers.

How does viscosity affect flow rate?

Higher viscosity decreases flow rate because it increases resistance to flow. Flow rate is inversely proportional to viscosity, so more viscous fluids flow more slowly under the same pressure difference.

Practice MCQs

  1. According to Poiseuille's law, flow rate is proportional to:
  2. The velocity profile in laminar pipe flow is:
  3. If pipe diameter is doubled, flow rate increases by:
  4. The average velocity in laminar pipe flow is:
  5. Poiseuille's law applies to:

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