Poiseuille's Law
Laminar flow rate through a pipe.
Formula
Variables
How it works
Poiseuille's law gives the steady volumetric flow rate for a Newtonian fluid moving in smooth, fully developed laminar flow through a long circular pipe. It comes from balancing the viscous resistance of concentric fluid layers against the pressure force driving the fluid forward, which produces a parabolic velocity profile and makes the flow rate extremely sensitive to the pipe radius. The r^4 dependence means a small change in radius has a much larger effect than the same fractional change in length, viscosity, or pressure difference.
When to use it
Use this for low-Reynolds-number flow in a straight cylindrical tube, such as blood flow in narrow vessels, liquid through capillary tubing, or lab-scale pipe-flow calculations when laminar assumptions are valid. It is not appropriate for turbulent flow, non-circular ducts, or situations where the fluid is not well approximated as Newtonian.
Worked example, step by step
A viscous oil flows through a straight tube with radius r = 2.0 mm, length L = 0.80 m, viscosity μ = 0.12 Pa·s, and pressure drop ΔP = 5.0 kPa. Step 1: Convert to SI units. r = 2.0 mm = 2.0 × 10^-3 m ΔP = 5.0 kPa = 5.0 × 10^3 Pa Step 2: Substitute into Poiseuille's law. Q = (πr^4ΔP)/(8μL) Q = [π(2.0 × 10^-3)^4(5.0 × 10^3)] / [8(0.12)(0.80)] Step 3: Evaluate the radius term. (2.0 × 10^-3)^4 = 16 × 10^-12 = 1.6 × 10^-11 Step 4: Compute the numerator. π(1.6 × 10^-11)(5.0 × 10^3) ≈ 2.51 × 10^-7 Step 5: Compute the denominator. 8(0.12)(0.80) = 0.768 Step 6: Divide. Q = (2.51 × 10^-7) / 0.768 ≈ 3.27 × 10^-7 m^3/s Final answer: Q ≈ 3.3 × 10^-7 m^3/s That is about 0.33 mL/s.
Common mistakes to avoid
- Using the formula for turbulent or highly disturbed flow, where Poiseuille's law does not apply
- Forgetting that the radius is raised to the fourth power, not the diameter
- Mixing units such as mm, kPa, or cP without converting everything to SI units first
Did You Know?
Poiseuille's law is why narrowed arteries force the heart to work dramatically harder to keep blood flowing.
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BasicForce per unit area.
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