Formula and Method for Poiseuille's Law
The Hagen-Poiseuille equation describes how a fluid flows steadily through a narrow, rigid, cylindrical tube when it is driven by a pressure difference between the two ends. First derived independently by Gotthilf Hagen and Jean Léonard Marie Poiseuille in the 1840s, it applies to laminar (smooth, non-turbulent) flow of a Newtonian fluid — one whose viscosity does not change with shear rate, such as water, air, or light oil. The volumetric flow rate is Q = πΔPr⁴ / (8μL), where ΔP is the pressure difference across the tube, r is the tube's internal radius, μ is the fluid's dynamic viscosity, and L is the tube's length.
How the calculation works
Enter the tube radius, tube length, the pressure difference driving the flow, and the fluid's dynamic viscosity, choosing whichever units match your data — the calculator converts everything to SI base units (meters, pascals, and pascal-seconds) before computing. It plugs the converted values into Q = πΔPr⁴ / (8μL) to get the volumetric flow rate, then divides by the tube's cross-sectional area (πr²) to get the average velocity. Because laminar flow through a pipe has a parabolic velocity profile, the fastest-moving fluid is at the center of the tube, moving at exactly twice the average velocity (v_max = 2 × v_avg). The calculator also reports the wall shear stress, τ_w = ΔPr / (2L), the frictional force per unit area the fluid exerts on the tube wall.
Common mistakes
- Using diameter instead of radius: the formula calls for radius, not diameter — if you only know the diameter, divide it by 2 before entering it, or the fourth-power term will overstate flow by a factor of 16.
- Ignoring the fourth-power radius sensitivity: because flow rate scales with r⁴, small measurement errors in radius produce large errors in the result — a 10% error in radius changes flow rate by roughly 46%.
- Applying the formula to turbulent flow: Poiseuille's law is only valid when the Reynolds number stays below about 2300 (laminar regime); above that, turbulence dominates and this formula understates the pressure needed to sustain a given flow.
- Mixing viscosity units: water's viscosity is about 1 mPa·s (1 centipoise) at room temperature, not 1 Pa·s — entering the wrong order of magnitude changes every result by a factor of 1000.
Real-world applications
- Medicine uses Poiseuille's law to size IV catheters and estimate infusion rates, and to model how vessel narrowing (stenosis) restricts blood flow.
- Microfluidics and lab-on-a-chip devices rely on it to design channel dimensions that deliver a target flow rate of reagents or samples.
- Hydraulic and pneumatic system designers use it to size tubing and estimate pressure losses in slow, viscous-dominated flows.
- Petroleum and chemical engineers apply it, or its turbulent-flow analogs, to estimate pumping pressure for viscous fluids in pipelines.