The Poiseuille's law calculator above works out the steady laminar flow of a viscous fluid through a straight circular tube. It is the exact analytical solution for that case, not an empirical fit, which is unusual in fluid mechanics and is why the result is worth trusting — but only inside the conditions the derivation assumes. This tool checks those conditions and tells you when you have left them.
Arb Digital publishes free engineering calculators that police their own range of validity. The Reynolds number in the grid is not decoration: above roughly 2,000 the flow stops being laminar and the fourth-power law simply stops describing reality.
What This Poiseuille's Law Calculator Does
You give it a pressure drop, an internal radius, a length, a dynamic viscosity and a density. It returns the volumetric flow rate in millilitres per second, with litres per minute and cubic metres per second underneath, since medical, laboratory and industrial users each want a different one.
The grid adds four numbers. Mean velocity is flow divided by cross-sectional area, and it is worth knowing that the centreline velocity in laminar tube flow is exactly twice this value. The Reynolds number classifies the regime. Flow resistance is the ratio of pressure drop to flow, the hydraulic analogue of electrical resistance, and it depends only on the tube and the fluid. Wall shear stress is the drag the fluid exerts on the tube wall, which matters in biology and in anything where a coating or a cell layer can be stripped off.
How to Use It
- Enter the radius, not the diameter. This is the single most common error on this page, and because of the fourth power it produces an answer sixteen times too large.
- Use a viscosity at your working temperature. Water is roughly three and a half times less viscous at 100 °C than at 20 °C, and oils change far more than that.
- Measure the pressure drop over the straight length you enter. If your gauges bracket a bend, a valve or a fitting, part of the drop is not viscous pipe loss and this law does not account for it.
- Read the Reynolds number before the flow rate. If it is above 2,000 the result is no longer physically meaningful, whatever the arithmetic says.
- Use the resistance figure for series calculations. Tubes in series add their resistances, exactly like resistors, which makes multi-segment systems quick to assemble.
The Formula: How Poiseuille's Law Is Derived
For steady laminar flow of a Newtonian fluid in a straight circular tube, the velocity profile is a paraboloid: zero at the wall, maximum on the axis. Integrating that profile across the cross-section gives Q = π ΔP r4 ÷ (8 η L), with flow in cubic metres per second, pressure drop in pascals, radius and length in metres and viscosity in pascal-seconds.
Written as a resistance, R = 8 η L ÷ (π r4) and Q = ΔP ÷ R, which is Ohm's law with pressure standing in for voltage and flow for current. OpenStax University Physics Volume 1, section 14.7 on viscosity and turbulence, gives that resistance form directly and also supplies the Reynolds threshold used here: laminar below about 2,000, turbulent above about 3,000, unstable in between. The continuity relationship that turns flow into a mean velocity is set out in OpenStax University Physics Volume 1, section 14.5 on fluid dynamics.
Work the defaults through. Water at 20 °C, η = 0.001002 Pa·s, through a 2 mm radius tube half a metre long under a 200 Pa drop. The numerator is π × 200 × (0.002)4 = π × 200 × 1.6 × 10−11 = 1.005 × 10−8. The denominator is 8 × 0.001002 × 0.5 = 4.008 × 10−3. Dividing gives 2.508 × 10−6 m³/s, which is 2.51 mL/s or 0.15 L/min.
The cross-section is π × (0.002)2 = 1.257 × 10−5 m², so the mean velocity is 0.200 m/s and the centreline velocity is 0.399 m/s. The Reynolds number, ρvd ÷ η, is 998 × 0.200 × 0.004 ÷ 0.001002 = 795, comfortably laminar.
The Fourth Power Is the Whole Story
Every other term in the equation is linear. Double the pressure and the flow doubles. Double the length and it halves. Double the viscosity and it halves. Radius alone carries an exponent of four, and that changes how you should think about the whole system.
A 20 per cent reduction in radius cuts the flow by 59 per cent, because 0.8 to the fourth power is 0.41. A 50 per cent reduction leaves one sixteenth. To restore the original flow through a half-radius tube you would need sixteen times the pressure, which is usually impossible.
The clinical consequence is stark and is the reason this law appears in every physiology course. Arterial narrowing does not reduce blood supply in proportion to the blockage; it collapses it. A vessel narrowed to 70 per cent of its diameter passes about a quarter of its original flow at the same driving pressure.
The engineering consequence is the same in reverse. When a system is flow-limited, going one tube size up is nearly always more effective than raising the pump pressure, and it costs no energy to maintain. If the geometry is fixed, the pressure converter at least lets you see what pressure you would need in familiar units.
When Poiseuille's Law Stops Being True
The derivation makes five assumptions, and real systems break them regularly.
The flow must be laminar. Above a Reynolds number of about 2,000 the parabolic profile breaks down into eddies, resistance rises much faster than this law predicts, and pressure drop starts scaling roughly with the square of flow instead of linearly. Once that happens you need a friction-factor method, which is what the pipe flow calculator uses.
The fluid must be Newtonian, meaning its viscosity does not depend on how fast it is sheared. Blood is not: it thins as shear rate rises, so it flows better in narrow vessels than a fixed viscosity predicts. Polymer solutions, slurries and paints are worse offenders still.
The profile must be fully developed. Near the entrance the fluid has not yet organised into a paraboloid, and over that entrance length the pressure drop is higher than the law gives. For laminar flow that length can be tens of diameters, so a short wide tube may never develop at all.
The tube must be rigid, straight and circular. A flexible tube widens under pressure, raising flow faster than linearly. A bend adds loss. A non-circular duct needs a different geometric constant entirely. And the fluid must be incompressible, which rules out gases at anything but small pressure ratios — the air preset here is only usable for gentle flows.
Where This Sits Beside Our Other Flow Tools
This page solves the laminar viscous case exactly, where flow is linear in pressure and controlled by the fourth power of radius. That is a genuinely different regime from a turbulent pipeline, which the pipe flow calculator handles with a Darcy friction factor, and from a concentrated restriction, where the orifice flow calculator gives flow proportional to the square root of pressure.
Check the regime boundary directly with the Reynolds number calculator, and convert an unfamiliar viscosity unit with the viscosity converter. For plain continuity between area, velocity and flow, use the flow rate calculator, and for output units use the flow rate converter.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering a diameter where a radius is asked for — the fourth power turns that slip into an answer sixteen times too large.
- Using room-temperature viscosity for a hot or cold fluid — water alone changes by a factor of three and a half between 20 and 100 degrees, and oils change far more.
- Ignoring the Reynolds number — above about 2,000 the flow is no longer laminar and the law does not apply, however confident the arithmetic looks.
- Applying it across bends and fittings — the derivation is for a straight tube. Every fitting adds a loss the equation cannot see.
- Treating blood as Newtonian — its viscosity falls as shear rate rises, so real flow in small vessels exceeds what a fixed viscosity predicts.
Related Free Tools From Arb Digital
Confirm the flow regime with the Reynolds number calculator before trusting anything here. For turbulent pipe runs use the pipe flow calculator, and for a hole or nozzle rather than a long tube, the orifice flow calculator. Free-surface channels are covered by the open channel flow calculator, and the general streamline energy balance by the Bernoulli equation calculator. Convert your inputs and outputs with the viscosity converter, the pressure converter and the flow rate converter. The full catalogue is at the free online tools hub.
Frequently Asked Questions
Two effects multiply. A wider tube has more cross-sectional area, which scales with radius squared, and the fluid in it also moves faster on average because more of it is far from the wall, which contributes another factor of radius squared. Together they give the fourth power.
Once the Reynolds number passes about 2,000 the flow ceases to be laminar, the parabolic velocity profile breaks up, and resistance rises much faster than the law predicts. It also fails for non-Newtonian fluids, in the entrance region before the profile develops, and around bends and fittings.
It is pressure drop divided by flow rate, equal to eight times viscosity times length divided by pi times radius to the fourth. It depends only on the tube and the fluid, so it behaves exactly like electrical resistance: segments in series add, which makes multi-part systems easy to combine.
Radius. Poiseuille's law is written with radius raised to the fourth power, so entering a diameter instead makes the calculated flow sixteen times too large. If your specification gives a bore diameter, halve it before entering it here.
Through viscosity, which falls steeply as fluids warm. Water is about 0.001 pascal-seconds at 20 degrees Celsius and roughly 0.00028 at 100, so the same tube under the same pressure passes about three and a half times more hot water than cold. Density barely changes by comparison.
Only as an approximation. Blood is non-Newtonian: its viscosity drops as shear rate rises, and in vessels narrower than about 300 micrometres the red cells migrate toward the centre and lower the effective viscosity further. The law still shows the right qualitative behaviour, particularly the fourth-power sensitivity to vessel narrowing.
Exactly twice it. The laminar velocity profile in a circular tube is a paraboloid with zero velocity at the wall, and integrating a paraboloid gives a mean of half its peak. That is why the fluid on the axis of a narrow tube moves noticeably faster than the average suggests.
This tool is provided for educational and preliminary engineering use. It assumes steady, fully developed, laminar flow of an incompressible Newtonian fluid in a straight rigid circular tube, and it is not a substitute for measurement or for review by a qualified engineer or clinician.
