Laminar Flow in Circular Tubes

The Hagen–Poiseuille equation is a foundational result in fluid mechanics describing how a Newtonian liquid moves through a long, straight, circular tube when the flow is steady and laminar. In many engineering situations—such as intravenous drips, oil pipelines operating at low Reynolds numbers, microfluidic chips, or groundwater movement through porous rock—the velocity profile across the pipe’s cross section becomes parabolic and viscous shear dominates over inertial effects. Under these conditions the volumetric flow rate is directly proportional to the pressure difference applied across the ends and the fourth power of the tube radius, and inversely proportional to the fluid’s dynamic viscosity and the tube’s length. Small changes in radius therefore have an outsized influence on flow, a fact exploited in applications ranging from medical catheters to inkjet printer heads.

Derivation of the equation begins with the Navier–Stokes equations for incompressible fluids in cylindrical coordinates. Assuming no variation in the angular direction and a uniform pressure gradient along the tube axis simplifies the momentum equation to a balance between pressure forces and viscous shear. Integrating twice with respect to the radial coordinate gives the velocity profile, which is maximum at the centerline and drops to zero at the wall. The volumetric flow rate is obtained by integrating this velocity profile over the cross-sectional area, yielding the celebrated formula:

Here Q is the volumetric flow rate, ΔP the pressure difference between tube ends, r the tube radius, μ the dynamic viscosity, and L the length. Because the radius term appears to the fourth power, doubling the radius increases flow sixteenfold if all other factors remain unchanged. Conversely, doubling viscosity or length halves the flow. The expression contains no density term because inertial effects are negligible in laminar regimes; flow is driven solely by viscous resistance.

Assumptions Behind the Formula

While the Hagen–Poiseuille relation is elegant, it hinges on several restrictive assumptions. The fluid must be Newtonian so that shear stress is proportional to strain rate. Liquids like water, air, or light oils meet this criterion, but blood, paint, or polymer melts often exhibit non-Newtonian behavior with viscosity changing under shear. The flow must be fully developed, meaning the velocity profile no longer changes with distance along the tube; this typically requires an entry length of several diameters. The tube walls are presumed rigid, smooth, and perfectly circular. Finally, the Reynolds number Re = ρ v d / μ, based on diameter and average velocity, must remain below roughly 2,000 so that the flow remains laminar. If turbulence develops, friction losses increase dramatically and the equation overpredicts the flow rate.

Despite these caveats, the model is immensely useful for preliminary design and for systems intentionally operated in laminar regimes. Microfluidic devices, with channels narrower than a millimeter, almost always satisfy the assumptions due to their small dimensions and low velocities. Engineers designing such systems frequently use the Hagen–Poiseuille equation to size pumps, predict residence times, and balance flow in complex networks. In biomedical contexts, the equation explains why a slight swelling of an airway or blood vessel can dramatically restrict flow, since resistance increases as the inverse fourth power of radius. Understanding this sensitivity is crucial for medical treatments involving stents or anti-inflammatory therapies.

Using the Calculator

This interactive tool applies the formula to user-provided values. Enter the pressure difference in pascals, the tube radius in millimeters, the tube length in meters, and the dynamic viscosity in pascal-seconds. Typical room-temperature water has μ ≈ 0.001 Pa·s, while glycerin is around 1 Pa·s. After clicking “Calculate Flow,” the script converts the radius to meters, plugs the inputs into the equation, and reports the volumetric flow rate in cubic meters per second along with a convenient liters-per-minute figure. The output updates instantly, allowing you to explore the dramatic effects of radius and viscosity. Halving the radius from 5 mm to 2.5 mm cuts the flow by a factor of sixteen. Reducing viscosity through heating or dilution can significantly boost flow without changing geometry.

For reference, a typical IV drip might operate with a pressure difference of a few thousand pascals and a radius of about half a millimeter, delivering flow on the order of milliliters per minute. A household faucet with a radius near 5 millimeters and a pressure difference of 50,000 pascals yields several liters per minute. Oil moving through kilometers of pipeline experiences much larger pressure drops but also faces higher viscosities, so flow rates are carefully managed to maintain laminar conditions where possible. The table below illustrates how flow varies with radius for water over a one-meter tube at a constant 10 kPa pressure difference.

Radius (mm)	Flow (L/min)
1	0.24
2	3.86
3	19.5
4	61.7
5	150

Limitations and Extensions

In practice, engineers often need to account for more complex behaviors. Entrance effects can add significant resistance if the tube is short, and noncircular geometries require different shape factors. For turbulent flows, the Darcy–Weisbach equation with empirical friction factors provides better predictions. Some fluids exhibit temperature-dependent viscosity, requiring iterative solutions that couple energy and momentum balances. Nevertheless, understanding the Hagen–Poiseuille equation provides a vital baseline against which such complexities are measured. It also forms the theoretical underpinning for permeability concepts in porous media, where the flow through a network of tiny tubes is approximated using similar mathematics.

When you experiment with the calculator, consider how each parameter might vary in real systems. Pressure difference could stem from gravity in a siphon, from pump head, or from capillary action. Length might represent a single tube or the equivalent length of a more complex network. Viscosity changes with temperature—heating oil by 20 °C can reduce μ by half. Even the choice of units matters; a mis-converted radius can lead to errors of several orders of magnitude due to the fourth power relationship. Engineers guard against such mistakes with careful unit tracking and by validating calculations against experimental measurements.

By internalizing the lessons of the Hagen–Poiseuille equation, practitioners gain intuition about laminar flow systems. The equation encapsulates the essence of viscous resistance and highlights the importance of geometry and fluid properties. Whether optimizing a microfluidic chip, designing a heating system, or studying physiological flows, the ability to estimate how pressure, viscosity, and dimensions interact is indispensable. This calculator offers a convenient, client-side implementation so you can explore scenarios and build that intuition directly in your browser without external dependencies.