Vary the pipe radius, length, pressure drop, and viscosity to visualize the volume flow rate and parabolic velocity profile of fully developed laminar pipe flow in real time. The Q ~ R^4 Delta P relation is shown directly on the chart.
Parameters
Pipe radius R
mm
Pipe length L
m
Pressure drop ΔP
kPa
Dynamic viscosity μ
Pa·s
The fluid is assumed to be water (rho = 1000 kg/m^3). The formula is rigorous in the laminar regime (Re below 2300).
Results
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Volume flow rate Q
—
Mean velocity
—
Max velocity (center)
—
Reynolds number
Pipe section and velocity profile
Left: inlet (high pressure) / Right: outlet (low pressure) / Two thick lines = pipe walls / Arrows = parabolic laminar velocity profile (max at center, zero at wall)
Volume flow rate Q vs pressure drop ΔP
X = ΔP [kPa] / Y = Q [mL/s] / Linear Q ~ ΔP relation in the laminar regime (yellow dot = current value)
Theory & Key Formulas
The Hagen-Poiseuille law gives the volume flow rate of fully developed laminar flow in a circular pipe analytically. It is derived from the Navier-Stokes equations and is rigorous for a Newtonian fluid in laminar flow through a constant-area circular pipe.
Here $R$ is the pipe radius [m], $L$ the pipe length [m], $\Delta P$ the pressure difference [Pa], $\mu$ the dynamic viscosity [Pa·s], $\rho$ the density [kg/m^3], and $r$ the radial distance from the centerline.
What is the Hagen-Poiseuille Flow Simulator?
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I have heard the drip rate of an IV depends on the needle bore, the drug viscosity, and the bag height. Is there a closed-form formula for that?
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Yes — that is exactly Hagen-Poiseuille's law, $Q = \pi R^{4}\Delta P / (8\mu L)$. It is proportional to the fourth power of the radius R and the pressure difference ΔP, and inversely proportional to the needle length L and the viscosity μ. The R^4 dependence is dramatic: halving the bore reduces the flow rate to 1/16. That is why "thin needle vs thick needle" is such a critical clinical decision.
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Wait, the fourth power of the radius? So shrinking the pipe a little kills the flow!
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Exactly. That is also why a slight narrowing of an artery from atherosclerosis causes such a dramatic drop in blood flow. Try sliding R from 2 mm to 1 mm in the simulator — the flow rate drops to exactly 1/16. The pressure ΔP, by contrast, is only first power, so doubling it only doubles the flow. Making the pipe larger is far more effective than raising the pressure.
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In the velocity-profile picture, the arrows are longest in the center and shortest near the wall. What does that show?
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That is the famous parabolic velocity profile of laminar flow, $u(r) = V_{\max}(1 - (r/R)^{2})$: maximum at the center r=0 and zero at the wall r=R. The wall value of zero is the no-slip boundary condition, a basic axiom of viscous flow. Averaging over the cross section gives exactly half the centerline value, so $V_{\max} = 2\,V_{\mathrm{avg}}$. With the default values (R=2 mm, L=10 m, ΔP=10 kPa, μ=0.001 Pa·s) the simulator gives V_avg = 0.5 m/s and V_max = 1.0 m/s.
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Why is the Reynolds number shown? This is supposed to be the laminar formula.
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Good catch. The Hagen-Poiseuille law is only rigorous when the flow is actually laminar. So you need to monitor Re to confirm that you are still in the laminar regime — as a rule of thumb Re below 2300 is laminar, 2300 to 4000 is transitional, and above 4000 is turbulent. With the defaults the simulator gives Re = 2000, right next to the laminar boundary, so as you raise ΔP the Re increases proportionally and eventually leaves the laminar regime. For turbulent flow you need the Darcy-Weisbach equation with the Moody chart — try the "Moody Diagram" simulator next.
Frequently Asked Questions
In laminar flow the inertial term in the Navier-Stokes equations is negligible, so the momentum balance reduces to a pure viscous-vs-pressure-gradient problem that admits a closed-form solution. That solution is the parabolic velocity profile, and integrating it across the cross section gives Q = pi R^4 Delta P / (8 mu L). In turbulent flow the inertial term produces large turbulent stresses (Reynolds stresses) and no analytical solution exists. Empirically the transition happens around Re = 2300, so Hagen-Poiseuille's law is used in the laminar regime; above that, the linear Q ~ Delta P relation breaks down toward Q ~ sqrt(Delta P).
In large arteries (the aorta and so on) the flow is pulsatile and the formula is not strictly valid, but in capillaries, arterioles, and venules the Reynolds number is low (a few to a few tens) and Hagen-Poiseuille is a good approximation that is widely used in physiology and medicine. Blood, however, is non-Newtonian: its apparent viscosity rises at low shear rates because of the red-blood-cell concentration. Modern blood-flow simulations therefore start from Hagen-Poiseuille and add non-Newtonian, pulsatile, and elastic-wall corrections layer by layer to improve fidelity.
At the pipe inlet the velocity profile is nearly uniform (plug-like). Wall viscosity then propagates inward, and after a certain entrance length the flow settles into the parabolic Poiseuille profile. That state is called "fully developed." The laminar entrance length is roughly L_e ~ 0.06 Re D, where D is the diameter; for Re=2000 and D=4 mm this gives L_e ~ 0.48 m. The simulator computes values for the post-entrance region. In design you assume a pipe much longer than L_e, or add an entrance-loss coefficient separately for short pipes.
Not directly, but laminar analytical solutions exist for rectangular, triangular, and annular ducts as well. For a square duct of side a, for example, Q = K a^4 Delta P / (mu L) with K about 0.0351. In practice it is common to use the hydraulic diameter D_h = 4A/P (A = area, P = wetted perimeter) and substitute it into the circular formula, which keeps the laminar error to roughly 10-20 percent. For precise calculations use the cross-section-specific analytical solution, or a CFD numerical solution.
Real-World Applications
Medicine: IV drips, infusions, and catheter design: The drip rate of an IV is set by the needle bore and the bag height (head pressure), and the design equation is exactly Hagen-Poiseuille. Because radius enters as R^4, the choice of standard needle gauge (18G, 20G, 22G, ...) is the dominant factor in delivery rate. Hemodialysis blood circuits, continuous anesthetic infusion, and CT/MRI contrast injection are all designed by relating flow rate, viscosity, and catheter size with this formula.
Microfluidics and lab-on-a-chip: Microfluidic chips have channels of tens to hundreds of micrometers, where Re is almost always below 1, so laminar flow is the default. Hagen-Poiseuille (and its rectangular-duct analog) is the standard design equation for bio-devices including PCR, cell culture, and protein crystallization. Narrowing the channel makes the required Delta P explode as R^4, so the trade-off between pump pressure and channel size is a central design problem.
Groundwater and porous-media flow: Darcy's law q = -(k/mu) grad P for flow through porous media is derived by averaging Hagen-Poiseuille flow over a pore-size distribution (the starting point for the Kozeny-Carman equation). Groundwater modeling, oil and gas reservoir simulation, contaminant transport in soil, and CO2 geological storage are all essentially collections of Hagen-Poiseuille flows.
Lubrication and tribology: Lubricant flow in the gap of a bearing or gear is also handled within the Hagen-Poiseuille framework as a thin-gap laminar flow (a superposition of Couette and Poiseuille flows). It is the starting point of the Reynolds equation, and is used to estimate friction loss and heat generation from oil-film thickness, viscosity, and rotation speed.
Common Misconceptions and Pitfalls
The most common misconception is that "raising the pressure can increase the flow rate without bound." Hagen-Poiseuille looks like Q ~ Delta P, but only as long as the flow stays laminar. Raise the pressure and Re will eventually exceed 2300, and the flow transitions to turbulent, where Q ~ sqrt(Delta P) — doubling the pressure roughly multiplies the flow rate by only 1.4. In the simulator, once you raise Delta P far enough that statRe goes above 2300, the displayed Q is the theoretical "what if it stayed laminar" value; the actual flow rate would be lower.
The next is to confuse "radius vs diameter" and "mean vs maximum." The R in the Hagen-Poiseuille formula is the radius, not the diameter D. Mixing it up with the diameter form Q = pi D^4 Delta P / (128 mu L) gives a factor-of-16 error. Likewise, Re = rho V D / mu is in terms of the diameter, and the mean velocity V_avg is half the maximum V_max — small definition mismatches matter. This tool is consistent throughout: input radius R, mean velocity V_avg, and diameter-based Re.
Finally, beware of the assumption that "the fluid is water, so I can ignore viscosity." Water has a small viscosity of about 1.0 mPa·s at 20 C, but this changes substantially with temperature and additives. Glycerin is 1500 mPa·s (1500 times water), engine oil is 100 to 500 mPa·s. The flow rate is inversely proportional to mu, so a tenfold viscosity increase reduces Q tenfold. Slide mu from 0.0010 (water) up to 0.01 and 0.1 (oil) to feel the dramatic effect. A viscosity input error is one of the classic causes of wildly wrong CAE results.