physicsme.ir Fluid Mechanics · Article

Hagen-Poiseuille Flow — Why the Fourth Power of Radius?

When water flows from a tap, blood circulates through vessels, or air escapes from a narrow tube,
all obey one remarkable law: the Hagen-Poiseuille law.
It states that the volumetric flow rate — the volume of fluid passing per unit time — is proportional
to the fourth power of the tube radius. Not the square, not the cube — the fourth power.
This article derives that R⁴ dependence using tools from high-school calculus and first-year university physics.

The Formal Question

Why is the flow rate through a narrow tube proportional to r⁴ and not r² or r³?

Consider a viscous fluid flowing through a cylindrical tube of radius R and length L.
The pressure difference between the two ends is ΔP.
The flow is calm and orderly (laminar). We will show that:

$$Q = frac{pi R^4 Delta P}{8eta L}$$

1. What Is Viscosity?

Honey pours far more slowly than water. This difference is due to viscosity — a fluid’s
internal resistance to flow. At the molecular level, viscosity arises from friction between
adjacent layers of fluid sliding past one another.

Newton’s Law of Viscosity

When two adjacent layers of fluid move at different speeds, the shear force (friction) between
them follows:

$$frac{F}{A} = eta frac{dv}{dy}$$

Fluids that obey this law are called Newtonian fluids. Water, air, and motor oil are Newtonian.

Viscosity Comparison for Common Fluids

2. Laminar vs. Turbulent Flow

The Hagen-Poiseuille law is only valid for laminar flow.
But when is flow laminar? The answer is given by the Reynolds number:

$$Re = frac{rho v D}{eta}$$

Re is a dimensionless number representing the ratio of inertial forces to viscous forces:

3. The Velocity Profile — Why Parabolic?

In laminar pipe flow, the fluid near the wall moves slowly and the fluid at the centre moves fastest.
This velocity distribution has the shape of a paraboloid — not a flat plug.

Why Is Velocity Zero at the Wall?

This is called the no-slip condition: fluid molecules right at the wall have zero velocity
because of molecular adhesion between the fluid and the solid surface.
This is an experimental fact confirmed in every careful measurement.

Deriving the Velocity Profile

Consider a thin cylindrical shell of fluid with radius r and length L.
The pressure force on its face: (F_P = Delta P cdot pi r^2)

The viscous shear force on its curved surface: (F_eta = eta frac{dv}{dr} cdot 2pi r L)

In steady state (constant velocity), these two forces balance:

$$Delta P cdot pi r^2 = -eta frac{dv}{dr} cdot 2pi r L$$

(The negative sign appears because velocity decreases as we move away from the centre.)
Simplify:

$$frac{dv}{dr} = -frac{Delta P}{2eta L} cdot r$$

Integrate:

$$v(r) = -frac{Delta P}{4eta L} r^2 + C$$

Boundary condition: at (r = R), velocity is zero. Therefore (C = frac{Delta P}{4eta L} R^2).
The final profile:

$$v(r) = frac{Delta P}{4eta L}left(R^2 – r^2right)$$

At the centreline ((r = 0)) the velocity is maximum:

$$v_{max} = frac{Delta P cdot R^2}{4eta L}$$

4. Computing the Volumetric Flow Rate — Where Does R⁴ Come From?

The volumetric flow rate Q is the sum of flow through all the thin annular rings of the cross-section.
A thin ring at radius r with width dr has area (dA = 2pi r,dr).
The flow through this ring is (dQ = v(r) cdot 2pi r,dr).

Integrating over the entire cross-section:

$$Q = int_0^R v(r) cdot 2pi r,dr = int_0^R frac{Delta P}{4eta L}(R^2 – r^2) cdot 2pi r,dr$$

$$Q = frac{pi Delta P}{2eta L} int_0^R left(R^2 r – r^3right) dr$$

Evaluating this integral — a straightforward polynomial integration:

$$int_0^R left(R^2 r – r^3right) dr = left[frac{R^2 r^2}{2} – frac{r^4}{4}right]_0^R = frac{R^4}{2} – frac{R^4}{4} = frac{R^4}{4}$$

And the final result:

$$boxed{Q = frac{pi R^4 Delta P}{8eta L}}$$

5. The Secret of the Fourth Power — An Intuitive Picture

Why the fourth power and not, say, the square? Let us think about it geometrically:

If we double the tube radius ((R to 2R)):

• Cross-sectional area: doubling the radius quadruples the area ((propto R^2))
• Mean velocity: with a larger radius the pressure gradient drives fluid over a wider region; the mean velocity also quadruples ((propto R^2))
• Total flow rate: Q = area × velocity → 4× × 4× = 16× ((propto R^4))

In other words: one factor of R² comes from the geometry of the circular cross-section,
and the other R² comes from the parabolic velocity profile. These multiply together to give R⁴.

Sensitivity of Flow Rate to Radius

6. Applications

6.1 Blood Flow in Blood Vessels

In the human body, blood flows from large arteries (aorta ≈ 2.5 cm diameter) down to the
smallest capillaries (≈ 5–10 μm diameter).
The Hagen-Poiseuille law explains why arterial narrowing (atherosclerosis) is so dangerous:

If a fatty plaque reduces an artery’s radius by just 20% ((R to 0.8R)):

$$Q_{new} = Q_{old} times (0.8)^4 = Q_{old} times 0.41$$

Blood flow drops to only 41% of its normal value. The heart must work much harder to compensate,
dramatically increasing the risk of heart attack or stroke.

6.2 The IYPT Funnel (Problem 7, 2027)

In Problem 7 of the International Young Physicists’ Tournament (IYPT 2027), air escapes through the
narrow stem of a funnel. The rate of volume change inside the funnel is described by Hagen-Poiseuille:

$$frac{dV}{dt} = -frac{pi r_s^4}{8mu L_s} Delta P$$

Notice the fourth power of r_s: a narrower stem offers vastly greater resistance.
This explains why blowing into the wide end of a funnel is so much harder than through a wide tube —
the narrow stem is the bottleneck.

6.3 Municipal Water Supply Networks

Civil engineers designing water distribution systems use this relation directly:
a pipe with twice the diameter can deliver 16 times the flow at the same pressure.
This is why main water supply pipes are very large — the cost of increasing pipe diameter
is far smaller than the cost of flow capacity shortfalls.

7. Limitations of the Hagen-Poiseuille Law

The law rests on several assumptions that, when violated, render the formula inaccurate:

8. Summary

The Hagen-Poiseuille law (Q = frac{pi R^4 Delta P}{8eta L}) follows from three steps:

1. Force balance on a cylindrical fluid element → differential equation for velocity
2. Integration of the differential equation with the no-slip boundary condition → parabolic profile (v(r))
3. Area integration of the velocity profile over the cross-section → flow rate Q with R⁴

The fourth power arises from multiplying two factors of R²: one from the geometry of the circular
cross-section, and one from the parabolic velocity profile. This R⁴ dependence makes flow
exquisitely sensitive to tube size — a fundamental lesson in fluid physics.

References

1. White, F. M. Fluid Mechanics, 8th edition, McGraw-Hill, 2016. (Chapter 6: Viscous Flow in Ducts)
2. Munson, B. R., Young, D. F., Okiishi, T. H. Fundamentals of Fluid Mechanics, 7th edition, Wiley, 2013.
3. Cengel, Y. A., Cimbala, J. M. Fluid Mechanics: Fundamentals and Applications, 4th edition, McGraw-Hill, 2018.
4. Sutera, S. P., Skalak, R. “The History of Poiseuille’s Law.” Annual Review of Fluid Mechanics, 25, 1–19, 1993.