How Far Must Water Flow Before It Becomes a Parabola — Entrance Length and Hagen–Poiseuille

The pipe entrance region, the Hagen–Poiseuille parabola, and the 64/Re friction factor

Open a faucet all the way and the stream stays glassy-smooth for a while. Something similar happens inside a pipe. Right at the inlet, the water barely feels the wall. Its profile is almost flat. But after flowing a bit, the cross-sectional velocity settles into a smooth parabola. The stretch in between is today's subject.

The Distance It Takes the Wall to Spread the News#

At the pipe wall the fluid does not slip (no-slip: zero velocity at the wall surface). That condition does not reach the whole cross-section instantly at the inlet.

A thin boundary layer (the near-wall layer where velocity changes sharply) forms only near the wall at first. As the flow moves on, this layer grows inward. The moment the top and bottom boundary layers meet at the pipe center, the profile stops changing. From that point on we call the flow fully developed.

The distance from inlet to full development is the hydrodynamic entrance length $L_h$ . For laminar flow it is set by the diameter $D$ and the Reynolds number (ratio of inertial to viscous forces).

\frac{L_h}{D} \approx 0.05\,\mathrm{Re}_D, \qquad \mathrm{Re}_D = \frac{\rho\,u_m D}{\mu}

Here $u_m$ is the cross-sectional mean velocity, $\rho$ the density, and $\mu$ the viscosity. At $\mathrm{Re}_D=2000$ the entrance length reaches 100 pipe diameters. Pick the wrong measuring station in a thin lab line and you end up measuring flow that is still developing.

Play with the Reynolds number in the simulation below.

Reynolds Re_D800

Raise Re from 100 up to 2000 and the orange "fully developed" line slides to the right. The flat inlet profile now needs a longer run to settle into its parabola.

The Parabola of the Developed Region — Hagen–Poiseuille#

Once the flow is fully developed, the axial velocity stops changing. We have $\partial u/\partial x = 0$ and radial velocity $v=0$ . The cylindrical momentum equation collapses.

\frac{1}{r}\frac{d}{dr}\!\left(r\frac{du}{dr}\right) = \frac{1}{\mu}\frac{dp}{dx}

The pressure gradient $dp/dx$ is constant. Apply $u(R)=0$ at the wall and $du/dr=0$ at the center, then integrate twice.

u(r) = -\frac{1}{4\mu}\frac{dp}{dx}\left(R^2 - r^2\right) = 2\,u_m\!\left(1 - \frac{r^2}{R^2}\right)

Here $R=D/2$ is the pipe radius and $r$ the distance from the center. The result is clean. The centerline velocity is exactly twice the mean. This parabola is Hagen–Poiseuille flow.

Why the Pressure Drops at a Constant Rate — the Friction Factor#

Averaging the parabola across the cross-section ties the mean velocity directly to the pressure gradient.

\frac{dp}{dx} = -\frac{32\,\mu\,u_m}{D^2}

A constant gradient means pressure falls linearly along the pipe. Bundle that loss into a dimensionless number and you get the Darcy friction factor $f$ .

f = \frac{-\,(dp/dx)\,D}{\tfrac{1}{2}\rho u_m^2} = \frac{64}{\mathrm{Re}_D}

The laminar friction factor depends only on the Reynolds number. Pipe roughness never enters. Below, slide the Reynolds number and move the point along the friction-factor curve.

Re_D1,000

regime: laminarf = 0.0640L_h/D ≈ 50.0

For $\mathrm{Re}_D<2300$ the point tracks the $64/\mathrm{Re}$ line exactly. Cross the threshold and the curve bends, switching to a different law.

The Threshold at 2300 — from Laminar to Turbulent#

Once $\mathrm{Re}_D$ climbs past about 2300, the once-smooth flow breaks up. Eddies mix in and it transitions to turbulence.

In turbulent flow the entrance length shrinks. $L_h/D$ stays around 10 to 60, versus a hundred for laminar. But friction rises. One empirical fit for a smooth pipe is the Blasius law.

f = 0.316\,\mathrm{Re}_D^{-1/4} \quad (4000 \lesssim \mathrm{Re}_D \lesssim 10^5)

The exponent softens from $-1$ to $-1/4$ . Raising the Reynolds number no longer drops friction as fast. You have to push the same flow rate with more pressure.

Reproducing Entrance Length and Friction in Python#

Let us trace one small pipe with numbers.

import numpy as np
 
def entrance_length(Re, D):
    # laminar hydrodynamic entrance length [m]
    return 0.05 * Re * D
 
def poiseuille_profile(r, R, u_mean):
    # fully developed laminar velocity profile (Hagen-Poiseuille)
    return 2.0 * u_mean * (1.0 - (r / R) ** 2)
 
def darcy_friction(Re):
    # laminar friction factor (Re < 2300)
    return 64.0 / Re
 
D = 0.02          # pipe diameter [m]
rho, mu = 1000.0, 1e-3
u_mean = 0.08     # mean velocity [m/s]
 
Re = rho * u_mean * D / mu
print(f"Re = {Re:.0f}")                                 # Re = 1600
print(f"entrance length = {entrance_length(Re, D):.2f} m")   # 1.60 m
print(f"centerline speed = {poiseuille_profile(0, D/2, u_mean):.3f} m/s")  # 0.160
dpdx = -32 * mu * u_mean / D**2
print(f"pressure gradient = {dpdx:.2f} Pa/m")           # -6.40 Pa/m
print(f"friction factor f = {darcy_friction(Re):.4f}")  # 0.0400

A 2 cm pipe, yet the entrance length is 1.6 m — 80 diameters. Most short benchtop tubing is handling flow that is still developing.

Next Time You Meet Pipe Flow#

The essence of pipe flow is "the wall's information seeping toward the center." Keep three things.

In the entrance region the profile keeps changing. The entrance length stretches to $0.05\,\mathrm{Re}\,D$ .
Fully developed laminar flow is a parabola. Centerline is twice the mean; friction factor is $64/\mathrm{Re}$ .
Transition hits near $\mathrm{Re}\approx2300$ . Turbulence develops fast but costs more friction.