Skip to content

Example · Hagen–Poiseuille Flow

Problem statement

A fluid of viscosity \(\mu\) flows steadily through a circular pipe of radius \(R\) under a constant pressure gradient \(dp/dx < 0\). Find the velocity profile and the volumetric flow rate.

Given information

  • Pipe radius \(R\), gradient \(dp/dx\), viscosity \(\mu\)
  • Steady, incompressible, fully developed, axisymmetric

Solution strategy

Cylindrical Navier–Stokes, axial component only; integrate twice; regularity on the axis + no-slip at the wall.

Step-by-step solution

  1. The reduced equation: $\(\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right) = \frac{1}{\mu}\frac{dp}{dx}\)$
  2. First integration: \(r\frac{du}{dr} = \frac{1}{2\mu}\frac{dp}{dx}r^2 + C_1\)
  3. Second integration: \(u(r) = \frac{1}{4\mu}\frac{dp}{dx}r^2 + C_1\ln r + C_2\)
  4. Velocity must stay finite at \(r = 0\) ⇒ \(C_1 = 0\). No-slip \(u(R) = 0\) ⇒ \(C_2 = -\frac{1}{4\mu}\frac{dp}{dx}R^2\).

Final answer

\[\boxed{\,u(r) = \frac{1}{4\mu}\left(-\frac{dp}{dx}\right)\left(R^2 - r^2\right)\,} \qquad Q = \int_0^R u\,2\pi r\,dr = \boxed{\,\frac{\pi R^4}{8\mu}\left(-\frac{dp}{dx}\right)\,}\]

A parabolic profile, and the celebrated \(R^4\) law.

Key takeaways

  • \(Q \propto R^4\) is brutal: narrow an artery by 20% and flow drops by ~59% at fixed gradient. This single exponent is why arterial stenosis and clogged pipes matter so much.
  • Dimensional analysis gets \(Q \sim \Delta P D^4/\mu L\) without solving anything — the exact solution supplies the \(\pi/128\) (in diameter form).
  • Valid while laminar: \(Re \lesssim 2300\) (Reynolds number); beyond that, turbulence steepens wall gradients and increases the effective resistance.

Couette flow · Navier–Stokes equation · Buckingham Pi theorem