Short Answer

The Hagen–Poiseuille equation gives the steady volumetric flow rate of a Newtonian, incompressible fluid in a long, straight circular pipe under laminar, fully developed flow. It relates pressure drop, pipe dimensions, fluid viscosity and pipe length:

or in terms of diameter :

This result is obtained by solving the simplified Navier–Stokes equation for axial flow with no-slip at the wall and symmetry at the axis, integrating the parabolic velocity profile to get the volumetric flow. The equation is valid only for steady, laminar, fully-developed flow of a Newtonian fluid in a circular pipe.

Detailed Explanation :

Hagen–Poiseuille equation

Goal and assumptions
We want the volumetric flow rate  produced by a pressure difference  across a straight horizontal circular pipe of length  and radius . Use these standard assumptions for the derivation:

• Steady flow (no time dependence).
• Incompressible Newtonian fluid (constant density  and viscosity ).
• Axisymmetric flow in a straight circular pipe.
• Fully developed flow (axial velocity profile does not change along pipe length).
• No body forces in axial direction.
• No-slip condition at the pipe wall: fluid velocity equals wall velocity (zero).
• Neglect entrance effects (pipe is sufficiently long so flow is fully developed).

Governing equation (reduced Navier–Stokes)

Under these assumptions the only non-zero velocity component is the axial velocity  which depends only on the radial coordinate . The axial momentum equation (steady, incompressible, axisymmetric, fully developed) reduces to a balance between pressure gradient and viscous diffusion:

Here  is constant along  for steady flow with constant .

Solve the ordinary differential equation

Rewrite:

Divide by  and integrate once with respect to :

Regularity at the centreline requires finite  as . That forces  (otherwise  would be singular). So

Integrate again:

Apply no-slip at the wall :

Thus the velocity profile is parabolic:

Because  is negative for flow in the positive -direction (pressure decreases along the pipe),  is positive.

Volumetric flow rate

Integrate the axial velocity over the pipe cross-section to get :

Substitute :

Evaluate the integral:

so the bracketed integral is . Thus

Replace  by  where  is the pressure drop across length  (positive for flow from 1 to 2). Then

Expressed with diameter :

Other useful results
Mean (average) axial velocity:

Velocity at centreline:

Wall shear stress:

Validity and limitations

Hagen–Poiseuille holds only for steady, laminar, fully-developed flow of Newtonian fluids in circular pipes. It fails if flow is turbulent (high Reynolds number), if fluid is non-Newtonian, for developing flow near the pipe inlet, or for non-circular ducts (different prefactors appear). Surface roughness and entrance disturbances can alter the transition point to turbulence but do not change the derivation assumptions.

Conclusion

The Hagen–Poiseuille equation is a fundamental analytical result giving a simple and exact relation between pressure drop and volumetric flow for laminar, fully developed, Newtonian flow in a circular pipe:

It follows directly from the balance between pressure forces and viscous diffusion and produces a parabolic velocity profile. This law underpins many practical calculations in hydraulics, biomedical flows and microfluidics, while its assumptions must be checked before application.