# Poiseuille's Law Derivation From the Navier-Stokes Equations

### Key Takeaways

• Poiseuille’s law relates the steady laminar flow rate through a capillary.

• The flow rate is based on a net force calculation, where it is assumed the only relevant forces are externally applied pressure and viscous forces.

• Poiseuille's law can be derived using the basic definitions of a pressure-induced force and viscosity.

Blood flow in arteries follows Poiseuille’s law

Laminar flow has several useful relationships and empirically observable results that nicely illustrate the drivers of fluid flow. One of these is Poiseulle’s law, which describes the flow rate of a fluid through a capillary of circular cross-sectional area. The material properties of the fluid also determine the flow rate due to the viscous forces they exert on the fluid during motion. Poiseuille’s law is primarily involved in discussions of capillary flow at high pressure, where the flow rate is driven in the laminar regime entirely by a pressure gradient. In this article, we’ll briefly show how to derive Poiseuille’s law from first principles, namely from the Navier-Stokes equations for an incompressible fluid.

## Poiseuille's Law Assumptions

Poiseuille’s law essentially makes a statement about the laminar flow rate through a capillary with circular cross-section, namely that the volumetric flow rate is proportional to the fourth power of the capillary’s radius. As one would expect for laminar flow, the volumetric flow rate is found to be inversely proportional to the dynamic viscosity of the fluid and directly proportional to the pressure gradient that drives the laminar flow. As such, the volumetric flow rate can be summarized in a simple equation as shown in the graphic below.

Poiseuille’s law describing the volumetric flow rate of a fluid

This flow rate result can be observed under the following assumptions or conditions:

• Steady flow: Here, we only consider steady laminar flow without changes in any of the material parameters over time and space. This leads to the 2nd important assumption in Poiseuille’s law (below).
• Incompressible flow: Poiseuille’s law is only valid for incompressible viscous laminar flows, although an analogous formula could be derived for a compressible fluid assuming the adiabatic index of the fluid was known.
• Pressure and viscosity dominate: In this type of capillary flow, we assume that gravity or any other forces that might drive transverse flow are much smaller than viscosity and pressure. Therefore, we can ignore gravity or any other body forces that might drive fluid flow.

Under these two simple assumptions, we can start from the first principles and directly derive the flow rate according to Poiseuille’s law.

## Poiseuille’s Law Derivation

Poiseuille’s law derivation begins from the Navier-Stokes equations. Note that we cannot start from the Euler equations, as we must consider the viscosity. According to the Euler equations, the flow rate would be infinite due to zero viscosity, something which cannot be observed in reality. Therefore, we must keep the viscous forces in the Navier-Stokes equations in the forthcoming derivation.

The Navier-Stokes equation of motion for an incompressible fluid is as follows:

Here, the material derivative is reduced to an ordinary time derivative because the density is constant everywhere, thus the flow rate is divergenceless. We’ve kept the viscosity term to account for viscous forces acting on the fluid, and the sum of external forces F are left ambiguous for the moment.

According to the above assumptions, we are dealing with steady flows and negligible body forces, so the time derivative and F are both zero. When these are dropped from the equation, we arrive at the following reduced equation of motion:

Going further requires breaking the above equation into its components and solving these individually, as is the case in standard fluid dynamics problems. As we are concerned with flow in a pipe or tube, we work in cylindrical coordinates and break the above equation into its components. As we are addressing flow along the axis of a pipe driven by a pressure gradient, with no forces along the radial and polar directions, one would naturally expect the radial and angular components to be zero.

Applying the continuity condition (divergenceless flow) reduces the above components to a purely radial derivative along the z-axis:

Here, the viscous force has been rewritten using the dynamic viscosity by eliminating the density for brevity. From here, we can integrate along the radial direction and evaluate the flow at the boundary using the no-slip condition to arrive at the following intermediate result:

One additional integration step over the cross-sectional area gives the result that is best known as Poiseuille’s law:

Poiseuille’s law for the volumetric flow rate of a fluid in laminar flow

An alternative method relies on dimensional analysis, which requires assuming the form of the above flow rate equation in terms of a power function. Using the standard SI unit definitions for each of the above quantities, one easily arrives at the final result for Poiseuille’s law.

### Extending Poiseuille’s Law to Compressible Flow

The above set of equations only applies to an incompressible fluid, which implies a flow involving a liquid. However, the primary result for Poiseuille’s law can be extended to the case of an ideal gas undergoing slow (isothermal) compression. The results for Poiseuille’s law in this case is:

Poiseuille’s law for the volumetric flow rate for an ideal gas

If compression is fast and the material properties change due to heating, the viscosity may change and the above result may not hold any longer. Accounting for any deviations from the above Poiseuille’s law derivation in a complex system requires CFD simulations to determine the flow rate and its dependence on pipe cross-sectional area.

In more complicated systems, use the complete set of CFD simulation features in Omnis from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented in Cadence’s CFD simulation tools.