# Can you use the Hagen-Poiseuille equation for a pipe which radius is in the sub-millimeter region?

As this will depend on Pressure drop $\Delta p$, assume that it does not leave the range from 0 to 100 bar. The Hagen-Poiseuille equation for an incompressible fluid is defined as:

$$\dot{V} = \frac{\pi R^4 \Delta p}{8\eta L}$$

I realize that it wont be applicable for very small (nm) diameters, so this question is in the context of microfluidics. Fluids of interest in this case have a kinematic viscosity of 1 cSt to 10000 cSt.

• You didn't name the substance (though even if you did, I couldn't offer an answer.) Jan 26 '15 at 12:18
• @dcorking So, you're want to know the viscosity of interest? Since it is an incrompressible fluid, this would be the only physical quantity which would change. Of course, if you leave non-newtonian fluids aside. Kinematic viscosities of interest would be between 1 cSt and 10000 cSt. Jan 26 '15 at 12:24
• You're only dealing with a single fluid phase, right? If you had two fluids in contact with each other, surface tension effects would prevent Hagen-Poiseuille from being applicable.
– Paul
Jan 26 '15 at 15:47
• @dcorking Thanks, I look into it. If you transfered her reasoning to this case, the Hagen-Poiseuille equation would be not applicable when you reach diameters comparable to the size of the water molecules. Jan 26 '15 at 16:51
• @Paul Yes, there is only a single fluid phase present. Jan 26 '15 at 16:51

A) Limits of continuum mechanics:

The continuum model of fluid dynamics is valid only till the fluid behaves as a continuous medium. This is characterized by the Knudsen number. The Knudsen number is given by $Kn = \frac{\lambda}{l_s}$, where $\lambda$ is the mean free path and $l_s$ is the characteristic dimension of the channel (diameter in the case of the circular pipe). Non equilibrium effects start to happen if $Kn > 10^{-3}$. Modified slip boundary conditions can be used for $10^{-3} < Kn < 10^{-1}$, and condinuum model completely breaks if $Kn > 1$. (Fun fact: because the distance between two vehicles on a crowded road is much smaller than straight portion of the road itself (length scale in $1d$ flow), we can model the traffic flow with a PDE! However it will not work if there is only one car on a long stretch of road)

Coming back to water, as the water molecules are not freely moving and are loosely bound, we consider the lattice spacing $\delta$ for computing $Kn$. For water $\delta$ is about $3 nm$. So continuum theory will hold good for a tube of diameter, $300 nm$ or larger $^*$. Now this is a good news!

$^*$ Reference: Liquid flows in microchannels

B) Applicability of Hagen Poiseuille equation:

Since your tube is in sub-millimeters range, it is much larger than the minimum diameter required (sub-micrometer)for the continuity equation. However, depending on the shape of cross section of the tube, the results will differ (Link to ref.). Liquid flows are much simpler to analyse since they are characterized by much smaller Reynold's number and velocities. Also density essentially remains constant. So there should not be a problem in considering the theory to be valid. Now since the Hagen Poiseuille flow is derived from the Navier Stokes equations, it follows the assumption of continuity.

If your flow is through a porous medium, you might have to consider effects like electrokinetic effect. There might be other complications in straightforward application of H-P equations to microfluidic flows, but I am unable to comment since do not know much in this field.

C) Some examples

In a report on "microfluidics networking", Biral has used the continuum theory for modeling and simulation (in OpenFOAM) of the microfluidic flows.

Fillips discusses more about the Knudsen number in his paper- Limits of continuum aerodynamics.

This report clearly mentions that HP equation is applicable even to microfluidic flows

This document on PDMS Viscometer gives derivation of HP equation for microfluidic flows.

Finally here is a YouTube video discussing about matrix formalism for solving the Hagen-Poiseuille law in microfluidic hydraulic circuits.

Based on these references, it should be safe to assume that H-P equation can be applied to microfluidic flows. However, experts are welcome to enlighten us in this regards.

Cheers!

• Wow, what a well thought out answer! I knew the knudsen number in the context of vacuum technology, but didn't realize that you can - of course - use it in this case. Jan 27 '15 at 10:02

Same answer as above, "Yes BUT..."

This is from a practical point of view:

Other than HPLC, nearly all "microfluidics" applications are far into the laminar region. Poiseuille's law works very well most of the time.

When going below 100$$\mu$$m dimensions, the problems come.

• lack of stiffness -> violates assumption that fluid into a network segment equals flow out of it. (easy to model)
• thermal effects -> varying viscosity and thermal expansion of fluid and solids. (easy to model)
• violation of the assumption of a single homogeneous phase (the difficult problem, in my opinion).

To expand on the final point, it is mainly from presence of gas or bubbles, but also from "old" fluid not having fully cleared out, if multiple fluids are involved. Focusing on gas/bubbles, it has two effects. First of all, "bulk modulus" again. More significantly, surface tension. It can dominate everything else at low dimension scales and low flows, and can often be intermittent (such as if resulting from dissolved gases vs temperature variation etc), leading to much confusion.