I am trying to determine the force it would take to fully empty a syringe using a long nozzle, that contains a high-viscosity fluid.

Specifications I'm using:


I'm trying to determine the force through Hagen-Poiseuille equation:

Hagen-Poiseuille equation

However the results I am getting from physical testing show are very different. I have tested with 3 different flow rates, and difference between them is minimal, while Hagen-Poiseuille predicts it will be substantial, especially at higher flow rates:

Graph 1

Physical testing showed results around 1kN lower than expected. I must be missing something. Syringe and nozzle are both highly polished - is there a way to take it into account?

Any thoughts would be welcome.

  • $\begingroup$ What is the density of the fluid? $\endgroup$
    – NMech
    May 5 '21 at 11:36
  • $\begingroup$ I would expect such formulas are very sensitive to tiny differences in just about every parameter - smoothness of surface, exact composition of the material itself (e.g. there are many variations of "stainless steel" ) . Probably the biggest troublemaker is viscosity as a function of temperature. $\endgroup$ May 5 '21 at 12:23
  • 1
    $\begingroup$ Did you check if the flow through the needle is laminar flow - the Hagen-Poiseuille equation is only valid for laminar flow. I personally wouldn't think the Hagen-Poiseuille equation would work for these small diameters based on what I learned about fluids, but I could be wrong. $\endgroup$
    – J. Ari
    May 5 '21 at 13:20
  • $\begingroup$ what's the exact fluid? what's the nozzle outlet diameter? what's the angle? $\endgroup$
    – Pete W
    May 5 '21 at 15:47

I will make two assumptions from your question:

  1. horizontal orientation of nozzle + piston in order to discard hydrostatic pressure, since a vertical orientation would modify the force required to move the piston due to the fluid's weight;

  2. the graph showed is obtained by experimental measurements.

We can estimate the Reynolds number $Re=\frac{\rho \frac{Q}{A} D}{\mu}$ considering $\rho \approx1000$ $kg/m^{3}$ and its order of magnitude is way lower than 1, indicating a laminar flow. @Pete W also brought this observation, ruling out surface roughness. We can also consider a long nozzle/pipe, since the ratio of length/diameter is higher than $\frac{Re}{48}$ (https://www.tec-science.com/mechanics/gases-and-liquids/energetic-analysis-of-the-hagen-poiseuille-law/).

However, if the fluid is initially at rest and the piston exerts force to generate flow, you must consider the acceleration phase, which the steady Hagen-Poiseuille model doesn't consider. Using assumption 2), the experimental graph shows a higher forces at initial times (transient or non-steady phase) to generate the flow, i.e. to accelerate and overcome the friction forces. After some time, the force is fully balanced by the friction in order to maintain the same flow rate, reaching the steady state. The graph shows it presenting an asymptotic behavior pointing out a force near $700 N$. Thus, if I correctly understood the problem, I would say that the disagreement is due to the transient phase which is not considered by the Hagen-Poiseuille equation.


Hard to say. Although the pressure is very high (up to 5kN/0.001m^2 = 5MPa), the velocity is super slow and Reynolds number is very low, deep laminar region. I wouldn't worry too much about roughness.

I assume you have ruled out friction, by running the syringe dry and finding negligible force.

Possible explanation: diameter variation

1kN error on 5kN reading is 20%. With 1/D^4 scaling, lets look for a possible 5% diameter error. It could be coming from elasticity, but only if you have a particularly soft/flexible material.


  • material=FEP, modulus=0.54GPa, poisson=0.45, OD=12mm x ID=9.5mm x LG=210mm
  • I am getting about 0.0075/MPa diameter expansion (thick walled cylinder internal pressure)
  • => 3.75% at 5MPa, which is in the ballpark for the resulting error in back pressure


  • material=Polypropylene, modulus=2GPa, poisson=0.33, OD=12mm x ID=9.5mm x LG=210mm
  • I am getting about 0.002/MPa diameter expansion (thick walled cylinder internal pressure)
  • => 1% at 5MPa, which seems too low to be a plausible explanation

  • possible explanation: flowrate measurement/control error? 20% seems large
  • possible explanation: viscosity changes due to dissolved gas
  • possible explanation: temperature coefficient of viscosity
  • possible explanation: viscosity changes at high pressures
  • possible explanation: non-newtonian (e.g. gel like, shear thinning) fluid... would check that too, with biological/industrial/cosmetic/food/medical fluids especially

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