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The picture below shows a piece of equipment dentists use, called a "water scaler". As you can see, there's water spraying out of it through a very small hole right in front of the bend.

I've been told that while the water is spraying, bacteria can flow up through that hole and into the equipment due to "capillary forces". We're not talking about bacteria entering the hole after the spray is turned off, but when it's actually spraying.


Capillary forces can make liquid flow without assistance, opposite to external forces such as gravity. I understand (broadly speaking) how it works when looking at trees, clay bricks etc, but I can't understand how those principles apply here.

I thought it might have something to do with a combination of the no-slip condition and turbulence, but I still can't see how.

Can bacteria flow up through the hole while the water is spraying?

enter image description here

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  • $\begingroup$ Your question surprises me as I thought that these scalers clean teeth by using ultrasound and the debris was just washed away. Where did you get the information about the back flow of bacteria? $\endgroup$ – Farcher Feb 1 '18 at 9:30
  • $\begingroup$ @Farcher you're right. There are both sonic and ultrasonic scalers, where the main difference is power. The sonic ones vibrates at 3-9 kHz, while the ultrasonic ones vibrate at >25 kHz. The water (or air) is used to wash away debris. I received it as second hand information, passed down from the manufacturer (of similar equipment, not the one in the picture). $\endgroup$ – Stewie Griffin Feb 1 '18 at 9:56
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It would be extremely unlikely.

A typical bacteria is about 1 µm diameter - they come in all kinds of shapes, but let's assume a "spherical bacteria" (this is the microscopic equivalent of the spherical cow).

The drag force depends on the Reynolds number. Recall that

$$\rm{Re} = \frac{u\ell}{\nu}$$

Where $u$ is the velocity, $\ell$ the "typical length scale" (1 µm), and $\nu$ the kinematic viscosity (about $8.9\cdot 10^{-7} m^2/s$ for water at 25°C). For such a small object, the Reynolds number will be very small and flow around the bacteria will most likely be laminar.

This means that the drag is given by Stokes' equation:

$$F = -6\pi \eta r \mathbf{v}$$

Here, $\eta$ is the dynamic viscosity ($8.9\cdot 10^{-4} Pa\cdot s$) - so $F=- 8.4\cdot 10^{-9} \mathbf{v} ~\rm{N}$.

With a mass of $5\cdot 10^{-16}$ kg, the slightest difference in velocity (between the bacteria and the liquid) will immediately result in an enormous acceleration. In other words - there is no way the bacteria can "swim upstream".

That leaves the question - could there be a water flow in the opposite direction that could entrain the bacteria? It would be quite hard to deliberately design a nozzle that could do that - you can safely assume that this is not happening here.

The final question you can ask: what about the boundary layer? With viscous flow, the liquid at the boundary is stationary ... could the bacteria "crawl along" that?

Even if we assume a parabolic velocity profile, there is a significant velocity gradient at the wall of the nozzle (where the velocity is lowest). Quite apart from the fact that the flow is most likely turbulent, given the small diameter of the nozzle it seems likely that there is significant water velocity even at 1 µm from the wall (and that's how far the bacteria would "stick out" into the flow). That is almost certainly enough, given the above calculation, to prevent "crawling upstream along the wall".

I can think of lots of things that can go wrong at the dentist's office - I don't see how this can be one of them.

But I have been wrong before.

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