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I'm working on a project about Plateau–Rayleigh instability for liquids. But I've a question.

we can examine the fluids with high $Re$ number that the influence of viscosity is negligible, but what about liquids with low $Re$ number like honey some kind of oils with high viscosity?

Does Plateau–Rayleigh instability happens in these kind of fluids too?

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The Reynolds number only gives the relative importance of inertial effects to viscous effects: $$\mathrm{Re}=\frac{\rho u u}{\mu u/l}=\frac{\rho u l}{\mu}$$ It doesn't say anything about the relative importance of viscous effects to surface tension effects which are typically what causes the Plateau-Rayleigh instability. This ratio is known as the Capillary number: $$\mathrm{Ca}=\frac{\mu u/l}{\sigma/l}=\frac{\mu u}{\sigma}$$

I would think that as long as surface tension effects are significant, i.e. $\mathrm{Ca}\ll1$, the PR instability will occur even for low Reynolds number flows as viscosity cannot effectively dampen the surface pertubations.

From this wiki article:

Honey is sufficiently viscous that the surface perturbations that lead to breakup are almost fully damped from honey threads. This results in the production of long filaments of honey rather than individual droplets.

and:

The pitch drop experiment is a famous fluid breakup experiment using high viscous tar pitch. The rate of breakup is slowed to such a degree that only 11 drops have fallen since 1927.

There is your answer on honey, but it doesn't mean that every low Reynolds number flow does not show fluid thread break-up as shown by the pitch drop experiment.

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  • $\begingroup$ so, how can we know whether the viscosity is that high to damp the perturbations or not??? $\endgroup$
    – David 2000
    Commented Dec 17, 2015 at 10:43
  • $\begingroup$ @David2000: I have added some information on the capillary number but i am not enough of an expert to say exactly what the viscosity dependence is. The best i can do is an order-of-magnitude estimation; for $\mathrm{Ca}<0.1$ you can be fairly sure the viscosity won't dampen the pertubations. I suggest looking through the linked wiki article on fluid thread break, perhaps it gives some insight. $\endgroup$
    – nluigi
    Commented Dec 17, 2015 at 10:54

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