# Is it possible to calcualte the yield-stress of a fluid by measuring the smalles bubble that rises through it?

Consider a yield-stress liquid in gravity. I assume that the buyouncy of a small gas-bubble will not be enough to overcome the yield stress (so the liquid doesn't behave liquid), thus leaving the bubble trapped. Is this so in theory, or is there a flaw in my thinking?

Sme argue that there are no yield-stress liquids in sense of one stress, below wich there is no yield, but rather a very high apparent viscosity. Would this imply that even miniscule bubbles will rise, but very slowly?

Lastly, have there been tries to measure viscosity or apparent viscosity or some measure for yield stress from rising bubbles?

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AFAIK yield stress can be reasonably defined only for solid materials. How can you deform liquid plastically? –  Pygmalion May 29 '12 at 13:13
@Pygmalion: many liquids have a viscosity that rises with decreasing shear. Shampoo is a good example. A yield stress means the viscosity rises to infinity at a finite stress. There is some debate about whether this ever really happens: see my answer for details. –  John Rennie May 29 '12 at 14:01
@JohnRennie OK, I am aware that we have yield stress in terms of rheology, but yield stress is defined as a point when material starts to behave like liquid and not opposite. Or am I missing something? –  Pygmalion May 29 '12 at 14:49
Assuming there's no hysterisis (not always a safe assumption!) the yield stress is the same whether you define it as the lowest stress above which a solid behaves like a liquid or the highest stress below which a liquid behaves like a solid. Industrial rheometers normally start at the high stresses and ramp the stress down, so we think of the yield stress as the point at which the liquid stops flowing. Which is fine, but as I've said below for most non-Newtonian fluids the yield stress probably doesn't exist i.e. the liquid never behaves like a solid. –  John Rennie May 29 '12 at 15:09
So I guess I agree with you :-) –  John Rennie May 29 '12 at 15:09

Yes, your method is a perfectly good way of measuring viscosities. It's a bit limited because it's hard to measure the bubble diameter exactly, but assuming you can do this you can calculate the viscosity of the fluid using Stokes' law.

However note that I haven't mentioned yield stress. That's because yield stress is a slippery thing to define. Indeed a rheologist friend of mine, Howard Barnes, maintains that there is no such thing as a yield stress. The problem is that for non-Newtonian fluids the viscosity often rises very rapidly as the shear decreases and it may get too high for you to measure it. So how do you you tell the difference between a true yield stress and a viscosity too high for your equipment to measure?

If you shake up your non-Newtonian liquid to get some bubbles in it then come back in an hour or so you'll see some apparently stationary bubbles, and you can use these to calculate a "yield stress". But if you come back a bit later you may find the bubbles that you thought were stationary have moved a bit. You could wait a bit longer, but you can't run the experiment for very long because Ostwald ripening will be changing the bubble sizes as you watch. That means there's inevitably a limit to how high a viscosity you can measure. Unless the yield stress is below this limit you won't be able to measure it.

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great answer, but by my understanding of ostwald ripening the ripening happens slower in higher (apparent) viscosities, so why the upper limit to measurable viscosity? –  mart May 29 '12 at 14:19
Suppose your standard error in measuring the particle position is 1mm, then to measure the particle velocity to 95% accuracy you need it to move 3mm. Suppose you can't do the experiment for more than one day (e.g. because of Ostwald ripening) then the smallest velocity you can measure is 3mm/day or $3.5 \times 10^{-8}$m/sec. If you feed this velocity into Stokes' law you'll get the highest viscosity you can measure. If the viscosity is higher than this it will look as if you have a yield stress when actually the viscosity is still finite. –  John Rennie May 29 '12 at 14:34
You can argue that maybe you can measure more accurately than 1mm, or wait longer than a day, but unless you can measure with infinite precision or wait and infinite time there will always be an upper limit to the viscosity you can measure. This is a real issue in industrial rheology. In the lab it's hard to do really long experiments because it's hard to keep things stable for long periods. However the bottle of shampoo may sit in the warehouse for several months. –  John Rennie May 29 '12 at 14:38
I was thinking along another line. Say you point a camaera at your liquid and measure (and count) bubbles as they surface. In an ideal yield stress liquid (Bingham plastic) you'd have a lower limit to the bubble size that actually surface, in a Herschel-Bulkely fluid (I guess?) you'd still have a distribution that disvavours small bubbles because they'd struggle with a higher apparant viscosity. The result would be harder (but my guess is not impossible) to interpret quantitativly for a fluid that's not an ideal yield stress liquid. –  mart May 30 '12 at 8:15
You'd still have a problem because you wouldn't know if the small bubbles were simply taking too long to reach the surface rather than never reaching the surface. I feel like I'm being a bit negative, but even with top end kit it's hard to prove the existance of a yield stress, so it's not surprising that it's even harder with a simpler method like bubble motion. –  John Rennie May 30 '12 at 8:28