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A few months ago, I tried to tackle the problem of finding the shape, the height and other properties of a puddle (whenever a few mL of some liquid are dropped in a "uniform way" so that the puddle remains approximately circular). Trying to find the height, dimensional analysis with density, gravity and superficial tension gives the "famous" (appears in most of the books about this problem) formula:

$$ H = \sqrt{\frac{\sigma}{\rho g}} $$

But as soon as I found this, I discarded it (at this point I hadn't researched anything about the problem) because it seemed to contradict the experiments I'd done so far! I had dropped a few mL of water and of honey, and, as I expected, the honey puddle was higher than the water one (3 mm vs 1.5 mm approx). But this is contradictory! The superficial tension of honey is similar to that of water, but the density is significantly higher, so the height should be lower.

What does this mean? Is it because the viscosity of the honey makes it "stop" from spilling on the ground and makes it stick on top of the honey that's already on the ground? I've tried a different approach which leads me to a formula that agrees much more with the experiments, but it's completely different from this one and doesn't have the theoretical justification that this one does.

Any suggestions or ideas? Thanks in advance.


EDIT: Here's what I obtained so far and some of the experiments I did (keep in mind that I didn't have much precision due to lack of any sophisticated instruments). (All measurements where taken about 20-25 degrees Celsius)

$$ H = \kappa \sqrt[\leftroot{-1}\uproot{2}\scriptstyle 15]{\frac{\eta^2 \sigma^6}{g^7 \rho^8}} $$ Where $\eta$ is the viscosity, $\sigma$ the superficial tension, $g$ is gravity and $\rho$ is density. The value of the constant is $\kappa = 0.89 \pm 0.24 $. With this I can "predict" the height of a puddle of water, honey, glycerin and alcohol. $$ H_{water}^{theory} = (1.1 \pm 0.4) mm \leftrightarrow H_{water}^{experiment} = (1.5 \pm 0.5) mm$$ $$ H_{alcohol}^{theory} = (0.8 \pm 0.3) mm \leftrightarrow H_{alcohol}^{experiment} = (0.8 \pm 0.5) mm$$ $$ H_{honey}^{theory} = (3.0 \pm 1.1) mm \leftrightarrow H_{honey}^{experiment} = (3.0 \pm 0.5) mm $$ $$ H_{glycerin}^{theory} = (2.4 \pm 0.9) mm \leftrightarrow H_{glycerin}^{experiment} = (2.6 \pm 0.2) mm$$

Also, the formula I've obtained for the shape of the drop is (in two dimensions): $$ H(r) = H_m(1-e^{\frac{tan{\theta_C}}{H_m}(\left|r \right| - R)}) $$ With $R$ the radius of the puddle, $\theta_C$ the contact angle and $H_m$ the maximum height of the puddle (can be obtained by the formula before) and the radius is approximately $R = \sqrt{\frac{V}{\pi H_m}} $. An example with R=10, the angle 64 degrees and the height 2.8.

I have too a formula for the contact angle, but requires extreme precision, so it's not of much use itself.

$$\theta_C = \arctan \left(\frac{\pi R H_m^2}{\pi R^2 H_m - V}\left(1+ \sqrt{\frac{2V}{\pi R^2 H_m} -1} \right) \right) $$

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    $\begingroup$ Your honey probably dries out and gets a skin before it spreads out. Did you measure the surface tension of your materials or are you just using textbook values? I don't think that's much of a precision experiment, either. $\endgroup$ – CuriousOne Jul 24 '16 at 23:30
  • $\begingroup$ are you sure theres not some disclaimer about newtonian fluids there? Honey is hardly an ideal liquid $\endgroup$ – anon01 Jul 24 '16 at 23:32
  • $\begingroup$ I don't really have a lab, I'm just a student in his house, i get the surface tension and the viscosity from internet tables but of course i keep the interval of values that my liquids can have. This is more of an order of magnitud estimate. And about the newtonian fluids thing, as far as I can remember they don't really specify it, and use it quite generally. I've seen it used in actual Phd papers for different approaches with simulations and non-uniform surfaces $\endgroup$ – Rafa Jul 25 '16 at 0:03
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    $\begingroup$ It is highly commendable that you are pursuing your curiosity through experiments. The problem which I see here is that we have no details about the experiments which you have performed, so in trying to explain why they don't agree with theory we are "shooting in the dark." For example : Did you give the honey sufficient time to spread out? Is there any consistency in the results you have obtained? $\endgroup$ – sammy gerbil Jul 25 '16 at 13:31
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    $\begingroup$ I would try various oils. They often have very well-characterised properties as engineers care a lot about this, and are not very variable for the same reason. $\endgroup$ – tfb Jul 25 '16 at 20:25

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