Soy Sauce

I have a bottle of soy sauce with a small, about [latex\]3mm[/latex] in diameter, hole on its top, covered with a lid. More or less, it looks like this

If I open the lid and then turn it upside down, a few drops of soy sauce runs out through the hole and, almost immediately, the flow stops! If I then turn it back upside up and invert it again, some more runs out.

In less than a second some of you may think that the explanation must be related to viscosity or surface tension. However, Mr. Further Investigation says “it’s not that simple, this effect was demonstrated using quite aqueous soy sauce”. I am sure that, with such a low viscosity, the viscous force will not be enough to slow down the flow considerably. Surface tension force seems to be a plausible answer, but let’s see how much it can do. From the shape of the soy sauce-air boundary when equilibrium is achieved, we know that the water-plastic adhesion (the perimeter of the hole is made of plastic) must be larger than soy sauce’s cohesion. Hence, if the weight of the soy sauce cannot be sustained, the first thing that will happen is that the soy sauce-air surface boundary will be torn out, due to the lack of cohesion force, as opposed to soy sauce near hole perimeter loses its grip. Let’s check out how much weight such cohesive force can sustain. Equilibrium can occurs if the following equation is satisfied

P\pi r^2=2\pi r\gamma \cos\theta

[latex\]\rho ghr=2\gamma\cos\theta[/latex]

Where [latex\]P=\rho gh[/latex] is the hydrostatic pressure due to the soy sauce’s weight, assuming the pressure of gas inside the bottle doesn’t differ from the atmospheric pressure.  [latex\]r=1.5 mm[/latex] and [latex\]\theta[/latex] are the radius of the hole and the angle of contact.[latex\]\gamma\approx 30 mN/m[/latex] and [latex\]\rho=1200 kg/m^3[/latex] are the soy-sauce air surface tension and the density of soy sauce found in google. Using [latex\]\theta=0[/latex]  just for order of magnitude checking, we get that the height of soy sauce that can be sustained by surface tension is only about [latex\]3 mm[/latex]. So clearly this is not enough to sustain the whole weight of soy sauce in the bottle. There must be something else that helps top support the weight. We seem to be careless in writing the first sentence of this paragraph, we did not provide any reason why the pressure of gas in the bottle must be equal tot he atmospheric pressure.

            It turns out that the pressure of gas inside the bottle must change significantly due to Boyle’s Law. When the bottle is turned upside down, the weight of the soy sauce wins against the cohesive force as calculated above, and some soy sauce comes out. As the soy sauce is coming out, a small amount of volume is missing from the bottle. Since remaining volume of soy sauce in the bottle does not change, the air inside the bottle must fill in for the missing volume. The air’s volume must increase and its pressure decreases. Thus, a pressure difference has been created. There is less pressure inside the bottle than outside. Note that this pressure difference can only be maintained if air does not bubble up through the hole. Which means surface tension also plays an important role and the hole radius has to be small enough to ensure this. The pressure of gas inside the bottle before a few drops runs out must be around the magnitude of the atmospheric pressure, which is extremely huge. So, even a slight change in volume is enough to give a considerable change in pressure. This must be enough to support most of the soy sauce’s weight.

Quantitatively it goes as follows. Let’s apply Boyle’s law to the air trapped in the bottle for one inversion, taking the bottle as cylindrically-shaped with length [latex\]L[/latex]. If just after being inverted, the bottle contains soy sauce up to the height [latex\]h[/latex], and the height falls to [latex\]h-x[/latex] soon after that, then we have

P_{atm}(L-h)=[P_{atm}-\rho g(h-x)](L-h+x)

[latex\]x=\frac{ghL\rho}{P_{atm}}(1-\frac{h}{L})[1-\frac{gL\rho}{P_{atm}}(1-\frac{2h}{L})][/latex]

For a half filled soy sauce bottle with [latex\]L=2h=20 cm[/latex]. The expression above gives x=1.2mm. A few drops are more than enough!

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