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I have a long straw ("small" diameter tube of constant cross-section) hanging vertically above me. The straw is filled with liquid (water). I blow a finite volume of air (gas) into the lower end of the straw. The input of gas into the lower end of the straw is such a way that the gas spans the entire diameter of the straw. The gas lifts the liquid column above it, but does not "bubble up through" the liquid. The gas and liquid are now in hydrostatic equilibrium; a column of gas at the bottom, with a liquid column sitting on top of the gas.

I now imagine a "funnel" or "cone" shaped straw, i.e., one where the diameter of the straw is "large" at the top $(D_{top}>>D_{bottom})$ and gradually gets smaller towards the bottom $(\Delta D/\Delta L<<1)$, where the diameter of this straw at the bottom is same as the previous scenario. Following the same scenario as previously described, I input air into the lower (smaller) end of the straw. I would imagine that as I put a little bit of gas volume in, the entire liquid column would be displaced upwards, and the gas would remain under the liquid column. But I would also imagine that if I put enough air in, enough so that the top of the gas column reaches a certain (larger) diameter of the straw, the gas at this interface will break away as a bubble and rise through the liquid column.

Is this intuition correct? If "yes" what are the underlying physics? What determines if the gas bubbles up through the straw or not?

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A tube full of fluid through which air bubbles can rise to the top at the same time the liquid falls out the bottom is said to support two-phase flow. So, your question can be translated as, What conditions are required to support two-phase flow?

Two-phase flow is controlled by the balance between surface tension forces in the tube and gravity, pulling on the fluid in the tube. The magnitude of the surface tension forces is set by the radius of curvature of the liquid in the tube, which therefore involves the tube's inside diameter. Two-phase flow requires strong gravity, low surface tension, and a large tube diameter. Small tubes inhibit it, as do weak gravity and high surface tension.

In the case of small enough tubes, weak enough gravity and high enough surface tension, the liquid and air have to take turns occupying the tube because they cannot squeeze past one another in the tube. In this case it is also common to see air bubbles alternate with globs of fluid along the length of the tube; in this case, the surface tension forces dominate over gravity.

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  • $\begingroup$ Thank you. Is there any equation or resource you can point me to? When you say fluid, do you mean liquid? Also, I'm not sure if earth's gravity is considered "strong" gravity, but the following assumes 1g gravity. When a gas (hydrocarbon) well is drilled (i.e. "large diameter tube"), is it the tall liquid column on top of the gas (i.e. "large" liquid hydrostatic head or pressure) that prevents the gas from bubbling up through the liquid or is it because the well diameter is still too small or is it due high surface tension (or some combination thereof)? $\endgroup$
    – Armadillo
    Sep 17, 2020 at 18:30
  • $\begingroup$ There is a nondimensional number (like the Reynolds number, for example) which allows you to predict this but I cannot remember its name or the formula for it, which I once knew in a previous lifetime as an engineer. $\endgroup$ Sep 17, 2020 at 19:12
  • $\begingroup$ Perhaps it is the Bond number? $\endgroup$
    – Armadillo
    Sep 17, 2020 at 19:54
  • $\begingroup$ could be... but i do not know. $\endgroup$ Sep 17, 2020 at 22:50

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