Which liquids stack, and which just mix?

In one line: the oil ends up on top whatever order you pour, because it never mixes with the others; honey, balsamic and water are all water-based, so they only sit in density order for a while, and stirring turns them into one liquid for good.

Denser sits lower

Density is mass per volume: how much stuff is packed into the same space. When two liquids don't mix, the denser one ends up underneath, because the pair has less energy that way (the heavier-for-its-size liquid gets as low as it can, and pushes the other one up). The numbers the sim uses, all at room temperature (20 °C):

• Vegetable oil 0.92 g/mL. In the US "vegetable oil" is usually soybean oil. The international food standard for named oils (Codex CXS 210) gives soybean oil a relative density of 0.919 to 0.925, which is 0.917 to 0.923 g/mL.
• Water 0.998 g/mL (CRC Handbook).
• Balsamic 1.30 g/mL. Nico's is the thick, syrupy aged kind. The rules for traditional balsamic from Modena say it must be at least 1.24 g/mL at 20 °C, and a study of such vinegars (Falcone and others, 2017) finds their texture changes near 1.32. That rule is for the traditional kind only, so it's a reference point, not a measurement of his bottle. I picked 1.30. Thin supermarket balsamic is less dense (the page says 1.1 to 1.3, "it depends on the bottle"); it would still sink through the oil and sit below the water.
• Honey 1.42 g/mL. It depends on how much water the honey holds: 1.44 g/mL at 13% water down to 1.39 at 21% (White, 1975, in the FAO's beekeeping handbook). I picked a typical 17%.

The glass holds 59 mL (¼ cup) of each liquid. The page gives no amounts and no glass, so the sim borrows the tall glass from Nico's volcano page (Libbey City, 14.3 oz, 3 × 6.1 in): about 29 cm² inside, so ¼ cup makes a layer about 2 cm thick. Every layer in the drawing is its volume divided by that area.

Which ones mix, and why

The oil mixes with none of them. Water molecules cling tightly to each other (hydrogen bonds). Oil molecules can't join in, so the water keeps squeezing them out into their own layer. That's why the oil-to-water line is sharp, and why the oil comes back up after you stir.

Honey, balsamic and water all mix with each other. Honey is sugar with a little water; balsamic is grape sugar and vinegar acid in water, cooked down thick; water is water. Sugar and vinegar acid dissolve in water, so these three dissolve into each other. There's no real line between them, only a blurry zone that slowly gets wider.

So why do they sit in layers at all? Because the denser one is underneath and nothing stirs them. Dissolving on its own (diffusion) is very slow: sugar spreads by about 3 mm in an hour, which is the blurry band drawn between them in the finished picture. (That's sugar in plain water. Inside thick honey or balsamic it spreads far more slowly, so the real band sits mostly on the water side; take it as a rough scale.) The page says it for honey: "Poured slowly, honey can sit underneath for a while." The sim draws that careful, unstirred pour. A splashy pour would stir some of them together at once; the sim doesn't model that.

Pour order doesn't matter here (for where the layers end up)

Honey poured into water goes down as a thick ribbon and settles at the bottom (the page's step 4). Water poured onto honey sits on top. Oil poured first gets pushed up when the others go in under it; oil poured last floats. Try the upside-down order: the layers end the same. The sim draws each pour as a straight stream down to where that liquid settles; really honey goes down as a rope, and water poured through oil breaks into blobs on the way.

That works for these four because of what they are. The oil can't mix with the others. Honey is about ten thousand times thicker than water (about 19 Pa·s; FAO beekeeping handbook, Table 2.3), so poured into water it goes down as a rope and hardly mixes. And water, the only runny one, is also the least dense, so it is never poured on top of something less dense that it could stir into. Pour a runny, denser liquid onto one it mixes with (salt water onto plain water, or thin balsamic onto water) and it falls through in fingers and stirs itself in on the way: then the order matters, which is why density towers are poured densest first, or slowly over a spoon. Even honey poured into water leaves a bit more honey dissolved in the water than the careful order does; the sim draws the same blurry line both ways.

The upside-down order's end picture, with the balsamic puddles on top of the honey, is what you'd see after minutes or more: the honey has to work its way under them. By then the water poured onto the drops will have mixed some of them away. The drops are the one real exception to "order doesn't matter": whether they get held at the oil–water line depends on whether there is an oil–water line when they go in.

Stir it

Stirring breaks the oil into drops, but they float back up and join into one layer again, the same thickness as before: oil and water "un-mix" on their own. The water-based ones don't: they become one liquid, and its density is the volume-weighted average of what went in. With ¼ cup each of honey and water: (1.42 + 1.00) ÷ 2 = about 1.21 g/mL. (That assumes the volumes just add up; real sugar solutions shrink very slightly when mixed.) The size of the stirred-up oil drops and how fast they rise are drawn, not modelled; the total oil stays the same.

The balsamic drops, step by step

Through the oil. A drop sinks because it's denser than the oil, and the oil's drag holds it to a steady speed. The starting drop, 1/4 tsp of balsamic (1.23 mL, a ball 13 mm across), reaches about 17.2 cm/s. In open oil it would reach 21.4 cm/s; in a glass this narrow the walls slow it by about 20% (Di Felice's wall correction, 1996), and the sim includes that for every drop and droplet. (The simple Stokes formula would say 67 cm/s; it only works for slow, small things, and at this size the flow around the drop has momentum, so the sim uses the standard drag curve for spheres, Clift and Gauvin's, as given in Clift, Grace and Weber's textbook.) Syrupy balsamic is thick: about 1 Pa·s (Falcone and others, 2012), about 18 times thicker than the oil (54 mPa·s, Noureddini and others, 1992). A drop that much thicker than what it falls through hardly stirs inside, so it falls like a solid ball: the Hadamard–Rybczynski formula for a liquid drop says only about 2% faster than a solid ball. So the sim treats it as a solid ball.

At the oil–water line: held, or through? To sink into the water wrapped in oil, the drop has to pull the oil–water surface down around itself until the oil pinches off behind it. Pierson and Magnaudet (2018, part 1, section 4) worked out what holds a sphere up at a line like this. Three things: the surface pulls back like a stretched sheet (at most 2π × the drop's radius × the surface's tension; Vella, 2015, says the same); the part of the drop that's down in the water is buoyed up more than it would be in oil; and the oil sitting in the dent above the drop has to be held up too, so it helps. The sim uses their formula (their equation 4.4). Vegetable oil against water has a tension of about 23 to 26 mN/m (Fisher and others, 1985), and bottled oils drift lower with time (Gaonkar, 1989), so the sim uses 20 to 30 mN/m. The smallest drop on the slider, 1/64 tsp (5.3 mm), weighs only about a third of what the line can hold, so it stays.

The drop wears a coat of oil. Where it meets the line, the drop is not bare balsamic: it is wrapped in oil, and it takes that coat down with it. So the sim treats it as a ball of balsamic inside a shell of oil, and works out the hold for the whole coated ball (its full radius, which the line pulls on and the water buoys up), with the push being the coated ball's weight in oil. (The coat is oil, so it adds nothing to that push, but it makes the ball wider, and a wider ball is easier to hold.) How thick is the coat? Nobody measured it for balsamic. The sim uses 1 mm, an estimate from the physics of liquid dragged along by a moving object (Landau and Levich, 1942; de Ryck and Quéré, 1998), which for this oil at these speeds gives about 1 to 3 mm. Pierson and Magnaudet saw spheres drag down oil "of the same order as" their own size; a 1 mm coat on the starting drop is 52% of the balsamic's volume, so the sim is on the cautious side.

Speed helps too: a drop arriving fast dents the line deeper. How much isn't known well. The sim adds its own rough estimate (not from either paper): for the starting drop, its speed adds about a third to its weight. The one study that measured it saw speed lower the size needed by up to about half in some cases. So the cut-off is a band: balls of balsamic somewhere between about 6 and 10 mm across (about 0.11 to 0.5 mL). The top is the strongest tension with no help from speed; the bottom uses that study's biggest help, and it's the least sure part (at the weakest tension it could be as low as 5.5 mm). The coat's thickness moves the band too: with a 0.5 mm coat the bottom drops to about 5.5 mm, with a 2 mm coat the top rises to about 11.5 mm. In the band a drop could go either way. The sim draws the middle numbers: drops up to 1/32 tsp (6.7 mm) held, 1/16 tsp (8.4 mm) and up through.

Drop size matters. Small drops get held at the oil–water line. Where they sit follows from the same balance: the smallest ride high, only dimpling the line; bigger ones sit lower, in a dent they push into the line (drawn to scale, coat included). They stay until the oil film under them breaks, then mix into the water right there, in dark wisps. Big drops are heavy enough to pull through, wrapped in oil. Nico saw both: drops "half vinegar, half oil, stuck in the middle, not moving or only really slowly", and a drop that fell through the water as a ball. The sim doesn't model what a held drop does next: how long it stays; right around the cut-off a real drop can sit a while and then slowly sink; a held drop can sag and slip through later; and held drops near each other drift together (like cereal in milk) and can sink as one bigger drop.

Why 1/4 tsp to start? We don't know how much balsamic Nico dropped, only that his drops were big. The starting drop is a spoonful-scale drip, a quarter of a teaspoon: a ball of balsamic 13 mm across, in a 15 mm coated ball once it has its oil coat. That's a guess, not a measurement. It is well above the whole band, so it goes through at any tension in the range, as Nico's did. A drop that drips off a bottle or spoon on its own is much smaller, about 4 to 6 mm across (Tate's law: a drop falls off when its weight beats the surface tension holding it to the rim), about the slider's smallest step, and most of those would be held. That could be why Nico saw both: drips right around the size where the line can just hold them, some held and some through.

Through the water, still wearing oil. This is what Nico saw with his syrupy balsamic: the drop fell through the water as a ball, still coated in oil, and didn't mix on the way down. Why not? Here's the likely reason (it fits what he saw, but nobody filmed it): before the vinegar can touch the water, the oil between them has to be squeezed out. The water does push on the front of the falling drop: it's the same kind of push as the drop later gives the bottom, though by the end of the fall only about a third as hard (the drop reaches about 14 cm/s of the 25 cm/s it would need for its weight to be fully carried by the water's push). The real difference is time: the fall lasts only about 0.16 s, and squeezing out an oil film takes much longer; drops resting on an oil–water surface wait seconds or more before they merge (Charles and Mason, 1960). At the bottom, the push is focused on a small flat patch and goes on as long as it takes, so the film finally thins and breaks. (Speeds in the water use the same drag curve and wall correction; that wall formula is used at every speed, and at the water's speeds it may slow the drop a few percent too much.)

How much oil rides down? The coat: a shell 1 mm thick, drawn to scale. In Pierson and Magnaudet's study of spheres crossing an oil–water line, most of the oil pulled down pinches off near the sphere and rides on its top; the sim spreads the same oil evenly round the drop instead (a simplification: it keeps the amount, not the shape). The coat makes the falling ball wider and less dense, so it falls about 21% slower through the water than bare balsamic would; its density drops by 10%. The oil taken down leaves the oil layer (you can see it get a little thinner) until it comes back.

At the bottom. The drop stops when the bottom of its coat touches the honey (honey is denser, and Nico's glass had honey: "the oil + water + honey") or the glass. The sim draws the film breaking at once; really it takes a moment. The balsamic spreads into a small puddle about 3 mm thick. It's denser than water, so it stays low and mixes in slowly, from the bottom up. By the plain-water number, sugar would spread out of a puddle that thin in about 4.6 hours; really it takes much longer, because sugar moves far more slowly through the syrup itself.

The oil climbs back. The freed oil gathers into one droplet (about 11 mm across for the starting drop's coat), which is less dense than water, so it floats up at about 14.5 cm/s and rejoins the oil layer in about 0.2 s. That's the page's "Then pieces climb": what climbs is oil, not vinegar. The oil layer ends exactly as big as it started.

With no oil in the glass, the drop has no oil coat and no line to cross; a real one would smear out as it falls, which the sim doesn't model.

Assumptions: room temperature (densities at 20 °C; the oil's thickness was measured at 24 °C), level ¼ cups, spherical drops touched onto the surface from a spoon.