Why mixing two liquids in a channel where nothing is turbulent

Why small channels will not mix on their own, and what is done about it.

Two liquids entering a narrow channel side by side will run down it for a long way still recognisably separate, because the only thing capable of mixing them at that scale is slow molecular diffusion across the boundary between the two streams, and a channel narrow enough to avoid turbulence offers that boundary nothing faster to work with.

Toothpaste stripes in a laminar channel

Squeezing two colours of toothpaste onto the same brush produces two neat stripes side by side, and they stay separate until a finger or the brushing itself works them together. Holding them near each other blends nothing, because nothing is forcing material across the boundary. Two liquid streams flowing side by side down a narrow channel are in the same position, with plain molecular diffusion as the only process available to smear the boundary away. Looking down such a channel under a microscope with one stream dyed shows exactly this: a sharp colour edge down the middle near the inlet, blurring only gradually into a soft band further along, and staying two-toned for far longer than anyone used to stirring a cup of tea would guess.

Mixing in a kitchen or a full-sized pipe happens through turbulence, the chaotic tumbling of a fast-moving fluid that tears streams apart and interleaves them, which is why stirring coffee blends it within a second or two. Whether a flow is turbulent or moves in smooth, non-crossing layers, called laminar flow, depends on the flow's speed, the channel's width and the fluid's resistance to flowing, combined into a single figure known as the Reynolds number. Channels inside a small-volume manifold are narrow enough, and their flows slow enough, that this figure sits firmly in the laminar range, so the turbulence a kitchen tap relies on is never available at this scale.

Five seconds, or five centimetres

Diffusion moves molecules from high concentration to low purely by random motion, and the distance it covers grows only with the square root of time, so covering twice the distance takes four times as long. A small dissolved molecule in water takes about five seconds to diffuse across a tenth of a millimetre. Since the streams are moving forward all that time, the wait converts directly into channel length: at a centimetre per second, a straight channel would need about five centimetres before diffusion had finished, which can be longer than the manifold block the channel is cut into. A channel ten times wider would need a hundred times longer, closer to eight minutes. The square-root rule is the whole difficulty in one line, because it punishes every increase in width twice over, and a designer cannot make molecules wander faster without changing the liquid or heating it, neither of which is usually allowed.

Why the coarse plumbing still mixes

Widen the channel enough, or push the flow fast enough, and the Reynolds number climbs back into the turbulent range, where two streams meeting fold into each other as common sense expects. The wider tubing carrying bulk reagent from a reservoir can therefore mix well at a junction, while the manifold's internal channels, narrower by a factor of ten or more, stay laminar throughout. Two halves of the same instrument can be governed by different physics purely because of scale, a trap for anyone who has only designed plumbing at the larger, more forgiving size.

Because a straight channel cannot mix anything within a sensible length, manifold designers change the geometry so that diffusion has less distance to cover. Folding, splitting and recombining a flow, or forcing it along a winding path, shortens the distance any molecule has to travel before meeting liquid it has not yet met. Raising the flow speed to force turbulence is rarely an option in a device whose small channels exist to handle a small, precisely controlled volume, so the geometric trick carries the whole job, and the next article in this set takes it up in detail.