FLOW: The Ink Is Not the Fluid

Erase every visible trace of ink. Has the fluid stopped?

It is an easy question to answer incorrectly, because the ink is usually the only part we see. The carrier becomes confused with the thing it carries. FLOW separates them in its state: velocity says how material moves; dye concentration says how much tracer occupies a location.

Two fields, two jobs

Velocity is a vector: in this two-dimensional model it has a horizontal and a vertical component. Concentration is a scalar. A cell can contain almost no visible dye and still have nonzero velocity. Conversely, a bright region can sit in a fluid that is momentarily still.

The passive-tracer assumption matters. The dye is carried by the flow but does not push back on it. There is no extra buoyancy merely because a pixel is bright. A force input may change velocity and a source may add dye during the same gesture; those two operations remain conceptually distinct.

Move the time control, then hide the dye. The small marker continues following the same prescribed translation. This is an exact teaching example on a unit-periodic line, not a run of the FLOW solver. Its advantage is that the answer is available independently: position changes by speed multiplied by time, wrapped around the domain.

Advection is not diffusion

Advection transports a quantity with the velocity field. Diffusion smooths differences between nearby values. A uniformly translating dye pattern can move without broadening; a diffusing pattern can broaden without a net translation.

For passive concentration c, the model reads:

change of c + transport by u = κ × spatial diffusion of c + dye sources

Here κ is dye diffusivity. The viscosity ν belongs to the momentum equation. Both can smooth a field, but they act on different fields. Turning up ν does not mean the code secretly turns up κ, and neither parameter is a license to blur the rendered image and call it physics.

In the current periodic implementation κ is 0.00005. That is a model parameter, not a claim about ink in a particular real liquid. The series uses model lengths and times unless a physical calibration is explicitly supplied.

A gesture has an end

A common interface misunderstanding is to treat continued motion as continued forcing. A gesture can add momentum for a bounded interval. After the force ends, the state need not return to rest immediately. The velocity already present continues evolving under transport, viscosity and the incompressibility constraint.

That is what makes a coasting interval useful. Observe after the input has stopped; otherwise you cannot tell whether motion persists because of the state or because you are still driving it. The internal Make a Flow chapter checks this distinction with measured nonzero speed after its input history ends.

A screenshot alone cannot tell you when the force stopped. For a recorded run, the timeline and method must supply that information. For a live experiment, a useful interface needs a pause state and an input history, not just a moving picture.

A concentration value is not a speedometer

Suppose the brightest patch stretches into a long filament. You have learned something about the distribution of dye. You have not yet learned the maximum speed. The fluid may be fast where the dye is scarce, or may carry a bright patch nearly uniformly.

The distinction becomes practical when changing a display. A color adjustment can reveal a faint filament without changing concentration. A tracer method can change concentration without changing the velocity method. These are different interventions and need different checks.

The public recorded preview lets you read a fixed example and its guide. The chapter selection does not rerun the input. Use it to identify what is recorded, then carry the question into a field readout rather than guessing from brightness.

Hold onto this: dye makes motion visible; it does not define the motion. Before interpreting a change, name the field that changed.

Next: why viscosity smooths momentum, while ink keeps its own parameter.