FLOW: What Viscosity Changes—and What It Does Not

Two swirls begin alike. One loses its motion faster. It is tempting to say that its ink is “thicker,” then treat every later visual difference as a direct reading of viscosity. That shortcut merges three questions: how momentum changes, how dye is transported, and how either field is displayed.

A cleaner starting point is a flow simple enough to solve exactly.

A wave that explains itself

Consider a unit-periodic shear flow. Horizontal velocity varies with vertical position; vertical velocity is zero. Start with u(y,0) = sin(2πy). Because the flow has no x dependence and no vertical motion, its nonlinear self-advection vanishes. Viscosity then damps the mode without changing its shape:

u(y,t) = exp(−4π²νt) sin(2πy)
v(y,t) = 0

This is a special solution under specific assumptions, not a formula for every swirl. Its simplicity is its value: it gives us an independent answer against which an implementation or intuition can be checked.

Start at t = 0. Both lines coincide regardless of the chosen viscosity. Advance time. Larger ν produces a smaller amplitude in this example. Now make B equal to A: the difference disappears. The useful control is not dramatic; it tells you the experiment respects equality.

Energy is a different curve

Kinetic energy is quadratic in speed. If the mode's amplitude halves, its energy becomes a quarter, not a half. The explorer reports both ratios so that a visual impression of “half as much motion” cannot quietly become a numerical claim.

The periodic incompressible energy balance gives a broader guide. With no forcing, viscosity dissipates kinetic energy through velocity gradients. That does not establish a pointwise ordering between two arbitrary nonlinear flows with different viscosities. A local speed, a concentration peak or a vorticity value need not follow the same simple ordering as the amplitude of this exact shear mode.

This distinction matters in an article as much as in code. “Higher viscosity damps this mode faster” is a defensible statement here. “Higher viscosity makes every pixel move more slowly at every time” is not what the example demonstrates.

Keep the dye parameter still

In FLOW's viscosity comparison, the intention is to alter ν in chamber B while holding dye diffusivity κ fixed. If κ also changed, the dye could spread differently for two reasons. If the color scale changed, the pictures could differ without either physical field changing at all.

The current periodic backend also uses three common scalar transport subcycles in both branches. A branch-specific numerical schedule could introduce another difference into an experiment meant to change only viscosity. Sharing that schedule does not make the method exact; it removes one avoidable confound.

The public recording is historical v1. The current conservative-dye implementation is v2. Those labels describe numerical methods, not two values of viscosity. A comparison between v1 and v2 belongs in the conservation story, where changing the method is the intervention being investigated.

Predict before revealing

Before moving the time slider, predict what should happen under three changes. Changing the line style should leave every reported value untouched. Setting the two viscosities equal should make both solutions coincide. Increasing time with positive ν should reduce this unforced mode's amplitude.

These are different tests. One tests presentation independence, one tests the equal-parameter control, and one tests the analytic dynamics. A collection of attractive screenshots cannot replace them.

For broader algorithmic background, GPU Gems chapter 38 introduces field-based fluid calculations. FLOW's actual numerical method must still be read from its own implementation and records, rather than assumed to match that historical tutorial.

The intervention has a name: change viscosity, keep the other conditions fixed, then state precisely which observable you measured.

Next: the state that must be copied before two futures can be compared.