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term in the change of energy of the fluid:
Δ tpd,
which shows up in the rescaled linear system as an addition of V i,j,k d i,j,k to
the right-hand side, where V i,j,k is the volume fraction (i.e., scaled between
0 and 1) of the fluid inside the grid cell ( i, j, k ).
Note that adding an arbitrary divergence control inside a closed
domain—one without free surfaces—may lead to an incompatible linear
system: if we constrain the fluid to maintain its total volume (the con-
tainer's volume) but require it to expand or contract in the interior, we
end up with no solution. Therefore it is imperative to enforce the compat-
ibility condition, as discussed at the end of Chapter 4 for the right-hand
side.
Divergence control can be used much more liberally than just to ac-
count for mass balance in thermal expansion. For example, Feldman et
al. [Feldman et al. 03] introduced the technique for modeling a large class
of explosions: look for more on this, and other techniques that control di-
vergence, in Chapter 7. A constant, or time-varying, positive divergence
can be added to the source volume of smoke to make it billow out more;
negative divergence inside a target region of space can cause the smoke to
be sucked up into the target.
Before leaving the subject of controlling smoke, it should be pointed
out that there are, of course, other methods than modifying the diver-
gence. Users may define force fields as additional body forces to coax the
air to blow a particular way; these are simply added to the momentum
equation much like the buoyancy terms we began with. Fattal and Lischin-
ski [Fattal and Lischinski 04] provide an interesting class of force fields that
automatically move smoke to a desired target shape—a particularly useful
idea for dealing with characters made out of smoke.
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