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They introduce a driving force term that causes the fluid to carry the smoke towards a particular target and they also add a smoke gathering term which prevents the smoke from diffusing too much due to numerical disipation.
The core concept behind this technique is using the gradient of a blurred density field of a target shape as direction vectors to advect the velocity field of the smoke.
The resulting vector field looks like all the vectors are pointing towards the target shape.Because the start and end shape might have different densities or the smoke might lose denisty along the way,density is adder as it interpolates between the two shapes as can be seen in Figure 4.