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Let us assume there is a closed room with 2 people, only one of them is smoking cigarette.

What equation would describe the smoke from the cigarette spreading? Is it “diffusion”? If so, what are the parameters?

Is there a technique or device that I can use or build that allows me to control the direction that the smoke travels in? Something like a vacuum that sucks all the smoke even if it is far away.

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This is covered in the standard convection-diffusion type of equation:

$$ \frac{\partial C}{\partial t} + \vec{u} \cdot \nabla C= D \nabla^2 C$$

Where $C$ is the smoke concentration, and $D$ is the diffusion coefficient of smoke.

While the air may be stagnant initially, it will come to move due to buoyancy effects, thus giving some non-zero air velocity $\vec{u}$, enhancing the spreading of the smoke. Smoke from the cigarette has a higher temperature than the surrounding, giving it lower density, which makes it rise. As it rises, it cools down, which also decrease the net force on the smoke parcel. At the same time hotter smoke from below hits the smoke that is more stagnant. This is typically what you see happening when the smoke comes directly from the cigarette.

The easiest way to let the smoke go away is opening a window. This will create draft, i. e. a high air velocity across the room, convecting all the smoke with it. This is why you might want to open a window when your house smells bad.

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