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Let's assume a close room with 1-2 people who only one of them smoking cigarette. What is the equation describe the smoking spreading? is it diffusion? what are the parameters is so?

Is there a way/ device I can build or put that make the smoke go in some specific way, like a vacuum that will suck 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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