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Below is the equation of convective flux of an arbitrary property that is carried by particles. When particles move they carry their property along with them. Given a fixed surface in space, in order to calculate the amount of property crossing a surface S we just use the formula. Note that $\Psi$ is the amount of the property per unit mass, $\rho$ is the density of particles, $v$ is the velocity and $n$ is the normal vector of the surface.

$$\Phi =\int \rho\ \Psi\ v.n\ dS\ $$

Derivation of the formula is taken from

I don't understand why do we have to include the normal component of the velocity vector of the particles ($v.n$). I understand that the particles are crossing the surface, but why can't they just cross the surface while keeping their original direction?

By the way, the surface is not meant to be real, it is just virtual... that's why the particles are able to cross it after all.

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  • $\begingroup$ It's really just a fudge to get vectors and scalars to behave the way we want. Flux is a scalar. Velocity is a vector. The simplest way to get a useful scalar flux quantity is to take a dot product of the field velocity vector with the surface's unit normal vector. Now we have a bunch of scalar quantities inside the integral which makes for a simpler calculation and a compact notation. It also makes it very easy to choose arbitrary closed surfaces where most of the patches have zero flux. $\endgroup$ – Phil Sweet Mar 31 at 18:00
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The flux of a quantity is by definition the rate per time per area of a given quantity normal to the area in question. If you use the original direction of the particles and this direction is not normal to the surface in question, then you are not evaluating the flux through the surface. To use the original particle direction, you would need to use a surface orthogonal to this direction. This thread puts it well: https://physics.stackexchange.com/questions/346144/why-surface-normal-is-used-while-defining-flux-through-an-open-surface The normal component of the quantity is the only component that pushes quantity through the surface. The other components play no role in transporting particles through the normal surface. Mathematically, I believe this stems back to rules of the dot product.

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