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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 afterall.

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If you break the velocity into two components, the normal component crosses the surface and the other one does not.

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