Willingness to pay

Streamline diffusion, given an advection-diffusion equation, refers to all diffusion going on along the advection direction.

Explanation

If we take an advection equation, for simplicity of writing we have assumed $\nabla \cdot 𝐮 = 0$ , and $| | 𝐮 | | = 1$

\frac{\partial ψ}{\partial t} + 𝐮 \cdot \nabla ψ = 0.

we may add a diffusion term, again for simplicty, we assume the diffusion to be constant over the entire field.

D \nabla^{2} ψ

,

Giving us an equation of the form:

\frac{\partial ψ}{\partial t} + 𝐮 \cdot \nabla ψ + D \nabla^{2} ψ = 0

We may now rewrite the equation on the following form:

\frac{\partial ψ}{\partial t} + 𝐮 \cdot \nabla ψ + 𝐮 (𝐮 \cdot D \nabla^{2} ψ) + (D \nabla^{2} ψ - 𝐮 (𝐮 \cdot D \nabla^{2} ψ)) = 0

The term below is called streamline diffusion.

𝐮 (𝐮 \cdot D \nabla^{2} ψ)

Any diffusion orthogonal to the streamline diffusion is called crosswind diffusion, for us this becomes the term:

(D \nabla^{2} ψ - 𝐮 (𝐮 \cdot D \nabla^{2} ψ))