SurfaceKinetics docs and form

[KulaginVladimir]

Proposed changes

From discussions with @RemDelaporteMathurin, I suppose that there are mistakes in the variational form for SurfaceKinetics BC and API docs.

The current docs states that the BC for solute species is:
$-D \nabla c_\mathrm{m} \cdot \mathbf{n} = -\lambda_{\mathrm{IS}} \dfrac{\partial c_{\mathrm{m}}}{\partial t}- J_{\mathrm{bs}} + J_{\mathrm{sb}}.$
If $\mathbf{n}$ is the outward normal vector of the boundary, at steady-state 1D, this should yield: $D\dfrac{\partial c_\mathrm{m}}{\partial x}= -J_{\mathrm{bs}}+J_{\mathrm{sb}}$, what contradicts Pick & Sonnenberg. I assume, the correct form is:
$-D \nabla c_\mathrm{m} \cdot \mathbf{n} = \lambda_{\mathrm{IS}} \dfrac{\partial c_{\mathrm{m}}}{\partial t}+J_{\mathrm{bs}} - J_{\mathrm{sb}}.$

To check the implementation, here is a simple MMS test, based on the approach of @RemDelaporteMathurin

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Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.55%. Comparing base (1add5fc) to head (b3dc5e3).
Report is 113 commits behind head on main.

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[RemDelaporteMathurin]

Hey I'm just looking at the signs of terms when we have a negative gradient at x=0.

Then we should have $J_{\mathrm{sb}}>J_{\mathrm{bs}} $. So in the following expression:

$-D \nabla c_\mathrm{m} \cdot \mathbf{n} = J_{\mathrm{bs}} - J_{\mathrm{sb}}.$

The first term is negative, and the RHS is also negative. Sounds good to me!

[RemDelaporteMathurin]

Thanks for this fix.

When we take care of #798 maybe it would be beneficial to add a short derivation showing that, at x=0, steady state, then we have $D \frac{dc}{dx} = J_{\mathrm{bs}} - J_{\mathrm{sb}}$ which is what Pick & Sonnenberg have in Equation X

Also, if we are happy with the MMS test, it might be good to add it to the CI here?

[KulaginVladimir]

We can discuss the MMS. Personally, I don't like that it's defined on $t\in[0,5]$.

[KulaginVladimir]

@RemDelaporteMathurin, I added here (also in this repo) a new MMS. Let me know what you think of it.

[RemDelaporteMathurin]

Looks good to me! Nice to have a MMS case for this!

Feel free to merge