A Shallow water equations evaluation

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The discussion revolves around confusion regarding the momentum equation in shallow water equations (SWEs). The user is attempting to apply product rules to derive equations but encounters difficulties, particularly with the second line of their calculations. They reference a paper that presents a similar issue but struggle to understand the transition between specific equations in that paper. The user questions how expanding terms can lead to a non-conservative set of equations, emphasizing that mechanical manipulation should not alter the foundational assumptions of the equations. Clarification on these points is sought to resolve the confusion.
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Summary:: A little confusion on the momentum equation (I think).

According to Wikipedia (I know, I just need basic resources for now), the conservative SWEs are
:
9b9d481407c0c835525291740de8d1c446265ce2


If I use product rules, I am supposed to get:

6bb10fdfb320a6bc0f4011b08b6b616b2a95929e


For context, note that ρ is a constant and can be taken out (thus canceled out), and η(x,y) = H + h(x,y) (H is a constant).

I have no issues getting the first line. The second line however, I am facing issues. This is what I have:

1638011369713.png


I intentionally did not distribute the derivatives with respect to y so I could see the problem more clearer. Am I missing something here with regards to the expansion of the PDE? Thanks in advance.
 
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Just an update:

I found this paper that actually is the same 'issue' that I am facing, though they just presented it.

This paper:
https://doi.org/10.1063/5.0039545

It stated that:

1638033387994.png


I am having trouble understanding how the second equations have been made (second equation 4, to second equation 5).
 
The wikipedia entry on shallow water equations doesn't make sense to me. How could you, from equations derived using momentum and mass conservation, merely expand terms and get a non-conservative set of equations. No mechanical manipulation of equations changes the underlying assumptions they were derived from.
 
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