Is My Solution to the Momentum Conservation Equations Correct?

[Sabra_a]

Homework Statement

In textbooks one can find two forms of momentum conservation
equations. For example, for two-dimensional flow for U component
of flow velocity along axis x these two forms can be written as

and

It is a general practice in computational fluid dynamics (CFD) to
solve momentum equation in the form (2) for computational
efficiency, stability and convergence.
Using the mass conservation equation for variable density flows

demonstrate that these two forms of the momentum conservation
equations are equivalent.

Relevant Equations

momentum conservation equations
mass conservation equation

I attached a PDF file where it clearly show the question and I showed my solution because trying to type it here will be quite hard
I want to check if my solution is correct

 

[berkeman]

I attached a PDF file where it clearly show the question and I showed my solution because trying to type it here will be quite hard

https://www.physicsforums.com/help/latexhelp/

 

[dRic2]

looks good, even though you could have used a nicer notation ;)

 

[Sabra_a]

thank you, but do you have any advice on how to write it better

 

[dRic2]

It just a suggestion, it doesn't really matter as long as you get the concepts right.

BTW in most textbooks the vector notation is employed (using $Nabla$ and $\mathbf v$ instead of the sigle components).

A part from that there is nothing really to say.

 

[Sabra_a]

Thank you!