Conservation law form of Navier Stokes Equation

[aerograce]

I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes,
first, the density term with the pressure gradient dropped out.
and second, du^2/dx seems to be equal to udu/dx.
Why is it so? I attached my notes here for your reference.

 

[Chestermiller]

I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes,
first, the density term with the pressure gradient dropped out.
and second, du^2/dx seems to be equal to udu/dx.
Why is it so? I attached my notes here for your reference.
View attachment 109122
View attachment 109121

This is pretty straightforward to do. Integrating by parts, $$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=u\frac{\partial u}{\partial x}+\frac{\partial (uv)}{\partial y}-u\frac{\partial v}{\partial y}$$Next, adding and subtracting ##u(\partial u/\partial x)## gives:
$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=2u\frac{\partial u}{\partial x}+\frac{\partial (uv)}{\partial y}-u\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)$$
But, from the continuity equation, the last term in parenthesis in this equation is zero. So$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=2u\frac{\partial u}{\partial x}+\frac{\partial (uv)}{\partial y}=\frac{\partial u^2}{\partial x}+\frac{\partial (uv)}{\partial y}$$
The rest of the derivation is straightforward.