Steps for arriving at this PDE system solution

[kmot]

Hi. I cannot break through this derivation from left to right.
Could somebody help and give me an insight into how this is solved?

u' and h' are functions, everything else is a constant


Many thanks

 

[Deleted member 4993]

Hi. I cannot break through this derivation from left to right.
Could somebody help and give me an insight into how this is solved?

u' and h' are functions, everything else is a constant
View attachment 24013

Many thanks

Please "simplify" the right-hand-side and tell us what you find.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[kmot]

Sorry for not reading the rules throughly.

My original problem was not being able to eliminate all derivatives of "u" after combining both equations. Some "u" term always remained.
Anyway in the meantime I was able to find a solution. Here is the result.
(D/Dt = shorthand for material derivative)

 

[Deleted member 4993]

Sorry for not reading the rules throughly.

My original problem was not being able to eliminate all derivatives of "u" after combining both equations. Some "u" term always remained.
Anyway in the meantime I was able to find a solution. Here is the result.
(D/Dt = shorthand for material derivative)

View attachment 24016

What is the difference between:

u ......and..... \(\displaystyle \overline{u}\)

&

h ......and..... \(\displaystyle \overline{h}\)