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We are given a 2-D velocity flow incompressible, irrotational of the form:

--> ^ ^ ^ ^

V = u i + v j = [4y -x(1+x)] i + y(2x+1) j

and we are asked to find the flow potential which obeys the Laplace Eq. for 2-D incompressible, irrotational flow: dΦ = u dx + v dy

in other words:

∂Φ

---- = u

∂x

∂Φ

---- = v

∂y

I integrated the first one and then the second one and compared the two functions and combined the terms, but at the end the Φ does not satisfy the first equation only the second one.

Another technique, I integrated the first function with respect to x and Φ is expressed as 4xy - x^2/2 - x^3/3 + f(y) = Φ (x,y)

now I differentiate with respect to y and equate it to v:

4x - f'(y) = y(2x+1) which solves to f(y) = xy^2 + y^2/2 -4xy + C

Plug it in the above expression and get:

Φ (x,y) = xy^2 + y^2/2 - x^2/2 - x^3/3 + C

now the first partial diff. eq is not satisfied but the first is.

Can someone explain what is wrong here?

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# Help needed to find the flow potential function

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