- #1
Saketh
- 261
- 2
Problem
The velocity of a two-dimensional flow of liquid is given by
[tex]
\textbf{V} = \textbf{i}u(x, y) - \textbf{j}v(x, y).
[/tex]
If the liquid is incompressible and the flow is irrotational show that
[tex]
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
[/tex]
and
My Work
I evaluated [itex]\nabla \times \textbf{V} = 0[/itex] through a determinant, and ended up with this expression:
Through this, I was able to verify:
I could not verify the other expression. How can I verify the other expression - I've tried everything I can think of. It seems simple, but I am missing something.
Thanks in advance.
The velocity of a two-dimensional flow of liquid is given by
[tex]
\textbf{V} = \textbf{i}u(x, y) - \textbf{j}v(x, y).
[/tex]
If the liquid is incompressible and the flow is irrotational show that
[tex]
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
[/tex]
and
[tex]
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
[/tex]
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
[/tex]
My Work
I evaluated [itex]\nabla \times \textbf{V} = 0[/itex] through a determinant, and ended up with this expression:
[tex]\textbf{i}\frac{\partial v}{\partial z} + \textbf{i}\frac{\partial u}{\partial z} - \textbf{k}\left ( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial z} \right ) = 0
[/tex]
[/tex]
Through this, I was able to verify:
[tex]
\frac{\partial u}{\partial y} = -\frac{\partial{v}}{\partial x}
[/tex]
\frac{\partial u}{\partial y} = -\frac{\partial{v}}{\partial x}
[/tex]
I could not verify the other expression. How can I verify the other expression - I've tried everything I can think of. It seems simple, but I am missing something.
Thanks in advance.
Last edited: