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Curl and Cauchy-Riemann Conditions problem

  1. Nov 23, 2006 #1
    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

    [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]​

    Through this, I was able to verify:

    [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: Nov 23, 2006
  2. jcsd
  3. Nov 23, 2006 #2

    nazzard

    User Avatar
    Gold Member

    Hello Saketh,

    to get

    [tex]
    \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
    [/tex]

    you need to use the fact that div V = 0 for incrompressible liquids.

    This follows from the continuity equation:

    [tex]
    \frac{\partial \rho}{\partial t} + \nabla(\rho\,\textbf{V})= 0
    [/tex]

    and using the fact that the liquid is incompressible, hence the mass density [itex]\rho=const.[/itex]

    Regards,

    nazzard

    P.S.: There's a small typo in your current solution: one \frac command is missing
     
    Last edited: Nov 23, 2006
  4. Nov 23, 2006 #3
    Oh, I thought that incompressible just meant that the fluid flow is behaving ideally, and had no mathematical significance.

    Now that you tell me [tex]\nabla \cdot \textbf{V} = 0[/tex], the answer is obvious. Thanks!
     
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