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Non-rotating vector fields with non-zero Curl

  1. Sep 8, 2012 #1

    ShayanJ

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    Gold Member

    In some texts the author tries to interpret operations like Curl.
    Some say the curl of a vector field shows the amount of rotation of the vector field
    But some of them say,if you put a wheel in a fluid velocity field which is like the vector field at hand,if it can rotate the wheel,then it has non-zero curl.
    For example the field [itex] \vec{F}=z \hat{y} [/itex] does not rotate by itself but if it shows a fluid velocity field,it can rotate wheels inside it and it has non-zero curl
    But I'm just uneasy with the idea
    maybe seeing more vector fields like this,ones which do not seem to rotate but have non-zero curl,make me better But I can't find more of such fields
    Do you know one?
    Thanks
     
  2. jcsd
  3. Sep 8, 2012 #2

    HallsofIvy

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    The point is that the field itself is not "rotating" because all motion is in the x- direction. If this were a flow of water, every molecule would flow in a straight line. But the speed of the the water would vary and, if fact, all water molecules above the xy-plane would flow in the positive y direction, all molecules below would flow in the negative y diretction. That is, if we were to put a "water wheel" with axis along the x-axis, water flowing by above the xy-plane would push it in the positive y direction, water flowing by beneath would push it in the negatve z direction and so it would rotate clockwise- as seen looking at it form x< 0. That is an argument why curl f= [itex]\nabla f[/itex], which, here is
    [tex]\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial }{\partial y} & \frac{\partial}{\partial z} \\ 0 & z & 0\end{array}\right|= -\vec{i}[/tex]
    is non zero even though every water molecule is moving in a straight line.

    And, it works the other way. Imagine that you are holding a pan of water in front of you. Very slowly, so that you do not "slosh", move that pan around in a circle in front of you. Now, every water molecule in the pan is moving in a circle, relative to the ground but there is no "rotary motion". If you were to write out the equations for the motion of the water, in x, y, z terms, you would get curl 0.
     
  4. Sep 9, 2012 #3
    Think about what you are asking.

    The curl of any vector (a,b,c) is another vector (r,s,t)

    for zero curl this requires r=s=t=0 since for any non zero r,s or t the modulus of the curl vector is √(r2+s2+t2)

    However the velocity field in a fluid boundary layer is not (usaually) rotating, but has non zero curl. It may contain rotating vortices though.
     
    Last edited: Sep 9, 2012
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