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When can the velocity gradient be set to zero?

  1. Jun 14, 2017 #1
    1. The problem statement, all variables and given/known data
    An experimentalist has measured the u-velocity component of a two-dimensional flow

    field. It is approximated by

    u = (1/3)( xy) (y^2)
    It is also known that the v-velocity is zero along the line y=0.

    2. Relevant equations
    ∇V=du/dx+dv/dy (partial derivatives)

    3. The attempt at a solution
    The solution is found by setting du/dx+dv/dy=0 (partial derivatives), solving the differential equation and then using the boundary conditions at v=0, y=0 to find the constant.
    How can we set the velocity gradient equal to zero? Why is it safe to assume that there is no velocity gradient?
     
  2. jcsd
  3. Jun 14, 2017 #2

    mfb

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    The problem statement should state that the fluid is incompressible.
     
  4. Jun 14, 2017 #3
    The velocity gradient is not zero. Do you know the difference between the velocity gradient and the divergence of the velocity? If so, please elaborate.
     
  5. Jun 17, 2017 #4
    Thank you, so if the fluid is incompressible the layers do not have a velocity towards each other? If it was compressible the fluid layers would be able to compress towards each other?
     
  6. Jun 17, 2017 #5
    As I understand the velocity gradient is the velocity that the layers of a fluid move in respect to each other. The divergence of the velocity is kind of the total rate of change of the field, since it is a sum of the components partial derivatives.
     
  7. Jun 17, 2017 #6

    haruspex

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    If the divergence is nonzero there is a net flow in or out of the point. I.e. it is a sink or a source.
     
  8. Jun 17, 2017 #7
    A non-zero divergence also applies to a distributed source or accumulation.
     
  9. Jun 17, 2017 #8
    You have been confusing the gradient of the velocity vector with the divergence of the velocity vector. The divergence of the velocity is not the same as the gradient of the velocity. The gradient is the del operator applied to the velocity vector. The divergence is the del operator dotted with the velocity vector. In post #1, the right hand side of your equation is the divergence of the velocity vector, not the velocity gradient.
     
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