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Plotting steamlines?

  1. Aug 28, 2012 #1
    1. The problem statement, all variables and given/known data
    The velocity field in a flow is given by V = (x^2)*yi + (x^2)*tj. (a) Plot the streamline through the origin at times t = 0, t = 1, and t = 2. (b) Do the streamlines plotted in part (a) coincide with the path of particles through the origin? Explain.

    i&j are directional vectors.

    2. Relevant equations

    Providing this just in case you guys haven't heard of stream lines. "A streamline is a line everywhere tangent to the velocity vector at a given instant."

    3. The attempt at a solution

    dx/u=dy/v (equation from streamline from Fluid mechanics textbook)
    where u=x^2 * y
    v=(x^2)*t

    After plugging those in and differentiating I get (y^2)/2=tx+C
    where C is a constant.


    My problem is I don't know how to go about plotting "the streamline through the origin at t=0,1,2."

    Do I plug the 3 t's into the equation 3 times and plot those 3 equations I get? If so, how do I get C?
     
  2. jcsd
  3. Aug 28, 2012 #2

    vela

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    Different values of C correspond go different streamlines. You have to choose C so that the equation holds for x=0, y=0.
     
  4. Aug 28, 2012 #3
    Oh I see since the question is through the original that would make C=0 right for this equation?

    Which would give me (y^2)/2=tx. Then I just plug in t=0,1,2 into this equation and plot the 3 equations?
     
  5. Aug 29, 2012 #4

    vela

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    Yup.
     
  6. Aug 29, 2012 #5
    That would give me
    the equations
    y=0
    y=(2x)^.5
    y=2*x^.5

    The solutions seem to only have the streamlines plotted in the first quadrant. But doesn't the 2 square root functions exist in both the 1st and 4th quadrant? The first equation y=0 would also exist in all 4 quadrants.
     
  7. Aug 29, 2012 #6
    In fluid mechanics, the concept of streamlines only applies to steady state flow. This is a non-steadystate problem. In such problems, you can solve for the pathlines of particles, but the pathlines change with time. For your problem, the pathlines are determined by:

    dx/dt = vx = x2y

    dy/dt = vy= x2t

    You need to solve this coupled set of ODEs for sets of initial values of x, y, and t.

    Chet
     
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