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Math Help - Conservative fields - line source

  1. #1
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    Conservative fields - line source

    We were given some practice questions from our last lecture but from what I could tell our prof did not touch on the subject and also the book only has one brief example about the topic and I really have no clue how to do the question.

    In 3 space filled with an incompressible fluid, we say that the z axis is a line source of strength m if every interval (gradient)z along that axis emits fluid at a volume rate dV/dt=2*pi*m*(gradient)z. The fluid is then spread out symmetrically in all directions perpendicular to the z axis. Show that the velocity field of the flow is

    v=(m/x^2+y^2)(xi+yj)

    Again I have no clue really even how to start. I have done some vector field questions not at all like this and some other questions asking if a given function is conservative and to find its potential. Any help is really appreciated.
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  2. #2
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    Re: Conservative fields - line source

    First, it should be easy to see that a vector of the form a(xi+ yj), for any number, a, points directly from (0, 0) to (x, y) so does give a "radial flow", directly away from the z-axis. Second, I think you have written the result incorrectly. In what you have the multiplier (my "a" above) is \frac{m}{x^2}+ y^2 when I am sure you mean \frac{m}{x^2+ y^2}. Finally, I think you have the equation wrong. You have "dv/dt= 2pi (gradient)z" when I think you mean \frac{\partial v}{\partial t}= 2\pi m grad v= 2\pi m (\frac{\partial v_x}{\partial x}i+ \frac{\partial v_y}{\partial y}j). That reduces to the two equations \frac{\partial v_x}{\partial x}= 2\pi m\frac{\partial v_x}{\partial x} and \frac{\partial v_y}{\partial t}= 2\pi m \frac{\partial v_y}{\partial y}
    Last edited by HallsofIvy; April 1st 2013 at 08:49 AM.
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  3. #3
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    Re: Conservative fields - line source

    Firstly, yes you are right. I meant to write the multiplier as, m/(x^2+y^2). However the question specifically says that the axis emits fluid at a volume rate dV/dt = 2*pi*m*grad(z). I am just confused at what are the steps in this question as we haven't touched on fluid flow stuff so I am worried that it could be on our final coming up without us really knowing about it. The book also is not much help.
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