# Conservative fields - line source

• Apr 1st 2013, 07:47 AM
lytwynk
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.
• Apr 1st 2013, 08:45 AM
HallsofIvy
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 $\displaystyle \frac{m}{x^2}+ y^2$ when I am sure you mean $\displaystyle \frac{m}{x^2+ y^2}$. Finally, I think you have the equation wrong. You have "dv/dt= 2pi (gradient)z" when I think you mean $\displaystyle \frac{\partial v}{\partial t}= 2\pi m grad v= 2\pi m (\frac{\partial v_x}{\partial x}i+ \frac{\partial v_y}{\partial y}j)$. That reduces to the two equations $\displaystyle \frac{\partial v_x}{\partial x}= 2\pi m\frac{\partial v_x}{\partial x}$ and $\displaystyle \frac{\partial v_y}{\partial t}= 2\pi m \frac{\partial v_y}{\partial y}$
• Apr 1st 2013, 06:10 PM
lytwynk
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.