Flux through a sphere

  1. Here's the question:

    A fluid has density 2 and velocity field [​IMG].
    Find the rate of flow outward through the sphere [​IMG].

    So far I've found n, which is

    and F dot n gives z^2.

    I converted to spherical coordinates and z^2 is equal to 4cos^2[phi].

    My integral is set up as:
    4*(int[0-2pi] int[0-pi] (cos^2[phi]*sin[phi]dphi dtheta.

    The first integral is -1/3cos^3[phi] from 0-pi which is 1/3 - - 1/3 = 2/3

    The second integral gives 2/3*2*pi, so the entire thing is 4*2/3*2*pi.

    I thought I was just supposed to multiply that by 2 (the density) but that's not the right answer. Can someone tell me what I did wrong or what I'm supposed to do with the density?

    Thanks a lot.
  2. jcsd
  3. StatusX

    StatusX 2,567
    Homework Helper

    One way to verify the answer is by Gauss' law (aka the divergence theorem). The divergence of the velocity field is seen to be 2, so the flux out of the sphere is 2*(4/3*pi*2^3), which is off by a factor of 4 from what you got. It seems like you forgot the r^2 factor in the surface area element. What you did with the density is correct since it's constant here.
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