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Problems with divergence

  1. Jan 16, 2013 #1
    Hello,

    I am new to calculus, and am having problems with divergence, I was reading something to explain the physical interpretation of divergence and i got stuck in the very first part.

    it says that if we have a small volume dxdydz at the origin, and that a fluid flowing into this volume from the positive x-direction per unit time, the the rate of flow in is
    = ρvx|x=0 = dy dz,
    where ρ is the density at (x, y, z), and vx is the velocity of the fluid in the x-direction, what does "|x=0 = dy dz" part mean.

    Thanks Alot.
     
  2. jcsd
  3. Jan 16, 2013 #2

    Erland

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    Science Advisor

    Re: Divergence

    "|x=0" probably means that the function, here ρvx, should be evaluauted at x=0. So they imagine the volume element as a cuboid with one vertex at the origin and the sides as dx, dy, and dz in the posotive directions. One face of the cuboid is then contained in the plane x=0.

    The flux into the volume through this face is then ρvxdydz, so something seems wrong in what you wrote anyway.

    Btw, you wrote "from the positive x-direction", but I assume that you meant "along the positive x-direction", so that positive vx is directed to the right.
     
  4. Jan 16, 2013 #3
    Re: Divergence

    yes what i meant is "along the positive x-direction"

    but about the flux into the volume, i agree that it should be 'ρvxdydz' but i am pretty sure this is what the book says 'ρvx|x=0 = dy dz', i think its a mistake.

    I have another question please, it then says that the flux out of the opposing face is
    ρvx|x=dx dydz
    which i understand but then it equates this equation with the following
    [ρvx + ∂ (ρvx)/∂x dx] dydz
    which i do not understand, how was the partial derivative introduced and why?

    thanks,
     
  5. Jan 17, 2013 #4

    Erland

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    Science Advisor

    Re: Divergence

    Yes, it is a typo. It should be ρvx|x=0dxdy.

    The second expression is a linear approximation of the first one.

    Recall that (ρvx|x=dx - ρvx|x=0)/dx ≈ ∂(ρvx)/∂x|x=0.
     
  6. Jan 17, 2013 #5
    Re: Divergence

    i think it is talking about mass flow rate... density X velocity X area=flow rate..or more precisely mass flow rate..it says rho X V X x such that at x=0 area is dy dz..for differential element the area perpendicular to flow in x direction is dydz...
    the second thing is taylor's theorem is being applied here..which means when fluid has flowed a length dx its mass flowrate has changed depnding upon dx..that is why patial derivative is introduced here..
     
    Last edited: Jan 17, 2013
  7. Jan 17, 2013 #6
    Re: Divergence

    Ok I get it now, thanks guys.
     
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