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Navier-Stokes Equation

  1. Sep 30, 2009 #1
    I was wondering if someone could help me this Navier-Stokes Equation.

    f[(δv/δt) + v.Dv] = -DP + Dt + f

    Could someone maybe explain the symbols and what it means.
    I'm not sure but I think Navier-Stokes equations describe fluid motion.

    (The P could be ρ. I'm not too sure)

  2. jcsd
  3. Sep 30, 2009 #2
    You are correct that the equations describe fluid motion. I have a beautiful proof that solutions always exist in three dimensions, but unfortunately it is too big for this marginal comment.
  4. Oct 1, 2009 #3

    Andy Resnick

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    I can't completely parse what you wrote, but some of it I can decipher:

    The term in [] (not sure what that 'f' is doing there), is the "total" or "convective" derivative. It simply means that a spatial quantity (the velocity field vector 'v') is allowed to vary both in time and in space. I am assuming 'D' is a nabla (del) operator.

    The term DP could be the gradient of pressure term, but the Dt term is normally the divergence of the stress tensor, so those two terms are a little ambiguous. The stress tensor consists of both an isotropic part (the pressure) and the off-diagonal antisymmetric components (the shear stress). The final 'f' is used if there is an external body force: gravity, centripetal forces, electromagnetic, etc. etc.

    The Navier-Stokes equation you wrote is nothing more than ma=F for a continuous material. If ma(or []) = 0, then you have conservation of momentum.
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