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Divergence Operator on the Incompressible N-S Equation

  1. Mar 4, 2015 #1
    Hello All,
    If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation:

    $$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$

    In 2D cartesian coordinates (x and y), I am supposed to get:

    $$\nabla ^2P = -\rho \left[ \left( \frac {\partial u} {\partial x} \right) ^2 + 2 \left (\frac{\partial u}{\partial y} \frac{\partial v}{\partial x} \right ) + \left ( \frac {\partial v}{\partial y} \right )^2 \right ] $$

    Where does this term come from $$2 \left ( \frac{\partial u}{\partial y} \frac{\partial v}{\partial x} \right )$$?

    Any guidance would be helpful. Thanks!
  2. jcsd
  3. Mar 5, 2015 #2
    First, I think your starting equation is wrong: inside square brackets you have a scalar, and then you take a dot product?
    Try with ##\nabla^2P = - \rho \nabla \cdot (V \cdot \nabla)V##
  4. Mar 5, 2015 #3
    The thing in brackets in the original post is not a scalar. ∇V is the velocity gradient tensor. The original expression has the same meaning as your representation.

  5. Mar 5, 2015 #4
    Thank you!
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