Divergence Operator on the Incompressible N-S Equation

[C. C.]

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!

 

[matteo137]

First, I think your starting equation is wrong: inside square brackets you have a scalar, and then you take a dot product?
Try with $\nabla^2P = - \rho \nabla \cdot (V \cdot \nabla)V$

 

[Chestermiller]

First, I think your starting equation is wrong: inside square brackets you have a scalar, and then you take a dot product?
Try with $\nabla^2P = - \rho \nabla \cdot (V \cdot \nabla)V$

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.

Chet

 

[matteo137]

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.

Thank you!