Laplacian term in Navier-Stokes equation

[Hypatio]

I am trying to derive part of the navier-stokes equations. Consider the following link:

http://www.gps.caltech.edu/~cdp/Desktop/Navier-Stokes%20Eqn.pdf

Equation 1, without the lambda term, is given in vector form in Equation 3 as \eta\nabla^2\mathbf{u}. However, when I try to get this from Eq. 1, I get 2\eta\nabla^2\mathbf{u}. I am getting 2\eta because

\eta\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)=\eta(\nabla \mathbf{u}+\nabla \mathbf{u}^T)=2\eta\nabla \mathbf{u}
and then the taking the divergence gives \nabla\cdot 2\eta\nabla\mathbf{u}=2\eta\nabla^2\mathbf{u} for constant viscosity \eta

I suspect that my second step in the first line is wrong. But I don't get it.

Thanks in advance.

 

[olivermsun]

You might want to keep the divergence and the terms in the parentheses together to remind yourself that you only sum over the j subscript (which is repeated).

You should notice that the derivatives of the second term in the parentheses add up to the divergence of u.

 

[Hypatio]

I don't see a repeated subscript. Do you mean that \nabla\mathbf{u}^T=\nabla\mathbf{u}? That is in fact my assumption and thus why I get 2*eta, not eta. Is this not true?:

\eta(\nabla\mathbf{u}+\nabla\mathbf{u}^T)=2\eta\nabla\mathbf{u}

if so, what happened to the 2?

What rules am I missing??

 

[olivermsun]

I don't see a repeated subscript.

In the original Eq. 1:
$$\frac{\partial} {\partial x_j} \left[ \eta\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) + \ldots \right]$$ has the subscript $j$ both outside and inside the parentheses.

 

[Delta2]

The i-th component of the force (rewriting equation 1 with \lambda=0 ) is

F_i=\eta\sum\limits_{j}{\frac{\partial^2{u_i}}{\partial^2{x_j}}}+\eta \frac{\partial}{\partial{x_i}}\sum\limits_{j}\frac{\partial{u_j}} {\partial{x_j}}.

The second term is zero cause the divergence of u is zero hence all that is left is \eta\sum\limits_{j}{\frac{\partial^2{u_i}}{\partial^2{x_j}}}=\eta\nabla^2u_i.