# Laplacian term in Navier-Stokes equation

1. Sep 9, 2014

### 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 Eqn.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.

2. Sep 9, 2014

### 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.

3. Sep 9, 2014

### 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 happend to the 2?

What rules am I missing??

4. Sep 9, 2014

### olivermsun

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.

5. Sep 9, 2014

### 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 $\eta\sum\limits_{j}{\frac{\partial^2{u_i}}{\partial^2{x_j}}}=\eta\nabla^2u_i$.

Last edited: Sep 9, 2014