- #1
Hypatio
- 151
- 1
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 [itex]\eta\nabla^2\mathbf{u}[/itex]. However, when I try to get this from Eq. 1, I get [itex]2\eta\nabla^2\mathbf{u}[/itex]. I am getting [itex]2\eta[/itex] because
[itex]\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}[/itex]
and then the taking the divergence gives [itex]\nabla\cdot 2\eta\nabla\mathbf{u}=2\eta\nabla^2\mathbf{u}[/itex] for constant viscosity [itex]\eta[/itex]
I suspect that my second step in the first line is wrong. But I don't get it.
Thanks in advance.
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 [itex]\eta\nabla^2\mathbf{u}[/itex]. However, when I try to get this from Eq. 1, I get [itex]2\eta\nabla^2\mathbf{u}[/itex]. I am getting [itex]2\eta[/itex] because
[itex]\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}[/itex]
and then the taking the divergence gives [itex]\nabla\cdot 2\eta\nabla\mathbf{u}=2\eta\nabla^2\mathbf{u}[/itex] for constant viscosity [itex]\eta[/itex]
I suspect that my second step in the first line is wrong. But I don't get it.
Thanks in advance.