I'm currently trying to derive the Navier-Stokes equations in cylindrical coordinates through tensor analysis. I am only struggling with the last term on the right side, which is a vector Laplacian:
$$\nabla^2 \mathbf{u}$$
(1) In tensor notation, with the aid of the covariant derivative, this can be written as:
$$\nabla^2 \mathbf{u}=g^k\nabla_k\cdot g^j\nabla_ju^ig_i$$
(2) The contravariant components of the equation above are then given by:
$$[\nabla^2 \mathbf{u}]^i=g^{jk}\nabla_k \nabla_ju^i$$
Where:
$$g^{jk}=g^j\cdot g^k= \begin{bmatrix}
1 & 0 & 0 \\
0 & \frac{1}{r^2} & 0 \\
0 & 0 & 1 \\
\end{bmatrix}$$
(3) For $i=r$, the contravariant components of the Laplacian can then be expanded as:
$$[\nabla^2\mathbf u]^r=g^{rr}\nabla_r[u^r_{,r}+\Gamma^r_{rk}u^k]+g^{\theta\theta}\nabla_\theta[u^r_{,\theta}+\Gamma^r_{\theta k}u^k]+g^{zz}\nabla_z[u^r_{,z}+\Gamma^r_{zk}u^k]$$
(4)In the equation above, $\Gamma^k_{ij}$ are the Christoffel symbols, defined as:
$$\Gamma^k_{ij}=g^k\cdot g_{i,j}$$
(5) In this coordinate system, the only non-zero Christoffel symbols are:
$$\Gamma^r_{\theta\theta}=-r$$
$$\Gamma^\theta_{r\theta}=\Gamma^\theta_{\theta r}=\frac{1}{r}$$
So:
$$[\nabla^2\mathbf u]^r=\nabla_r[u^r_{,r}]+\frac{1}{r^2}\nabla_\theta[u^r_{,\theta}-ru^\theta]+\nabla_z[u^r_{,z}]$$
(6) Applying the remaining covariant derivatives, I got:
$$[\nabla^2\mathbf u]^r=[u^r_{,rr}-\Gamma^k_{rr}u_k]+\frac{1}{r^2}[u^r_{,\theta\theta}-\Gamma^k_{\theta \theta}u_k-r[u^\theta_{,\theta}+\Gamma^\theta_{\theta k}u^k]]+[u^r_{,zz}-\Gamma^k_{zz}]$$
(7) Replacing the Christoffel symbols and changing the notation:
$$[\nabla^2\mathbf u]^r=\frac{\partial^2 u^r}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2 u^r}{\partial \theta^2}+\frac{u_r}{r}-\frac{1}{r}\frac{\partial u^\theta}{\partial \theta}-\frac{u^r}{r^2}+\frac{\partial^2 u^r}{\partial z^2}$$
(8) Finally, substituting the contravariant velocity components by their physical counterparts:
$$[\nabla^2\mathbf u]^r=\frac{\partial^2 u^{(r)}}{\partial r^2}+\frac{u^{(r)}}{r}+\frac{1}{r^2}\frac{\partial^2 u^{(r)}}{\partial \theta^2}-\frac{1}{r^2}\frac{\partial u^{(\theta)}}{\partial \theta}-\frac{u^{(r)}}{r^2}+\frac{\partial^2 u^{(r)}}{\partial z^2}$$
This is not the correct solution. It should be:
$$[\nabla^2\mathbf u]^r=\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial u^{(r)}}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 u^{(r)}}{\partial \theta^2}-\frac{2}{r^2}\frac{\partial u^{(\theta)}}{\partial \theta}-\frac{u^{(r)}}{r^2}+\frac{\partial^2 u^{(r)}}{\partial z^2}$$
Somewhere along the process I must have done something wrong. Any ideas?
SOLUTION EDIT
I figured out what the issue was that I was treating $u^i_{,j}$ as a vector, when it is in fact a tensor. Because of this, the covariant derivative of it can be expressed as:
$$\nabla_ku^i_{,j}=u^i_{,jk}+\Gamma^i_{kp}u^p_{,j}-\Gamma^p_{jk}u^i_{,p}$$
With this in mind, the process above can be repeated and the correct answer is obtained for all coordinate components. Retaking my previous solution at step (6), and writing the expression properly, I obtained:
$$[\nabla^2\mathbf u]^r=[u^r_{,rr}-\Gamma^k_{rr}u^r_{,k}+\Gamma^r_{rk}u^k_{,r}]+\frac{1}{r^2}[u^r_{,\theta\theta}-\Gamma^k_{\theta\theta}u^r_{,k}+\Gamma^r_{\theta k}u^k_{,\theta}-r[u^\theta_{,\theta}+\Gamma^\theta_{\theta k}u^k]]+[u^r_{,zz}-\Gamma^k_{zz}u^r_{,k}+\Gamma^r_{z k}u^k_{,z}]$$
(7) Replacing the Christoffel symbols:
$$[\nabla^2\mathbf u]^r=[u^r_{,rr}]+\frac{1}{r^2}[u^r_{,\theta\theta}+ru^r_{,r}-ru^\theta_{,\theta}-r[u^\theta_{,\theta}-ru^r]]+[u^r_{,zz}]$$
(8) Finally, substituting the contravariant velocity components by their physical counterparts:
$$[\nabla^2\mathbf u]^{(r)}=[u^{(r)}_{,rr}]+\frac{1}{r^2}[u^{(r)}_{,\theta\theta}+ru^{(r)}_{,r}-\frac{ru^{(\theta)}_{,\theta}}{r}-r[\frac{u^{(\theta)}_{,\theta}}{r}-ru^{(r)}]]+[u^{(r)}_{,zz}]$$
It is now trivial to show that the equation above is correct when comparing it to:
$$[\nabla^2\mathbf u]^{(r)}=\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial u^{(r)}}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 u^{(r)}}{\partial \theta^2}-\frac{2}{r^2}\frac{\partial u^{(\theta)}}{\partial \theta}-\frac{u^{(r)}}{r^2}+\frac{\partial^2 u^{(r)}}{\partial z^2}$$
Best Answer
I figured out what the issue was that I was treating $u^i_{,j}$ as a vector, when it is in fact a tensor. Because of this, the covariant derivative of it can be expressed as:
$$\nabla_ku^i_{,j}=u^i_{,jk}+\Gamma^i_{kp}u^p_{,j}-\Gamma^p_{jk}u^i_{,p}$$
With this in mind, the process above can be repeated and the correct answer is obtained for all coordinate components.