I'm testing myself on my knowledge from this book by taking the material derivative of velocity in cylindrical coordinates:
$$\frac{D\mathbf u}{Dt}=\mathbf u\cdot \nabla \mathbf u$$
Which, in tensor notation, can be written as:
$$\frac{\partial u^ig_i}{\partial t} + u^jg_j\cdot g^k\nabla_k u^i g_i$$
Which can be simplified to:
$$\frac{\partial u^ig_i}{\partial t} + u^j\delta^k_j\nabla_k u^i g_i$$
And further:
$$\frac{\partial u^ig_i}{\partial t} + u^j\nabla_j u^i g_i$$
Finally, the contravariant components can be written as:
$$\frac{\partial u^i}{\partial t} + u^j(u^i_{,j}+\Gamma^i_{jk}u^k)$$
The expression above yielded the correct expressions for the $r$ and $z$ coordinates. For the $\theta$-coordinate, however, I get the following expression:
$$\frac{\partial u^\theta}{\partial t}+u^r(\frac{\partial u^\theta}{\partial r}+\frac{u^\theta}{r})+u^\theta(\frac{\partial u^\theta}{\partial r}+\frac{u^r}{r})+u^z(\frac{\partial u^\theta}{\partial r})$$
Which can be simplified to:
$$\frac{\partial u^\theta}{\partial t}+u^r\frac{\partial u^\theta}{\partial r}+u^\theta\frac{\partial u^\theta}{\partial r}+\frac{2u^r u^\theta}{r}+u^z\frac{\partial u^\theta}{\partial r}$$
The physical components to this velocities are:
\begin{align}
u^{(r)} & =u^r \\
u^{(\theta)} &= ru^\theta \\
u^{(z)} &= u^z
\end{align}
However, putting the physical components back into the last expression, I end up with:
$$\frac{1}{r}\frac{\partial u^{(\theta)}}{\partial t}+\frac{1}{r}u^{(r)}\frac{\partial u^{(\theta)}}{\partial r}+\frac{u^{(\theta)}}{r^2}\frac{\partial u^{(\theta)}}{\partial r}+\frac{u^{(r)} u^{(\theta)}}{r^2}+\frac{u^{(z)}}{r}\frac{\partial u^{(\theta)}}{\partial r}$$
Which is almost the definition for the material derivative, except that it is divided by $r$ for some reason. Any ideas where I might have gone wrong?
Best Answer
I believe I figured out the problem, but it would be nice if someone could confirm. What I have calculated above is the contravariant component of the material derivative expressed in terms of the physical velocity components. However, it does not represent the physical components of the material derivative. To get the physical components of the last expression, it needs to be divided by $|g^\theta|$ which is $1/r$, which gets rid of the problem. The reason this wasn't an apparent issue for the $r$ and $z$-coordinates is that $|g^r|=|g^z|=1$.