I'm trying to write out the 3D incompressible Navier-Stokes equations in cylindrical coordinates but got stuck. The Navier-Stokes equations: $$u_t – \nu\Delta u + u\cdot \nabla u + \nabla p=0$$ $$\nabla \cdot u =0 $$
Here both $u$ denotes the velocity vector field and $p$ is the pressure.
We consider the cylindrical coordiantes: $x=r\cos \theta; y=r\sin \theta; z=z$. I know then $$\Delta=\frac{1}{r}\frac{\partial}{\partial_{r}}(r\frac{\partial}{\partial_{r}}) + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial z^2} $$. So the velocity components are $(u_{r}, u_{\theta},u_{z})$.
My question is: why is the following $r$ momentum equation true?
$$\frac{\partial u_r}{\partial t} + u\cdot \nabla u_{r}-\frac{1}{r}u_{\theta}^2+p_{r}=\nu(\Delta u_{r}-\frac{u_{r}}{r^2}-\frac{2}{r^2}\frac{\partial u_{\theta}}{\partial \theta})$$
I have no idea where the term $-\frac{1}{r}u_{\theta}^2$ comes from. And for the term on the right hand side, what are the two terms besides the laplacian operator?
Cylindrical coordinates in PDE
analysiscalculuscylindrical coordinatespartial differential equationsvector analysis
Best Answer
Your confusion seems to stem from thinking $\mathbf{e}_r,\mathbf{e}_\theta$ are constants. They are not, and so when differentiating you need to take account of their change too.
Remember the operator $\nabla$ is $$ \nabla=\mathbf{e}_r\frac{\partial}{\partial r}+\mathbf{e}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+\mathbf{e}_z\frac{\partial}{\partial z} $$ and the vector laplacian is defined as $$ \nabla^2 \mathbf{A}=\nabla(\nabla\cdot\mathbf{A})-\nabla\times(\nabla\times\mathbf{A}). $$ In cylindrical coordinates, we have \begin{align*} (\mathbf{u}\cdot\nabla)\mathbf{u} &=\left(u_r\frac{\partial}{\partial r}+u_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+u_z\frac{\partial}{\partial z}\right)(u_r\,\mathbf{e}_r+u_\theta\,\mathbf{e}_\theta+u_z\,\mathbf{e}_z)\\ \end{align*} and note that $\frac{\partial}{\partial\theta}\mathbf{e}_\theta=-\mathbf{e}_r$, giving the extra term $-\frac1ru_\theta^2$ that you see in the LHS.
Similarly, when you expand the vector laplacian, you get $$ (\nabla^2\mathbf{u})\cdot\mathbf{e}_r = \nabla^2 u_r -\frac2{r^2}\frac{\partial u_\theta}{\partial\theta}-\frac{u_r}{r^2}. $$