I am starting with the following reduced form of the Navier-Stokes equations in polar coordinates.
$$u_r\frac{\partial u_r}{\partial r}+\frac{u_\theta}{r}\frac{\partial u_r}{\partial\theta}-\frac{u_\theta^2}{r}=-\frac{1}{\rho}\frac{\partial p}{\partial r}+\nu\left(\nabla^2u_r-\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_\theta}{\partial\theta}\right)$$ $$u_r\frac{\partial u_\theta}{\partial r}+\frac{u_\theta}{r}\frac{\partial u_\theta}{\partial\theta}+\frac{u_ru_\theta}{r}=-\frac{1}{r\rho}\frac{\partial p}{\partial \theta}+\nu\left(\nabla^2u_\theta+\frac{2}{r^2}\frac{\partial u_r}{\partial\theta}-\frac{u_\theta}{r^2}\right)$$
Using the definition of the streamfunction $\psi$ in polar coordinates, $$u_r=\frac{1}{r}\frac{\partial\psi}{\partial\theta},\;u_\theta=-\frac{\partial\psi}{\partial r}$$ and the vorticity $\omega$ $$\omega=\frac{1}{r}\frac{\partial}{\partial r}(r u_\theta)-\frac{1}{r}\frac{\partial u_r}{\partial\theta}=-\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial\psi}{\partial r}\right)-\frac{1}{r^2}\frac{\partial^2 \psi}{\partial\theta^2}$$ I need to combine the equations into the following single PDE.
$$u_r\frac{\partial\omega}{\partial r}+\frac{u_\theta}{r}\frac{\partial\omega}{\partial\theta}=\nu\nabla ^2\omega$$
The instructions for this process are given in this paper.
- Take the derivative of the $r$-component with respect to $\theta$. (I think this means $\frac{1}{r}\frac{\partial}{\partial\theta}$.)
- Take the derivative of the $\theta$-component with respect to $r$.
- Combine these "appropriately" and note that the pressure terms vanish.
I started this by rewriting both equations, and solving for the pressure term before taking the derivatives.
$$\frac{1}{r}\frac{\partial}{\partial\theta}\left(\frac{1}{\rho}\frac{\partial p}{\partial r}\right)=\frac{1}{r}\frac{\partial}{\partial\theta}(...)$$ $$\frac{\partial}{\partial r}\left(\frac{1}{r\rho}\frac{\partial p}{\partial \theta}\right)=\frac{\partial}{\partial r}(...)$$
However, I'm unclear on what to do with the $r$-derivative of the $\theta$ pressure term so that the two cancel out when subtracted. Similar issues occur when expanding the other derivatives. I've tried expanding out each term in terms of derivatives of $\psi$ but it gets very complicated. Is there an easier strategy to try and combine these into the desired form?
Addendum: I thought of this after posting the question.
Comparing the original two equations with the final goal, I noticed that the first two advective terms have a counterpart with the vorticity. That is there is a $u_r\frac{\partial\phi}{\partial r}+\frac{u_\theta}{r}\frac{\partial\phi}{\partial\theta}$ group where $\phi$ is either velocity or vorticity. The same is true for the Laplacian $\nabla^2\phi$.
My current thought is that writing the remaining terms out in terms of the streamfunction $\psi$, taking the respective derivatives, and then subtracting them will be zero. I am currently working on that but not having much success.
Addendum 2: 9/21/23 After doing some more reading, I think what the authors of the paper actually intended was for the derivatives of $u_r$ and $u_\theta$ to be calculated normally and inserted into the advective and diffusive terms using the stream function $\psi$. I am still looking to see if I can identify different terms and I've noticed a lot of cancellations but haven't been able to resolve it yet fully. However, the pressure terms do not appear to cancel out.