I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations $$u_t - \nu\,\Delta u + u\cdot \nabla u + \nabla p=0$$ We consider in cylindrical coordinates $u=(u^r, u^\theta,u^z)$. And we have the following vorticity equation in cylindrical form. $$\omega^r = \frac{1}{r}\frac{\partial u^z}{\partial \theta} - \frac{\partial u^\theta}{\partial z} = - \frac{\partial u^\theta}{\partial z}, \\ \omega^\theta = \frac{\partial u^r}{\partial z} - \frac{\partial u^z}{\partial r}, \\ \omega^z = \frac{1}{r}\frac{\partial}{\partial r}(r u^\theta) - \frac{1}{r} \frac{\partial u^r}{\partial \theta} = \frac{1}{r}\frac{\partial}{\partial r}(r u^\theta).$$ My question goes as follows: Suppose $J=\frac{w^r}{r}$, then how to derive the following steps?

\begin{align} & \int J(\omega^r \partial_r+\omega^z \partial_z)\frac{u^r}{r} r \, dr \, dz \\[8pt] = {} & \int [\nabla\times (u^\theta e_\theta)] \left(J \, \nabla \frac{u^r}{r}\right) r \, dr\,dz \\[8pt] = {} & \int (u^\theta e_\theta) \left(\nabla J\times\nabla \frac{u^r}{r}\right) r \, dr \, dz \end{align}

I'm assuming some integration by parts is involved but couldn't derive them; besides, where does the cross product come from? The equality above that bugs me is from the bottom of page 11 of this paper.


1 Answer 1


There's a typo in the last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.

Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions for the $\omega $ in terms of the $u$; in the second expression, note $$ \nabla \times (u^{\theta } e_{\theta } ) = e_z \partial_{r} u^{\theta } + e_z u^{\theta } /r - e_r \partial_{z} u^{\theta } $$ and $$ \nabla (u^r /r) = e_r \left( \frac{1}{r} \partial_{r} u^r -\frac{1}{r^2 } u^r \right) + e_{\theta } \frac{1}{r^2 }\partial_{\theta } u^r +e_z \frac{1}{r} \partial_{z} u^r $$ and multiply that out - no integration needed, already the integrands are identical. To get to the third expression, consider the volume integral of $$ \nabla \cdot \left[ (u^{\theta } e_{\theta } ) \times (J\nabla(u^r /r)) \right] $$ You don't say anything about the boundary conditions, but I suppose the field in the square brackets doesn't have net flux out of the volume - then this vanishes by Gauss' theorem. Also, the $u$ have to be regular enough so that different components of $\nabla $ commute on them ...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.