I'm having trouble with a boundary condition. In a fluid mechanics problem, I have flow at $z = \infty$ flowing into a solid plate at $z = 0$ and then flowing radially, and the problem is given as axisymmetric. I know that the velocity field has $v_r$ and $v_z$, and $v_\theta = 0$. How in the world would I express the symmetry boundary conditions without $\theta$?
1 Answer
Axisymmetry implies that there is no change in anything in the $\theta$ direction, i.e.
$$ \frac{\partial}{\partial\theta}(\text{anything}) = 0 $$
Which would mean
\begin{align} \frac{\partial p}{\partial\theta} &= 0 \\ \frac{\partial \vec{V}}{\partial\theta} &= 0 \\ \implies &\frac{\partial v_r}{\partial\theta} = 0 \\ \implies &\frac{\partial v_\theta}{\partial\theta} = 0 \\ \implies &\frac{\partial v_z}{\partial\theta} = 0 \end{align}
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$\begingroup$ I was trying to express the axisymmetry without utilizing theta, as mentioned above. Thanks for the response though. $\endgroup$ Commented Dec 9, 2013 at 19:30
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$\begingroup$ Axisymmetry is not a boundary condition, because there is no boundary in the $\theta$ direction. The conditions for axisymmetry, however, are used to simplify the Navier-Stoke equations and can not be denoted without $\theta$, because they are used to eliminate $\theta$ $\endgroup$– phoCommented Dec 9, 2013 at 22:36