I have a PDE equation with Robin Boundary condition in an annulus system and I should solve it by finite volume method:
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\begin{align} \frac{\partial T_f}{\partial t} - k \left(\frac{\partial^2 T_f}{\partial r^2} + \frac{1}{r} \frac{\partial T_f}{\partial r} \right) = -Q \end{align}

I do not have any problem in the discretization of inner points by finite volume method, but in $r_{ID}$ and $r_{OD}$, it has robin boundary condition which I do not know how to discretize it.

\begin{align} \left.\frac{\partial T_f}{\partial r}\right|_{r=r_{ID}} &= h(T_f-T_w ) \\ \left.\frac{\partial T_f}{\partial r}\right|_{r=r_{OD}} &= h(T_f-T_g ) \end{align}

(h&k are constant)
I would appreciate it if you could help me to discretize this boundary condition.

  • $\begingroup$ When you test the $\frac{\partial^2 T_f}{\partial r^2}$ term, you do integration by parts, which produces a surface integral that includes $\frac{\partial T_f}{\partial r}$ as part of the integrand, right? Just substitute your Robin BCs into that term. If you detail your discretization process, it might be easier to give an explicit answer. $\endgroup$
    – LedHead
    May 14, 2017 at 16:54


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