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Flux across a boundary as source term

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Good morning

I've a simple (i think) problem, but yet i can't find a proper solution. In general terms I've an equation of the kind: (i'm using the general pde module, but let's fix the idea and think is a fluid dynamic problem). Now i want define a source term () in a specific way: in my domain (let's say a big square) i've a region (a small circle), J is zero everywhere, but not in the circle where is equivalent to the mass flow that left this region. The total amount of liquind in the whole system keep increase, but locally in the circle is constant (the source "pump" the same amount of liquid that left at each time step). Mostly i need to compute the flux that left the circular domain and use it as source term.

Thanks in advice for the help


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