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The problem at hand is a bubbly flow in a cylinder: I'm using an FEM to identify how the walls effect the drag on bubbles in a flow. To test my results I want to set up an infinite cylinder with randomly distributed spheres and then average the Navier-Stokes equations over the entire length and polar angle.

I don't want to average over the radial direction as I want to solve the resulting "averaged equations" for moving walls: i.e I want to use the equations to find how the drag depends on distance from the wall.

I've attached two pictures to give an idea of what I'm talking about -- the second picture should be integrated through the polar angle too.

I can find almost no literature where the Navier-Stokes equations (or PDEs in general) are averaged in space over multiply connected domain. Can someone point me to an author/book which uses this method?

Cheers,

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# Spatially averaging the Navier Stokes equations on a multiply connected domain

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