Spatially averaging the Navier Stokes equations on a multiply connected domain

[cathalcummins]

Hi all,

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,

 

[jvicens]

As a fellow scientist, I find your research topic very interesting. Bubbly flows are known to have complex dynamics and understanding the effects of walls on drag is crucial in various industrial and natural processes.

I am not aware of any specific literature that uses the method of averaging the Navier-Stokes equations over a multiply connected domain, but I suggest looking into the work of Dr. John Kim from the University of Michigan. He has done extensive research on fluid dynamics and has published several papers on the topic of averaging PDEs in complex geometries. His work may provide some insights and references for your research.

Additionally, I recommend reaching out to other researchers in the field and discussing your approach with them. Collaboration and exchanging ideas can often lead to new and innovative solutions.

Best of luck with your research!