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Two-Way fluid coupling help

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Hi,

I am trying to get my head around the two-way coupling described by R.Bridson in his book "Fluid Simulation for Computer Graphics" (page 165-167...or equivalently the paper here: http://www.cs.ubc.ca/labs/imager/tr/2007/Batty_VariationalFluids/variationalFluids.pdf chapter 3).

I am struggling in understanding how to implement the additions to the matrix beside the additions to the rhs.
It says deltaTime * J^T * Ms^-1 * J is added to the matrix but I don't quite understand what J^T and J are. I thought it would be fluid and solid fractions accordingly (divided by voxeldelta) but I am quite confused.

Is anybody able to help or understand how this equation expresses itself codewise (for example for the right face of a cell)?
Thanks in advance
Best

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