We describe a generalization of the immersed boundary (IB) method for simulating fluid-structure interaction, which incorporates the effect of porosity into elastic immersed boundaries. The mathematical formulation of the IB problem can be altered to allow for porous leakage by incorporating an additional porous slip velocity term in the membrane evolution equation. We study how this change affects the linear stability of the underlying governing equations, comparing to previously published results for an impermeable membrane. We also address the impact of membrane permeability on numerical computations through a study of stability and time step restrictions for certain discrete formulations of the IB problem. All IB computations are characterized by an inherent volume loss which resembles a porous membrane leaking fluid -- we obtain an estimate of this ``inherent membrane permeability" and use it to devise a method for reducing the volume loss in computations.
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