Abstract
We derive a new computational model for the simulation of viscous incompressible flows bounded by a thin, flexible, porous membrane. Our approach is grid-free and models the boundary forces with regularized Stokeslets. The flow across the porous membranes is modeled with regularized source doublets based on the notion that the flux velocity across the boundary can be viewed as the flow induced by a fluid source/sink pair with the sink on the high-pressure side of the boundary and magnitude proportional to the pressure difference across the membrane. Several validation examples are presented that illustrate how to calibrate the parameters in the model. We present an example consisting of flow in a closed domain that loses volume due to the fluid flux across the permeable boundary. We also present applications of the method to flow inside a channel of fixed geometry where sections of the boundary are permeable. The final example is a biological application of flow in a capillary with porous walls and a protein concentration advected and diffused in the fluid. In this case, the protein concentration modifies the pressure in the flow, producing dynamic changes to the flux across the walls. For this example, the proposed method is combined with finite differences for the concentration field.
Highlights
We present a new model for a type of problem in which a thin flexible porous boundary, such as a membrane, interacts with an incompressible viscous fluid
This work was later expanded to consider Stokes flow through a pipe of temporally dynamic radius to simulate “pumping” mechanisms encountered in numerous biological systems [20]
We have presented a model for computing flows bounded by thin permeable membranes in which the amount of fluid that is filtrated across the membrane depends on local flow properties such as the pressure drop across the membrane, the concentration of a solute, and the membrane permeability
Summary
We present a new model for a type of problem in which a thin flexible porous boundary, such as a membrane, interacts with an incompressible viscous fluid. Layton [22] uses a Cartesian grid method combined with Mayo’s technique [25], similar to the immersed interface method [26], to incorporate jump conditions into the finite-difference stencils This formulation incorporates a solute concentration into the model so that the flux velocity across the membrane can be a function of the hydrostatic pressure jump or the solute concentration differences across the boundary, a condition often encountered in biological settings. This work was later expanded to consider Stokes flow through a pipe of temporally dynamic radius to simulate “pumping” mechanisms encountered in numerous biological systems [20] These analytical solutions assume that the position of the porous boundary is independent of the flow, as such they are useful for validation of numerical methods with a similar assumption. The latter is based on the notion that the flux velocity across the boundary can be considered as the flow induced by a fluid source/sink pair with the source on the high-pressure side of the boundary and magnitude proportional to the pressure difference across the membrane
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
