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Abstract This work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time‐dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin–Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for the coupled problem. The first method uses the second‐order backward differentiation formula (BDF2) for implicit integration, while treating the interface terms explicitly using a second‐order extrapolation formula. The second method is the Crank–Nicolson and Leap‐Frog (CNLF) method, where the Crank–Nicolson method is used to implicitly advance the solution in time, while the coupling terms are explicitly approximated by the Leap‐Frog integration. We show that the BDF2 method is unconditionally stable and uniformly stable in time, while the CNLF method is stable under a CFL condition. Both schemes are validated using numerical simulations. Second‐order convergence in time is observed for both methods. Simulations over a longer period of time show that the errors in the solution remain bounded. Cases when the structure is poroviscoelastic and poroelastic are included in numerical examples.
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A second-order partitioned method for bioconvective flows with concentration dependent viscosity
This work is focused on the mathematical and computational modeling of bioconvection, which describes the mixing of fluid and micro-organisms exhibiting negative geotaxis movement under the force of gravity. The collective population moves towards the surface of the fluid, generating a Rayleigh–Taylor instability, where initial fingers of organisms plummet to the bottom. The inherent drive to swim vertically generates large collective flow patterns that persist in time. We model the flow using the Navier–Stokes equations for an incompressible, viscous fluid, coupled with the transport equation describing the concentration of the micro-organisms. We use a nonlinear semigroup approach to prove the existence of solutions. We propose a partitioned, second-order, time adaptive numerical method based on the Cauchy’s one-legged ‘-like’ scheme. We prove that the method is energy-stable, and for small time steps, the iterative procedure in the partitioned algorithm is linearly convergent. The numerical results confirm the expected second-order of accuracy. We also present a computational study of a chaotic system describing bioconvection of motile flagellates.
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- PAR ID:
- 10524420
- Publisher / Repository:
- International Press of Boston, Inc
- Date Published:
- Journal Name:
- Annals of Mathematical Sciences and Applications
- Volume:
- 9
- Issue:
- 1
- ISSN:
- 2380-288X
- Page Range / eLocation ID:
- 141 to 184
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/AMSA.2024.v9.n1.a5
