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Computational Challenges for Simulating Strongly Elastic Flows in Biology

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Complex Fluids in Biological Systems

Part of the book series: Biological and Medical Physics, Biomedical Engineering ((BIOMEDICAL))

Abstract

Understanding the behavior of complex fluids in biology presents mathematical, modeling, and computational challenges not encountered in classical fluid mechanics, particularly in the case of fluids with large elastic forces that interact with immersed elastic structures. We discuss some of the characteristics of strongly elastic flows and introduce different models and methods designed for these types of flows. We describe contributions from analysis that motivate numerical methods and illustrate their performance on different models in a simple test problem. Biological problems often involve the coupled dynamics of active elastic structures and the surrounding fluid. The immersed boundary method has been used extensively for such problems involving Newtonian fluids, and the methodology extends naturally to complex fluids in conjunction with the algorithms described earlier in this chapter. We focus on implicit-time methods because the large elastic stresses in complex fluids necessitate high spatial resolution and long time simulations. As an example to highlight some of the challenges of strongly elastic flows, we use the immersed boundary method to simulate an undulatory swimmer in a viscoelastic fluid using a data-based model for the prescribed shape.

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Notes

  1. 1.

    When derived from the kinetic theory of dumbbells [2, 27] there is polymer stress diffusion; however the stress diffusion coefficient is proportional to the square of the ratio of the bead diameter (or polymer radius of gyration) to the flow length scale, and even in the context of micro-fluidics, it is minute, \(\mathcal{O}(10^{-9}),\) and is typically ignored.

  2. 2.

    The data are kindly provided by Paulo Arratia and Xiaoning Shen.

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Acknowledgment

The authors would like to thank Lisa Fauci and Michael Shelley for interesting discussions and suggestions on this work. We also acknowledge Paulo Arratia and Xiaoning Shen for allowing us to use their data. This work was supported in part by NSF grants DMS-1160438 and DMS-1226386 to RDG.

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  1. Department of Mathematics, University of California, Davis, 1 Shields Ave., Davis, CA, 95616, USA

    Robert D. Guy & Becca Thomases

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  1. Robert D. Guy
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    Saverio E. Spagnolie

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Guy, R.D., Thomases, B. (2015). Computational Challenges for Simulating Strongly Elastic Flows in Biology. In: Spagnolie, S. (eds) Complex Fluids in Biological Systems. Biological and Medical Physics, Biomedical Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2065-5_10

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