Computational Challenges for Simulating Strongly Elastic Flows in Biology

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. There are many different kinds of complex fluids in biology, and they frequently contain dynamic active or passive structures in the fluid. Numerical simulations of these complex flows can be a powerful tool in understanding these biological systems. Existing techniques in computational fluid dynamics are often sufficient for problems with weak flows and low elasticity. However, when elastic forces become large due to, for example, long relaxation times, extra forces from internal structures, or interactions with complex boundaries, more care must be taken to properly simulate these flows. This chapter is devoted to the challenges that R.D. Guy ( ) • B. Thomases Department of Mathematics, University of California, Davis, 1 Shields Ave., Davis, CA 95616, USA e-mail: [email protected]; [email protected] © Springer Science+Business Media New York 2015 S.E. Spagnolie (ed.), Complex Fluids in Biological Systems, Biological and Medical Physics, Biomedical Engineering, DOI 10.1007/978-1-4939-2065-5__10 359 360 R.D. Guy and B. Thomases arise when internal elastic forces are modeled in complex fluids with an eye towards recognizing, understanding, and properly treating the features of strongly elastic flows. We will focus on the Oldroyd-B model as one of the simplest closed continuum models of viscoelastic fluids. In the original derivation [1], Oldroyd set out requirements for constitutive equations so that the material properties would be frame invariant in a coordinate system which convected with the material. This procedure leads to the upper-convected time derivative (also called the Oldroyd derivative, see Eq. (10.5)) which gives the rate of change of a tensor property of a small volume of fluid written in a coordinate system rotating and stretching with the fluid. The most widely used Oldroyd model is the Oldroyd-B model, in part because this model can also be derived from a theory of dilute polymer solutions [2]. The Oldroyd-B model is given below for u the velocity of the fluid, p the pressure, and the deviatoric stress tensor; see also Chap. 1. From balance of momentum and mass conservation for an incompressible fluid we have