Interaction of Immersed Boundaries in Complex Fluids

1. Interaction of Immersed Boundaries in Complex Fluids Hector D. Ceniceros Department of Mathematics, USCB 2011 AMS Spring Western Sectional Meeting, Las Vegas

2. Collaborators Jordan E. Fisher (Ph.D. 2011) Courant Alexandre M. Roma (USP, Brazil)

3. Outline Introduction: elastic structures in fluids. The Immersed Boundary (IB) Method. Recent progress on IB Method (2D and 3D). Complex Fluids.

4. Applied Math at UCSB • Computational Science and Engineering Graduate Emphasis • Interdisciplinary Complex Fluids/Soft Materials Group • Copolymers • Liquid Crystalline Polymers • Bio-Polymers • Polymeric Solar Cells

5. Flow-Structure Interaction Structures can be simple or complex Flexible rod Heart Model Jellyfish Flow-structure interaction: structures react to flow and flow is affected by structure forces.

6. Flow-Structure Interaction Relevant to Reproduction Free swimmers (e.g. spermatozoa) Ciliary motion (e.g. in airways, oviduct) Channel with elastic contracting walls (Peristalsis) In these examples the fluid is viscoelastic (non-Newtonian) Fauci and Dilon, Ann. Rev. Fluid Mech. 38:371-394, 2006

7. Immersed Boundary Setting C. Peskin (70’s) Eulerian-Lagrangian representation

8. Immersed Boundary Method C. Peskin, 70’s Spreading Interpolation

9. Versatility of the IB Method • Vast structure-building capability: • from a simple link to complex fiber architecture • Easy implementation • readily available flow solvers and simple tracking • Has been used in many applications: • Cardiac fluid dynamics, swimming, insect flight, locomotion of cilia and flagella, peristalsis, particulate flows, bio-films, complex fluids, etc.

10. An Old Problem: Stiffness • Immersed structures can be very stiff and induce severe time-step restrictions for explicit methods (Peskin 77, Stockie and Wetton 95, 99). • Fully implicit discretizations seem too expensive for any practical use (Tu and Peskin 92, Mayo and Peskin 93). • Recent progress with semi-implicit method (Hou and Shi 2008) but limited to periodic interfaces.

11. Cartesian grids with mesh size Discretization Peskin’s lagged operators discretization, 1977

12. Stability and Robustness Neglect advection, Linear and self-adjoint negative def Unconditionally stable Newren, Fogelson, Guy, Kirby JCP, 222, 2007

13. Stiffness Problem Solved? Stiffness can be removed with suitable implicit discretization e.g. Peskin’s lagged operators discretization Caveat: solving the implicit discretization even in the linear case is too costly, impractical This has been known to the community for almost 40 years The problem has received renewed attention recently (Peskin & Mori, Newren et al, Hou & Shi, Griffith, Layton &Beale)

14. Recasting the Equations Fluid solve

15. In fact unless is in the kernel of the projection, i.e. a gradient field Eliminating un+1 Due to the spreading, the IB method fails to yield discrete gradient

16. Forward Euler/Backward Euler (FE/BE) Efficiency How to solve economically to produce non-stiff integration of IB Method for a wide range of practical immersed structure situations? • There are really two interrelated problems: • 1. Efficient computation of the flow-structure interaction Mn f • 2. Efficient iterative solution methods for Xn+1 Main cost is fluid solve. Any method requires bn which involves a fluid solve, appropriate to measure the cost relative to one FE/BE step

17. Spreading +fluid solve + interpolation • Matrix-vector multiplication Costs (2D) In the design of efficient iterative methods it is crucial to streamline the computation of quantities of the form i.e. Flow-Structure Interaction Operation Caveat: We need a matrix representation of Mn which is too costly to obtain directly

18. correspond to velocity that is obtained The entries by interpolating the values produced at Xi by spread unit forces located at Xj Tremendous savings if we assume Matrix Approximation At the continuum level G(Xi -Xj) Not true at discrete level due to spreading and interpolation

19. Cost Shifting to the Origin Fix the point where force is applied and evaluate effect on each Eulerian grid point. These Eulerian values can be precomputed (2 fluid solves!) For a each given Xj-Xi the corresponding velocity is obtained from interpolation of the Eulerian values

20. Idea of Proof The estimate follows from estimates on Gh and an identity for the discrete delta. Accuracy of Matrix Approximation

21. The Prototypical Test Initially elliptical interface Relaxes to a circle

22. Solving the Linear System Since the matrix is available it is easy to construct a wide class of iterative methods (e.g. weighted Jacobi, Gauss-Seidel, etc) Standard algebraic multigrid works the best in this case Example: Stokes flow

23. 2D Model of a Heart Valve 1. Valve. 2. Cushions 3. Hinges 4. Artery wall There are rigid structures, tethered points and crossed links

24. The Nonlinear System Challenges Because of the lack of positive definiteness CG does not converge and BiCG takes over 100 iterations. Difficult to find effective preconditioners

25. 1. 2. Solving the Nonlinear System Fixed point iteration Fails miserably! The eigenvalues of J are huge. One could consider the reversed iteration

26. Time step for FE/BE Numerical Results Imposed horizontal flow Re=50 CPU time FE/BE: 34714584* CPU time Semi-implicit 1907 Four orders of magnitude faster!

27. 3D Recall there are two main difficulties: 1. The heavy cost of computing the flow-structure interaction Mnf 2. Solving the nonlinear system Thus our matrix approach is impractical in 2D! Our solution: Adapt a Fast MultiPole Method (treecode) approach to the IB method

28. Two main ideas The idea is to use far field (multi-pole) expansions of Gh to compress the effect of clusters of fiber points

29. Treecode Approach

30. How to Select Win and Wout? Solution: binary partitioning, quadtree (2D), octree (3D) To evaluate loop over each panel P and calculate the far field expansion of all poles in P

31. Expansions

32. Results Flow past an immersed plate. Each fiber point X is tethered to a corresponding point T: Flow is induced with a time periodic forcing term. Re=10 We solve implicit system via CG CPU time in hours

33. Flow past a plate Depiction of the flow using streamlines

34. Oscillating Immersed Spheroid Velocity magnitude Six order of magnitude faster

35. Complex (Non-Newtonian) Fluids • Polymeric liquids • Gels, sols, emulsions • Foams • Liquid crystalline materials • Granular materials Swimmers in vicoelastic fluid There is a microstructure (e.g. long molecules) whose interaction with a flow leads to many phenomena not observed in Newtonian fluids

36. Generic Framework Low Re, Stokes approximation Polymeric stress Q e.g. molecules modeled as dumbbells If we can get an evolution eq for We eliminate configuration space. This is called Oldroyd B model

37. F.E.N.E. Model Hooke’s law implies dumbbells could be extended without limit!!!!! Finitely Extensible Nonlinear Elastic No longer possible to eliminate configuration space to compute stress Pre-averaging (FENE-P) and closure approximations are sometimes used to reduce the computational complexity We are working on multiscale approaches to effectively compute these type of flows in the presence of immersed flexible boundaries

38. Peristaltic Pumping in Viscoelastic Fluid Peristalsis: fluid transport that occurs when waves of contraction are passed along a fluid bearing tube Stokes-Oldroyd B system (Teran, Fauci, Shelley 2008) Flux very different from Newtonian We’re investigating for larger We=tp/tf and longer times both in 2D and 3D

39. Conclusions • There are two main difficulties: • computing the flow-structure interaction (Mn F) • solving the implicit system. • It is possible to expedite the computation of Mn F and to arrive a efficient solutions for the implicit system to produce enormous savings in 2D & 3D. • Current work is the application of these techniques to investigate the motion of immersed flexible boundaries in complex fluids.

40. Acknowledgements Partial support by National Science Foundation: DMS 0609996 and DMS 1016310 Special thanks to the American Mathematical Society for sponsoring this talk