Complex Creeping Fluids: Numerical Methods and Theory (17w5155)

Arriving in Oaxaca, Mexico Sunday, October 1 and departing Friday October 6, 2017


(Florida State University)

(University of Texas)


Many applications in biology, physics, and science require solving problems in complex fluids. A few examples of complex fluids include colloidal suspensions, blood flows, fumble and rigid particle suspensions, vesicle flows, Mizar solutions, bacterial solutions, and polymer solutions. They can be viewed as fluids that possess some kind of elastic microstructure, characterized by the presence of a microscopic length scale. Because of the scales of many of these complex fluids, the Reynolds number is small, and the flow is assumed to be creeping (i.e., governed by the incompressible Stokes equations). Simulations of complex fluids are particularly challenging because of fluid-structure interactions, large deformations, strong nonlinearities, non-local interactions, evolving interfaces, and multiple length and time scales with, typically, no clear scale separation. They are non-equilibrium, non-Hamiltonian systems and typically lack a variational characterization of their dynamics. As a result, complex fluids display an incredible range of structural and rheological properties.

Numerical methods for Stokesian flows can be classified to unstructured finite element methods, Cartesian-grid methods, and integral equation methods, and many groups have developed extremely efficient software for such simulations. Stochastic methods like Fokker-Plank equations, Stokesian dynamics, dissipative particle dynamics, and others have also been developed. This workshop aims to bring together experts from these diverse areas and catalyze advances in the simulation capabilities and our ultimate understanding of complex fluids. Recent advances which will be further discussed of the workshop include:

  • Flows in complex geometries: Many problems in complex fluids also involve complex geometries. A large suite of methods such as the immersed boundary method, cut cells, and integral equation methods, are being developed by many groups. The workshop will identify open issues, accuracy, and stability for concentrated suspensions and long-horizon simulations.
  • Fast algorithms and parallel architectures for complex fluids: In order to simulate realistic complex flows, efficient algorithms must be used to accelerate the time to solution. Algorithms such as fast summation techniques in modern computer architectures have allowed researchers to simulate increasingly complex problems. The workshop
  • will identify numerical challenges as we scale to larger problem sizes and additional possible acceleration schemes.

  • Upscaling and coarse-grained variables: Another important task when simulating complex flows is to identify coarse-grained variables. This allows models to be upscaled and can be used to study average quantities such as anomalous diffusion rates, transport, and the overall rheology. The workshop will focus on the selection of coarse-grained variables and their interaction in a mathematically rigorous way.
  • Improved surface representations and discretizations: Real-world applications often involve complicated geometries, interfaces, and boundary conditions. This is particularly true in three dimensions. There has been recent progress on high-order surface representations, which are essential to maintaining stable and accurate methods.
  • New and expanding areas of application: Complex fluids arise in numerous complex systems in science and engineering. Some of the current popular topics include active matter, interfacial dynamics, and non-Newtonian fluids. The workshop identify common mathematical challenges and opportunities for future research.

A BIRS workshop would be an ideal vehicle for advancing progress in this field. It would achieve several key objectives:

  • Catalyze new developments in algorithms and theory for complex fluids: Although significant progress has been made recently on simulations of complex fluids, many open problems remain especially in three dimensional systems of concentrated suspensions. The need for faster methods, proper upscaling, and effective statistical measures is critical in order to enable accurate simulation of complex fluids.
  • Facilitating interdisciplinary research: Many world-renowned researchers in complex fluids are in either a Mathematics or an Engineering department. This workshop will provide a venue where interdisciplinary work and different areas of expertise can be discussed
  • Develop young talent: The workshop is also intended to inspire and motivate younger researchers in the field, and to attract some of the most promising new talent. We have already begun to encourage our potential attendees to bring students and postdocs from their groups.

The workshop would be structured to organize the scientific discussion around four main themes, which are outlined below:

  1. Mathematical foundations of dynamics of complex fluids: A discussion of recent advances, possibilities for combining the new techniques, and an exploration of future development. Topics to be discussed could include: dynamics, phase transitions, mixing, Lagrangian coherent structures, anomalous diffusion, and constitutive laws for complex fluids.
  2. Direct numerical simulations: There is a large collection of methods that can be used to simulate complex fluids. Methods that will be discussed include integral equations, immersed boundary methods, phase field methods, time-integration, and complex geometries.
  3. Stochastic numerical simulations: As an alternative to directly simulating the governing equation, stochastic methods can be used. The discussion will include dissipative particle dynamics, Fokker-Planck equations, stochastic differential equations (dumbbell models), and molecular dynamics simulations.
  4. Applications: Additional applications to be discussed include colloidal suspensions, settling flows, blood flows, vesicle flows, intra-cellular flows, bacterial suspensions, and electrochemical flows.

The format of the workshop will consist of research seminars on recent advances and poster presentations. In addition, we plan to hold a forum to address special needs to students, postdocs, and junior faculty, and to provide guidance on prerequisites and possible research directions in our identified themes. We will plan for time to allow brainstorming by participants on important open problems, such as upscaling, mixing, and time integration.