Research

We use computational fluid dynamics, microscale modeling, dynamical systems theory, and machine learning to investigate the behavior of complex fluids and flows

 

The Graham group uses theory and computations to study problems in fluid dynamics, rheology and transport phenomena over a wide range of scales. We focus on problems that hold both fundamental interest in advancing basic principles as well as impact on applications. The group has two basic thrust areas, one in microscale flows and complex fluids and the other in the nonlinear dynamics of turbulent flows.

In the first area, we are interested in general in the dynamics of mechanically and geometrically complex objects suspended in a flowing fluid and the interplay between microstructure and flow. Specific examples under study include the dynamics of blood cells in flow, the interplay between cell geometry and mechanics in bacterial swimming, the deformations of thin deformable sheetlike particles in flow, and the rheology and fluid dynamics of dilute micellar surfactant solutions.

In the area of turbulent flows, the group aims to elucidate the complex interaction between rheology and fluid dynamics that leads to the phenomenon of turbulent drag reduction in polymer and surfactant solutions — this topic is a bridge to the group’s interest in microscale flows and rheology. We also apply ideas from nonlinear dynamical systems theory, data science and machine learning to elucidate the principles underlying the complex dynamics of turbulent shear flows, aiming toward development of control schemes that can manipulate turbulence to desired ends such as drag reduction.

Microscale and complex fluids

Complex and microstructured fluids are found in a vast array of commercial, industrial, and biological applications. The interactions and dynamics of microstructure (e.g., molecules and cells) within fluids can induce a variety of effects, from complex instability formation to drag reduction in turbulent fluids. In the Graham group, we are interested in using computational and analytical techniques to understand phenomena in complex fluids.

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Cell-level dynamics of blood flow

The dynamics of blood cells in the circulation are complex. Experiments have revealed that in blood flow, red blood cells (RBCs) tend to migrate away from the vessel walls, leaving a cell-free layer near the walls, while leukocytes and platelets tend to marginate towards the vessel walls. This segregation behavior of different cellular components in blood flow largely arises from their discrepancies in shape, size and deformability. In more complicated blood flow scenarios such as sickle cell disease (SCD), diseased red blood cells (RBCs) display substantially different physical properties than do healthy RBCs. The mechanism for endothelial dysfunction, a chronic complication associated with SCD, remains unclear. We are using computations and theory to develop a systematic understanding of the distribution and dynamic of blood cells and other particles in the bloodstream during flow in small blood vessels. Recent work focuses on understand the dynamics and distributions of sickle cells in the circulation. This work will help improve our understanding of the pathophysiology of hematologic diseases such as SCD, and may further shed light on the design of novel therapies.

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Recent papers in this area:

  • X. Cheng and M.D. Graham “Marginated aberrant red blood cells induce pathologic vascular stress fluctuations in a computational model of hematologic disorders.” bioRxiv, 2023. [Link]
  • X. Zhang, C. Caruso, W.A. Lam, and M.D. Graham “Flow-induced segregation and dynamics of red blood cells in sickle cell disease.” Phys. Rev. Fluids, 2020, 5, 05301. [Link]
  • X. Zhang and M.D. Graham “Multiplicity of stable orbits for deformable prolate capsules in shear flow.” Phys. Rev. Fluids, 2020, 5, 023603. [Link]

Rheology and fluid dynamics of dilute surfactant systems

Dilute wormlike (or rodlike) micelle solutions are formed from the self-assembly of amphiphilic surfactant molecules and can exhibit remarkable rheological properties, such as both shear-thickening and shear-thinning, reentrant (i.e., multivalued flow curves), and the formation of flow-induced structure. Wormlike micelle (WLM) solutions are widely used in a variety of industrial applications, particularly in gas and oil industries, closed-loop heating systems, and in the development and processing of hygiene and cleaning products. Notably, the addition of a small concentrations of surfactants that form wormlike micelles to turbulent flows can significantly reduce turbulent drag. Dilute WLM solutions also exhibit unique and poorly understood instabilities, such as vorticity banding and finger-like structure formation. Using a combination of continuum modeling and direct numerical simulations, our research aims to understand both the mechanisms behind turbulent drag reduction as well as the development and dynamics of instabilities in dilute WLM systems. We have previously developed a constitutive model, the reactive rod model (RRM), for studying WLM solutions that has shown strong agreement with experimental observations of these solutions, and we are actively extending the physical grounding of the RRM while using it to understand turbulence, FIS, and drag reduction in WLM systems.

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Recent papers in this area:

  • R.J. Hommel and M.D. Graham “Flow instabilities in circular Couette flow of wormlike micelle solutions with a reentrant flow curve.” JNNFM, 2024, 324, 105183 [Link]
  • R.J. Hommel and M.D. Graham “Constitutive modeling of dilute wormlike micelle solutions: Shear-induced structure and transient dynamics.” JNNFM, 2021, 295, 104606. [Link]
  • S. Dutta and M.D. Graham “Mechanistic constitutive model for wormlike micelle solutions with flow-induced structure formation.” JNNFM, 2018, 251, 97-106. [Link]

Thin sheet dynamics in flow

Thin sheets and films suspended in a fluid arise in many current and proposed applications. For example, graphene is an atomically thin sheet of carbon that has incredible mechanical and electronic properties. The incorporation of graphene sheets into fibers via spinning processes involves complex fluid-structure interactions such as graphene folding that are not well understood. In another examples, inspired by the art of origami, researchers have engineered polymer film to have creases and serve as precursors to complicated 3D structures. We are exploring the dynamics of a thin sheets in different flow fields. Mechanical properties such as deformability, orientation, and geometry of these sheets strongly influence their behavior in fluid flow. We are characterizing and understanding how these different properties influence the dynamics of sheets through simulations, with the goal of expanding knowledge in particle dynamics in fluids and furthering the field of materials design and application.

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Recent papers in this area:

  • Y. Yu and M.D. Graham “Wrinkling and multiplicity in the dynamics of deformable sheets in uniaxial extensional flow.” Phys. Rev. Fluids, 2022, 7, 023601. [Link]
  • Y. Yu and M.D. Graham “Coil–stretch-like transition of elastic sheets in extensional flows.” Soft Matter, 2021, 17, 543-555. [Link]
  • S. Dutta and M.D. Graham “ Dynamics of Miura-patterned foldable sheets in shear flow.” Soft Matter, 2017, 13 2620-2633. [Link]

Elastoinertial turbulence in dilute polymer solutions

The addition of small amounts of rheologically active additives, such as flexible long-chain polymers or surfactants forming wormlike micelles, can significantly reduce drag in turbulent flows. The most dramatic effect of the polymer additives on turbulence occurs in the near-wall region, weakening the turbulent eddies in this region. The key feature of these polymer solutions in drag reduction is the existence of the so-called maximum drag reduction (MDR) phenomenon, at which very high levels of drag reduction are achieved by polymer additives. The most intriguing observation for MDR is its universal mean velocity profile, the experimentally observed upper limit on the amount of drag reduction that can be achieved with polymer additives. This asymptotic limit is insensitive to changes in the polymer solution such as concentration, molecular weight or polymer type. Recent experiments and simulations suggest that turbulence in this regime has structure very different from Newtonian. We aim to to elucidate and exploit the mechanisms underlying turbulence in this regime of very high drag reduction.

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Recent papers in this area:

  • M. Kumar and M.D. Graham “Effect of polymer additives on dynamics of water level in an open channel.” JNNFM, 2023, 105129. [Link]
  • A. Shekar, R.M. McMullen, B.J. McKeon, and M.D. Graham “Tollmien-Schlichting route to elastoinertial turbulence in channel flow.” Phys. Rev. Fluids, 2021, 6, 093301. [Link]
  • A. Shekar, R.M. McMullen, B.J. McKeon, and M.D. Graham “Self-sustained elastoinertial Tollmien-Schlicting waves.” JFM, 2020, 897. [Link]
  • A. Shekar, R.M. McMullen, S-N Wang, B.J. McKeon, and M.D. Graham “Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence.” Phys. Rev. Letters, 2019, 122, 124503. [Link]

Data-driven Modeling of Dynamical Systems

High dimensional dynamical systems, such as turbulent fluid flows and complex weather systems, present a problem for computational forecasting due to the intractably large nature of the data. Many of these systems, however, have long-time dynamics that collapse onto finite-dimensional invariant manifolds, allowing the high-dimensional states of the systems to be characterized by a low-dimensional representation. In the Graham group, we apply machine learning techniques to generate data-driven models that learn how to optimally compress complex data into minimal dimensional states and learn how the dynamics of the system evolve in this low-dimensional representation. Our goal is to use data-driven learning to better understand, predict, and control these dynamical systems.

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Data-driven Manifold Dynamics

Our primary approach towards using machine learning to model dynamical systems is a technique known as “Data-driven manifold dynamics” or “DManD”. In this methodology, we combine two types of neural networks, autoencoders and neural ODEs , to generate a low-dimensional model for complex dynamical systems. Autoencoders are an hourglass-shaped neural network that combines an encoder network, which learns to compress the full state data to a low-dimensional latent space representation of the manifold, and a decoder network, which learns to reconstruct the full space data from the latent state. A second neural network, a neural ODE, learns the temporal evolution of the data along the manifold in the latent space representation. We then combine these neural networks to create a low-dimensional dynamical model of the system, which we then use for forecasting and controlling the full state.

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Recent papers in this area:

  • C. E. Pérez De Jesus and M. D. Graham “Data-driven low-dimensional dynamic model of Kolmogorov flow” Physical Review Fluids 2023 8, 044402 [Link]
  • A. J. Linot and M. D. Graham “Deep learning to discover and predict dynamics on an inertial manifold” Physical Review E 2020 101, 6, 062209 [Link]

Implicit Rank Minimizing AutoEncoder with Weight Decay

Optimally-compressed low-dimensional models can represent the dynamics of a system at the intrinsic dimension of the system’s invariant manifold.  Recently our group developed a new autoencoder architecture named IRMAE-WD (“Implicit Rank Minimizing AutoEncoder with Weight Decay”) that automatically estimates the manifold dimension. IRMAE-WD combines implicit regularization with internal linear layers and explicit weight regularization (weight decay) to automatically estimate the underlying dimensionality of a data set. We aim to apply IRMAE-WD to a library of dynamical systems, as well as understand its robustness with respect to dimension estimates.

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Recent papers in this area:

  • K. Zeng, C.E. Pérez De Jesús, A.J. Fox, and M. D. Graham “Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems” arXiv, 2023. [Link]
  • C.E. Pérez De Jesús, A.J. Linot, and M. D. Graham “Building symmetries into data-driven manifold dynamics models for complex flows” arXiv, 2023. [Link]

Chart and Atlases for Nonlinear data-driven Dynamics on Manifolds

For an arbitrary dynamical system, there is no guarantee that a single latent state representation of the full state can be compressed to the intrinsic dimensionality of the manifold. Our group has developed a technique known as “Charts and Atlases for Nonlinear data-driven Dynamics on Manifolds” or “CANDyMan” to overcome this by parametrizing the manifold as a combination of latent states. The CANDyMan method works by decomposing the system into separate charts containing distinct portions of the full state, learning latent state representations and dynamical models in each chart via individual autoencoders and time-mapping neural networks, then stitching the charts together to create a single atlas to yield a global multi-chart dynamical model. We have demonstrated that this technique allows us to optimally compress and model the full state of a dynamical system at its intrinsic dimensionality.

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Recent papers in this area:

  • A. J. Fox, and C. R. Constante-Amores, and M.D. Graham, “Predicting extreme events in a data-driven model of turbulent shear flow using an atlas of charts” Physical Review Fluids 2023 8, 9, 094401 [Link]
  • D. Floryan and M. D. Graham, “Data-driven discovery of intrinsic dynamics” Nature Machine Intelligence 2022 4, 12, 1113-1120 [Link]

Extended Dynamic Mode Decomposition – Dictionary Learning

Our standard approach to modeling dynamical systems using autoencoders and neural ODEs relies on the state space approach. Nevertheless, nearly a century ago, Koopman introduced an innovative perspective known as the function space approach. In this alternative framework, the focus shifts to examining the evolution of functions, or observables, within the state space, with a basis formed by eigenfunctions of the Koopman operator. The Koopman operator, a linear operator of infinite dimensions, takes center stage in this approach. It adeptly captures the intricate nonlinear dynamics of a system, propelling observables forward in time. Consequently, the spectral characteristics of the Koopman operator offer profound insights into a system’s dynamics. However, there is a nuanced trade-off in this paradigm. While linearizing nonlinear systems, the resultant learned linear mapping holds an infinite-dimensional nature, leading to a delicate interplay between the aspects of linearity and dimensionality.

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Recent papers in this area:

  • C.R. Constante-Amores and M. D. Graham “Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds” arXiv, 2023. [Link]
  • C.R. Constante-Amores, A.J. Linot, and M. D. Graham “Enhancing predictive capabilities in data-driven dynamical modeling with automatic differentiation: Koopman and neural ODE approaches” arXiv, 2023. [Link]