Numerical methods for the study of deformable bodies in viscous and viscoelastic flows
Keywords: Chemical Engineering.

Suspended soft particles in viscoelastic fluids are ubiquitous in biological applications and are being utilized with increasing frequency in microfluidic platforms. Biological fluids are often laden with cells or swimming microorganisms which are highly deformable, while the suspending fluid usually includes polymeric macromolecules that impart elasticity to the fluid. These highly elastic fluids can be found in the human body in mucus linings, in direct-write additive manufacturing applications, and even in injectable therapeutics. In this work we present the development of a high-fidelity simulation tool with general constitutive model implementations for both the viscoelastic fluid and deformable solid to understand the physics behind these complex systems. We discuss a modified version of the IFEM (Immersed Finite Element Method) that allows for the simulation of deformable particles in viscoelastic flows which minimizes the need for costly re-meshing operations and scales well in particle number. This simulation tool is validated for a number of simple Newtonian and viscoelastic cases to ensure the fidelity of the presented algorithm. Lastly, we consider a series of specific applications that demonstrate the breadth and scalability of the simulation platform. Specifically, we consider the slowdown of swimming microorganisms in viscoelastic fluids, the rheology of soft particles in viscoelastic shear flows, and the dynamics of red blood cells in small arteriole flow and AFM indentation. We emphasize the study of swimming behavior of undulatory and amoeboid swimmers in viscoelastic fluids. The undulating swimmer C. elegans is an excellent case study since it is both experimentally well-studied and the microorganism's motion resembles the behavior of many other biological structures, like cilia or flagella. Additionally, there is a well-known speed decrease as the Deborah number increases that has been experimentally observed but, to date, has not been studied numerically with a fully resolved three-dimensional simulation. In this work, we discuss the use of the IFEM with an added conformation-driven force or extra surface traction that allows the swimmer to evolve through an arbitrary set of specified shapes. We compare numerical results for C. elegans against experimental speed data provided by Shen and Arratia (2011) and the speed reduction as a function of Deborah number is presented with good agreement for Oldroyd-B fluids. A similar set of results is considered for the amoeboid swimmer which has never been numerically studied in viscoelastic fluids. The simulation tool is then further utilized to explore the underlying physical mechanism that drives swimming speed reduction in viscoelastic fluids, including comparison to other more simplified simulations/theories. The role of polymer stretch boundary layers near the surface of these swimmers is noteworthy, which demonstrates the need for fully resolved simulations which take into account the finite size of the microorganism. Additionally, we highlight the study of rheology of soft particles in viscoelastic flows. While studies to date have investigated the dynamics of soft solids and membranes in pressure driven flow as well as the shapes and dynamics of soft particles in simple shear flows, little work has been completed to examine the effective rheology of suspensions of these particles. In this work, we discuss the application of the IFEM to the motion of deformable Neo-Hookean solid particles in simple viscoelastic shear flows. We then discuss the interplay of fluid elasticity and particle elasticity and how this effects the key viscometric functions for shear flows. We break the viscometric measurables into contributing parts from the fluid (particle induced fluid stress) and the particle (stresslet) that show interesting trends and underlying physical principles. We find that all components of the particle induced fluid stress are nearly invariant to the deformation (within the parameter range studied), while the shear stress component of the stresslet rapidly decreases in magnitude as elasticity in the fluid increases. These rheological measures have widespread impact for the design of microfluidic devices and we believe that further investigation of these findings will aid in the design of engineered fluids. The simulation tool also has the capability of allowing the simulation of denser suspensions of particles and more complex geometries opening many new possibilities for future studies of soft matter in viscoelastic fluids.