Polymer stress growth in viscoelastic fluids in oscillating extensional flows with applications to micro-organism locomotion

Simulations of undulatory swimming in viscoelastic fluids with large amplitude gaits show concentration of polymer elastic stress at the tips of the swimmers.We use a series of related theoretical investigations to probe the origin of these concentrated stresses. First the polymer stress is computed analytically at a given oscillating extensional stagnation point in a viscoelastic fluid. The analysis identifies a Deborah number (De) dependent Weissenberg number (Wi) transition below which the stress is linear in Wi, and above which the stress grows exponentially in Wi. Next, stress and velocity are found from numerical simulations in an oscillating 4-roll mill geometry. The stress from these simulations is compared with the theoretical calculation of stress in the decoupled (given flow) case, and similar stress behavior is observed. The flow around tips of a time-reversible flexing filament in a viscoelastic fluid is shown to exhibit an oscillating extension along particle trajectories, and the stress response exhibits similar transitions. However in the high amplitude, high De regime the stress feedback on the flow leads to non time-reversible particle trajectories that experience asymmetric stretching and compression, and the stress grows more significantly in this regime. These results help explain past observations of large stress concentration for large amplitude swimmers and non-monotonic dependence on De of swimming speeds

Orientation dependent elastic stress concentration at tips of slender objects translating in viscoelastic fluids

Elastic stress concentration at tips of long slender objects moving in viscoelastic fluids has been observed in numerical simulations, but despite the prevalence of flagellated motion in complex fluids in many biological functions, the physics of stress accumulation near tips has not been analyzed. Here we theoretically investigate elastic stress development at tips of slender objects by computing the leading order viscoelastic correction to the equilibrium viscous flow around long cylinders, using the weak-coupling limit. In this limit nonlinearities in the fluid are retained allowing us to study the biologically relevant parameter regime of high Weissenberg number. We calculate a stretch rate from the viscous flow around cylinders to predict when large elastic stress develops at tips, find thresholds for large stress development depending on orientation, and calculate greater stress accumulation near tips of cylinders oriented parallel to motion over perpendicular.Comment: Supplementary information include

Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems

Symmetric factorization of the conformation tensor in viscoelastic fluid models

The positive definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.Comment: 7 pages, 5 figure

POD analysis of temporal flow patterns in different regimes

Proper Orthogonal Decomposition (POD) has been used broadly in analyzing turbulent flows at high Reynolds numbers, such as flow in a pipe. However, there exists a lack of knowledge in analyzing some other regimes which contain interesting temporal behaviors. We present two study cases with completely different flow regimes showing the advantages of analyzing them using POD. First, we describe an application in creeping flow (very low Reynolds number) in Non-Newtonian fluids. POD helps characterize the different bifurcations of the flow directly related to the movement of stagnation points of the problem. We have also proved the efficiency of this method to store data recovering 90% of the temporal evolution with only a few geometric modes (time-independent) and some temporal modes, which are a single value for each time. Second, we analyze experimental data of a wing tip vortex at moderate Reynolds numbers. The possible attenuation of this kind of vortices is a key criterion for any airport design. By using POD, we were able to describe the vortex and isolate a mode representing the global attenuation of the vortex.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Computational challenges for simulating viscoelastic fluid-structure interactions 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. I will describe recent work on micro-organism locomotion in viscoelastic fluids that highlights some of these challenges and discuss the specific modeling and numerical considerations needed for these types of problems.Non UBCUnreviewedAuthor affiliation: University of California, DavisResearche

JNNFM Complex Fluids Seminar - Quantifying fluid transitions for different strokes with applications to micro-organisms swimming in viscoelastic fluids

Non UBCUnreviewedAuthor affiliation: University of California, DavisFacult