Becca Thomases

Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

Abstract The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t −(3/2). While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data.

Emergence of singular structures in Oldroyd-B fluids

Numerical simulations reveal the formation of singular structures in the polymer stress field of a viscoelastic fluid modeled by the Oldroyd-B equations driven by a simple body force. These singularities emerge exponentially in time at hyperbolic stagnation points in the flow and their algebraic structure depends critically on the Weissenberg number. Beyond a first critical Weissenberg number the stress field approaches a cusp singularity, and beyond a second critical Weissenberg number the stress becomes unbounded exponentially in time. A local approximation to the solution at the hyperbolic point is derived from a simple ansatz, and there is excellent agreement between the local solution and the simulations. Although the stress field becomes unbounded for a sufficiently large Weissenberg number, the resultant forces of stress grow subexponentially. Enforcing finite polymer chain lengths via a FENE-P penalization appears to keep the stress bounded, but a cusp singularity is still approached exponentially in time.

LOCAL ENERGY DECAY FOR SOLUTIONS OF MULTI-DIMENSIONAL ISOTROPIC SYMMETRIC HYPERBOLIC SYSTEMS

The local decay of energy is established for solutions to certain linear, multidimensional symmetric hyperbolic systems, with constraints. The key assumptions are isotropy and nondegeneracy of the associated symbols. Examples are given, including Maxwells equations and linearized elasticity. Such estimates prove useful in treating nonlinear perturbations.

Immersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains

Abstract The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor. In a previous work we introduced the Immersed Boundary Smooth Extension (IBSE) method, a variation of the IB method that achieves high-order accuracy for elliptic PDE 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. In this work, we extend the IBSE method to allow for the imposition of a divergence constraint, and demonstrate high-order convergence for the Stokes and incompressible Navier–Stokes equations: up to third-order pointwise convergence for the velocity field, and second-order pointwise convergence for all elements of the stress tensor. The method is flexible to the underlying discretization: we demonstrate solutions produced using both a Fourier spectral discretization and a standard second-order finite-difference discretization.

Computational Challenges for Simulating Strongly Elastic Flows 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. We discuss some of the characteristics of strongly elastic flows and introduce different models and methods designed for these types of flows. We describe contributions from analysis that motivate numerical methods and illustrate their performance on different models in a simple test problem. Biological problems often involve the coupled dynamics of active elastic structures and the surrounding fluid. The immersed boundary method has been used extensively for such problems involving Newtonian fluids, and the methodology extends naturally to complex fluids in conjunction with the algorithms described earlier in this chapter. We focus on implicit-time methods because the large elastic stresses in complex fluids necessitate high spatial resolution and long time simulations. As an example to highlight some of the challenges of strongly elastic flows, we use the immersed boundary method to simulate an undulatory swimmer in a viscoelastic fluid using a data-based model for the prescribed shape.

The role of body flexibility in stroke enhancements for finite-length undulatory swimmers in viscoelastic fluids

© 2017 Cambridge University Press. The role of passive body dynamics on the kinematics of swimming micro-organisms in complex fluids is investigated. Asymptotic analysis of small-amplitude motions of a finite-length undulatory swimmer in a Stokes-Oldroyd-B fluid is used to predict shape changes that result as body elasticity and fluid elasticity are varied. Results from the analysis are compared with numerical simulations and the numerically simulated shape changes agree with the analysis at both small and large amplitudes, even for strongly elastic flows. We compute a stroke-induced swimming speed that accounts for the shape changes, but not additional effects of fluid elasticity. Elasticity-induced shape changes lead to larger-amplitude strokes for sufficiently soft swimmers in a viscoelastic fluid, and these stroke boosts can lead to swimming speed-ups. However, for the strokes we examine, we find that additional effects of fluid elasticity generically result in a slow-down. Our high amplitude strokes in strongly elastic flows lead to a qualitatively different regime in which highly concentrated elastic stresses accumulate near swimmer bodies and dramatic slow-downs are seen.

A DECAY THEOREM FOR SOME SYMMETRIC HYPERBOLIC SYSTEMS

In this short note, we consider smooth solutions to certain hyperbolic systems of equations. We present a condition which will ensure that no shocks develop and that solutions decay in L2. The condition is restrictive in general; however, when applied to the system of one-dimensional gas dynamics it is shown that if the condition is satisfied initially then it will be satisfied for all time and therefore one obtains smooth solutions which decay.